quantum markov semigroups and quantum flows

QUANTUM MARKOV SEMIGROUPS AND QUANTUM FLOWS Franco Fagnola Appeared in: Proyecciones 18, n.3 (1999) 1–144.

Preface A.N. Kolmogorov and J. von Neumann proposed in the 30’s two sets of axioms for the mathematical modelling of random phenomena. In the classical (Kolmogorov) models one introduces a triple (Ω, F, IP ), where Ω is the set of all possible results of the random phenomenon, F is a σ-algebra of subsets of Ω called events, IP a probability measure on F. In the quantum (von Neumann) one starts with a pair (A, ϕ) where A is a von Neumann algebra (of operators) whose projections are called events and ϕ is a state (probability) on A. The original scheme of von Neumann, however, covered only some fundamental ideas of what could be called a “non-commutative measure theory”, while more probabilistic aspects such as the notions of random variable and stochastic process, probability and conditional expectation, Markov chains and Markov processes, statistical independence, were absent from this scheme. Starting from the second half of the 70’s this program was developed systematically. The notions of “random variable” and “stochastic process” were stated in a purely algebraic way; a non-trivial notion of “quantum conditional expectation” allowing the construction of the first examples of quantum Markov processes was introduced by L. Accardi and extended to arbitrary von Neumann algebras by L. Accardi and C. Cecchini in [3]; the study of limit theorems, law of large numbers and central limit theorem, was started by W. von Waldenfels. At the beginning of the 80’s the notion of “quantum stochastic process” was formulated in the form nowadays generally accepted by L. Accardi, A. Frigerio and J.T. Lewis (see [5]). A quantum stochastic process on a *-algebra A with values in a *-algebra B is a family j = (jt)t≥0 of *-homomorphism jt : B → A. This generalises the notion of the flow of a classical stochastic process (X(t))t≥0 on a probability space (Ω, F, IP ) with values in a measurable space (E, E). Indeed a classical process can be described through the homomorphisms jt : L∞(E, E) → L∞ (Ω, F, IP ),

jt (f) = f(X(t));

for this reason quantum stochastic processes are also called quantum flows. Quantum stochastic processes also generalise evolutions of a closed quantum system represented by a Hilbert space h with Hamiltonian H. In this case a “random variable” X is a self-adjoint operator and the evolution of X is given by the homomorphisms jt (X) = U (t)XU (t)∗

jt : B(h) → B(h),

where B(h) is the von Neumann algebra of all bounded operators in h and (U (t))t≥0 is the unitary group generated by H. This evolution is considered reversible because j is a one-parameter group. The developement, almost at the same time, of the quantum stochastic calculus by R.L. Hudson and K.R. Parthasarathy [58] gave a strong impulse to the study of quantum stochastic processes (especially quantum Markov processes in the sense of [1]) both in the direction of applications to physics, especially to quantum optics, and to the theory of I

II continual measurements (A. Barchielli, E.B. Davies, G. Lupieri), and in the applications to classical probability (P.A. Meyer [69], K.R. Parthasarathy [74]). The homomorphisms of the most studied class of quantum Markov processes are of the form jt (X) = V (t)XV (t)∗ (1) jt : B(h) → B(H), where H is the tensor product of h and a Fock space representing the external “noise”, and (V (t))t≥0 is a family of unitary operators on H satisfying a quantum stochastic differential equation of Hudson-Parthasarathy type dV (t) = V (t) L1 dA+ (t) + L2 dΛ(t) + L3dA(t) + L4 dt , V (0) = 1l. (2) The noises A, A+ , Λ are the annihilation, creation and numbers process of Boson Fock quantum stochastic calculus (we wrote a quantum stochastic differential equation only for one-dimensional noise for simplicity), L1 , . . . , L4 are operators in the system space h and satisfy appropriated conditions for the operators V (t) to be unitary. The family (V (t))t≥0 is not a semigroup but it is a cocycle with respect to time shift. This property plays a key role when showing that j enjoys a quantum analogue of the Markov property. From the probabilistic point of view it is interesting to note that the family of operators (A+ (t) + A(t))t≥0 , (i(A+ (t) − A(t)))t≥0 can be interpreted as a pair of non-commuting Brownian motions and (Λ(t) + z¯A(t) + zA+ (t) + |z|2t)t≥0 (z ∈ C) I as a Poisson process with intensity |z|2. L. Accardi, A. Frigerio and Y.G. Lu showed that these quantum Brownian motions can be obtained as a weak coupling limit of quantum electrodymanics under the dipole and rotating wave approximation and the above quantum analogue of the Poisson process arises in a stochastic limit of a system interacting with a low density gas (see [6], [8]). These results showed rigorously under what conditions a quantum Markov semigroup and a quantum stochastic differential equation gives a good markovian approximation of an Hamiltonian evolution (see [54] for further references). Moreover, Barchielli [15] applied for the first time these quantum stochastic differential equations to the study of physical phenomena in quantum optics as, for instance, the electron shelving effect. At the same time quantum stochastic differential equations of the form (2) (and their natural generalisations driven by multidimensional or infinite dimensional noises) were studied systematically by many authors (see e.g. [37], [38], [39], [40], [41], [52], [71]) together with their applications to quantum Markov processes and semigroups in particular to the realisation of flows of classical stochastic processes as quantum flows in Fock space. The study of quantum stochastic differential equations as (2) is not simple when the operators L1, . . . , L4 are unbounded. In this case an associated semigroup T of operators (Tt)t≥0 on B(h) which is, roughly speaking, the infinitesimal generator of the process (ut )t≥0 plays a fundamental role. The operators Tt of this semigroup are normal and enjoy a fundamental property called complete positivity. Unboundedness of L1 , . . . , L4 makes it difficult to characterise the infinitesimal generator of T and to determine whether it is identity preserving or Markov. This is a crucial step for proving that solutions to (2) are family of unitary operators and almost the whole Chapter 3 will be dedicated to the study of this problem. In this work we give a self-contained exposition of several basic results scattered in the literature that has grown very rapidly in the last years and develop a general framework for constructing quantum Markov processes as quantum (Evans-Hudson) flows in Fock space. Moreover we collect several new results on quantum Markov semigroups and quantum stochastic differential equations that are necessary tools in the applications to the construction of special quantum flows.

III The main application is the realization of diffusion processes with smooth (C 4 , say) covariance and drift as quantum flows in Fock space. This was first done in our paper [41] for strongly elliptic diffusions. As P.-A. Meyer writes ([69] p.186), “this is much better than all previous results, though not yet completely satisfactory since ellipticity plays no role in Ito’s theory”. Here we remove this hypothesis by applying a new result on quantum Markov semigroups (see [28]) and some new results on quantum flows. In Chapters 1 and 2 we recall some definitions of quantum probability theory. The main goal is to show that the fundamental definitions and properties of classical probability and Markov processes are easily formulated in an algebraic language suitable for the study of Markov processes appearing in quantum theory. In particular the notion of complete positivity which is the appropriate generalisation of positivity in the non-commutative framework is discussed in detail. In Chapter 3 we give a self-contained exposition of the basic results on uniformly continuous completely positive semigroups on the Von Neumann algebra B(h) of all bounded operators in a Hilbert space h proving the well-known Lindblad’s theorem. Moreover we describe the construction of completely positive semigroups of normal operators on B(h) continuous for the σ-weak topology (the construction of the so-called minimal quantum dynalmical semigroup) and we give conditions - necessary and sufficient or simply sufficient in order for the semigroup to be identity preserving (i.e. Markov). In Chapter 4 first we study the following problem: given a classical markovian semigroup is it a non-trivial restriction to an abelian subalgebra of a quantum Markov semigroup? We show that the problem can be attacked in two steps: 1) represent the infinitesimal generator of the classical Markov process in a “Lindblad form” so that we can find immediately an infinitesimal generator for the candidate quantum Markov semigroup, 2) show that the minimal quantum dynamical semigroup associated with this infinitesimal generator is Markov. We shall see that the answer to the above extension problem is in the affirmative in several cases and prove a new result on quantum extension of classical semigroups of diffusion processes. Then we apply our results on quantum Markov semigroups to construct the solution to a class of quantum master equations. In Chapter 5 we discuss a general scheme for constructing quantum Markov processes through Markov operator cocycles (see Chapter 2, Section 3). In particular we show how to construct a quantum flow with unbounded structure maps satisfying a quantum stochastic differential equation and give a rather general sufficient condition in order for the restriction to an abelian subalgebra to be a commutative flow. As an application we show that diffusion processes with smooth coefficients can be realised as quantum flows in Fock space. Acknowledgements I would like to thank L. Accardi, K.R. Parthasarathy, R. Rebolledo and K.B. Sinha for many fruitful discussions. I would also like to thank H. Comman, R. Monte, K.B. Sinha and S. Wills who kindly readed and corrected portions of the manuscript. It is a great pleasure to thank also R. Soto and the Editorial Board of “Proyecciones” for publication of this unusually lengthy article.

Contents 1 Algebraic Probability Spaces 1.1 Fundamental definitions . . . 1.2 Classical probability spaces as 1.3 Quantum probability spaces . 1.4 Conditional expectation . . . 1.5 Topologies on algebras . . . .

. . . . .

3 3 4 4 8 9

2 Algebraic Markov Processes 2.1 Fundamental definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Completely positive linear maps . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 A quantum Feynman-Kac formula . . . . . . . . . . . . . . . . . . . . . . . .

13 13 16 24

3 Quantum Markov Semigroups 3.1 Fundamental definitions and examples . 3.2 Uniformly continuous QDS . . . . . . . 3.3 Minimal quantum dynamical semigroup 3.4 The resolvent of the minimal semigroup 3.5 Conservativity . . . . . . . . . . . . . . 3.6 Sufficient conditions for conservativity .

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27 27 33 39 48 50 58

4 Classical and Quantum Semigroups 4.1 Preliminary definitions and results . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Diffusion processes on IRd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 A quantum master equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 67 71 77

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5 Quantum Flows 5.1 Quantum stochastic calculus . . . . . . . 5.2 Time reversal and dual cocycles . . . . . . 5.3 Quantum stochastic differential equations 5.4 Unitary solutions . . . . . . . . . . . . . . 5.5 Inner quantum flows . . . . . . . . . . . . 5.6 Quantum diffusions . . . . . . . . . . . . . 5.7 Conclusion . . . . . . . . . . . . . . . . . A Results on semigroups

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83 83 90 93 101 106 110 115 117

1

2

CONTENTS

1

Algebraic Probability Spaces In this chapter we recall some definitions of quantum probability theory in the general framework developed by Accardi and several co-workers. Several expository papers in the series Quantum Probability and Related Topics and two monographes [69], [74] are already available in the literature. We will only illustrate the definitions by two examples which are included only with the aim to give a flavour of the relation with classical probability. The reader is supposed to be familiar with the basic language and facts on C ∗-algebras and von Neumann algebras. In Section 5 we recall only some fundamental definitions and results.

1.1

Fundamental definitions

Definition 1.1 A ∗-algebra is a complex algebra A with an involution, denoted by ∗, with the following properties: (λa + µb) (a∗ )∗

∗

= =

¯ ∗+µ λa ¯b∗ a

∗

=

b∗ a∗

(ab)

for all a, b ∈ A, λ, µ ∈ C. I An element a of A is called positive if there exists b ∈ A such that a = b∗ b. We shall consider often ∗-algebras with a unit denoted by 1l. Definition 1.2 Let A be a ∗-algebra with a unit 1l. A state ϕ on A is a linear map ϕ : A →C I with the properties: 1. (positivity) ϕ(a∗ a) ≥ 0, for all a ∈ A, 2. (normalisation) ϕ(1l) = 1. We can now introduce the notion of algebraic probability space and algebraic random variable according to Accardi, Frigerio and Lewis [5]. Definition 1.3 An algebraic probability space is a pair (A, ϕ) where A is a ∗-algebra with unit and ϕ is a state on A. 3

4

1. ALGEBRAIC PROBABILITY SPACES

Definition 1.4 Let (A, ϕ) be an algebraic probability space and let B be a ∗-algebra. An algebraic random variable on A with values in B is a ∗-homomorphism j : B → A. The above quite general definitions will be always applied in “concrete” cases when A and B are at least C ∗-algebras. In this case it is well known (see [21] Th. 2.1.10 p.60) that A and B are isomorphic to a norm-closed ∗-algebra of bounded operators in a Hilbert space.

1.2

Classical probability spaces as algebraic

A measurable space (Ω, F) clearly determines uniquely the commutative ∗-algebra A = L∞(Ω, F;C) I of F-measurable, bounded, complex-valued functions on Ω. A probability measure IP on F induces a state ϕ on A by Z ϕ(f) = f(ω)dIP (ω). Ω

Therefore the classical probability space (Ω, F, IP ) can be considered also as the algebraic probability space (A, ϕ). Let (E, E) be a measurable space. Classical random variables on Ω with values in E can be also interpreted as algebraic random variables. In fact, consider the ∗-algebra B = L∞(E, E;C). I A classical random variable x can be described as an algebraic random variable by the ∗-homomorphism j : B 7→ A, j(f) = f ◦ x. It is worth noticing here that each event can be represented by a projection in the ∗-algebra A through the identification with its indicator function.

1.3

Quantum probability spaces

Algebraic probability spaces appear as the basic structure in mathematical models for quantum mechanics. The mathematical structure however is (or at least can be assumed to be) richer with some more analytical properties on the algebra A and state ϕ. In order to distinguish an important class of algebraic probability spaces we give the following: Definition 1.5 A quantum probability space is a pair (A, ϕ) where A is a von Neumann algebra and ϕ a σ-weakly continuous state on A. An event in the quantum probability space (A, ϕ) is a projection operator in A. A quantum random variable in (A, ϕ) with values in a von Neumann algebra B is a σ-weakly continuous homomorphism j : B → A. Although one could define events in the same way also when A is only a C ∗ -algebra the set of events in this case might be very poor. Indeed, if A is the C ∗-algebra of complexvalued continuous functions on IRd , then the set of events is trivial. On the other hand a von Neumann algebra A is generated by projections in A. Notice that, contrary to the classical case, the intersection (product) of two events is no longer an event if the corresponding orthogonal projections do not commute. We will show now in which sense a self-adjoint operator X affiliated with the von Neumann algebra A of a quantum probability space (A, ϕ) can be considered both as a classical and a quantum random variable.

1.3. QUANTUM PROBABILITY SPACES

5

As we already noted we can assume that A is a sub von Neumann algebra of the von Neumann algebra B(h) of bounded operators on a Hilbert space h. Therefore X is a selfadjoint operator on h. Let B be the von Neumann algebra L∞ (IR;C). I By the spectral theorem in functional calculus form ([78] Th. VIII.5 p. 262), for all f ∈ B, we can define the element f(X) of A. The map j : B 7→ A, j(f) = f(X) is a *-homomorphism. This is clearly σ-weakly continuous because, for every increasing net (fα )α of positive elements of B with least upper bound f in B we have sup j(fα ) = sup fα (X) = f(X) = j(f). α

α

Therefore X defines a quantum random variable. Let Ω = IR and let F be the Borel σ-field on Ω. Since both j and ϕ are σ-weakly continuous we can define a probability measure on F by putting IP (B) = ϕ(1B (X))

(1.1)

where 1B denotes the indicator function of a B ∈ F. Thus we have constructed a classical probability space (Ω, F, IP ). We want to show now that X can be represented as a classical real random variable on (Ω, F, IP ). Fix a unit vector e in h and consider the closed subspace h0 of h generated by e and vectors of the form f1 (X)f2 (X) . . . fn (X)e with n ≥ 1, f1, . . . , fn ∈ B. Let U be the unique unitary operator U : h0 → L2(Ω, F, IP ) such that U e = 1,

U f1 (X)f2 (X) . . . fn (X)e = f1 f2 . . . fn .

The operator U is unitary because of the relation Z g(ω)dIP (ω) = ϕ(g(X)) IR

for g integrable with respect to IP which follows immediately from (1.1). It is easily checked that the following diagram commutes: h0   U y

X −−−−−−→

h0   yU

L2 (Ω, F, IP ) −−−−−−→ L2 (Ω, F, IP ) fX (where fX (x) = x). Precisely we can show that – v ∈ D(X) if and only if fX (·)(U v)(·) ∈ L2 (Ω, F, IP ), – if w ∈ U (D(X)), then (U XU ∗ w)(·) = fX (·)w(·). Therefore U XU ∗ acts on L2 (Ω, F, IP ) as the multiplication operator by a real function fX and the self-adjoint operator X defines a classical real random variable. Two non-commuting self-adjoint operators, however, cannot be represented as multiplication operators on the same Hilbert space L2 (Ω, F, IP ). This, roughly speaking, can

6


be summarized by saying that (A, ϕ) is a quantum probability space “containing infinitely many classical probability spaces.” The following are concrete example in which the above fact occurs. We will use these models to illustrate also how classical probabilistic notions appear naturally in the quantum context. Example 1.1 Spin matrices Let h = C I 2, and let A be the von Neumann algebra of 2 × 2 complex valued matrices and ϕ be any state on A. The pair (A, ϕ) is clearly a quantum probability space. The self-adjoint operators on h (also called spin matrices or Pauli matrices). 0 1 0 −i 1 0 σ1 = , σ2 = , σ3 = , 1 0 i 0 0 −1 represent three non-commuting quantum random variables. The above discussion shows immediately that σ3 can be represented as a classical real random variable x taking values {−1, +1} with probabilities 1 0 0 0 IP {X = 1} = ϕ , IP {X = −1} = ϕ . 0 0 0 1 Notice that −1 and 1 are the eigenvalues of σ3 and the events {σ3 = 1} and {σ3 = −1} are represented respectively by the orthogonal projections 1 0 0 0 , . 0 0 0 1 In a similar way the laws of σ1 and σ2 can be computed using the spectral decompositions 1/2 1/2 1/2 −1/2 σ1 = − 1/2 1/2 −1/2 1/2 1/2 −i/2 1/2 i/2 σ2 = − i/2 1/2 −i/2 1/2 Thus σ1, σ2, σ3 are three non-commutative random variables with values in {−1, +1}. Example 1.2 The harmonic oscillator Let h be the Hilbert space h = l2 (IN ),

IN = { 0, 1, 2, . . .}

with the canonical orthonormal basis (en ). We consider then the following operators: 1. annihilation operator D(a)

=

aen

=

{x = (xn)n ∈ h | √

X

n|xn|2 < ∞}

n

n en−1

if n > 0,

ae0 = 0,

2. creation operator D(a∗ ) a∗ en

= {x = (xn )n ∈ h |

n|xn|2 < ∞}

n

√ =

X

n + 1 en+1

for all n ≥ 0,

1.3. QUANTUM PROBABILITY SPACES

7

3. number operator D(N )

=

{x = (xn )n ∈ h |

X

N en

=

a∗ aen = nen

for all n ≥ 0.

n2|xn|2 < ∞}

n

4. momentum (or electric field) operator i p = √ (a∗ − a) 2

D(p) = D(a) = D(a∗ ), 5. position (or magnetic field) operator

1 q = √ (a∗ + a) . 2

D(q) = D(a) = D(a∗ ),

It is well-known that the operators N, p, q are self-adjoint. Therefore they can be considered as algebraic random variables. The canonical commutation relations (domains of the operators involved will be made precise later) [a, a∗] = aa∗ − a∗ a = 1l,

(N + 1)a = aN,

a∗ (N + 1) = N a∗

([·, ·] denoting the commutator) yield [q, p] = i1l,

[N, p] = iq,

[N, q] = −ip.

Therefore N, p, q can not be represented as random variables on the same classical probability space. The above model is called the Heisenberg representation of the canonical commutation relations over C. I It is well-known that it is unitarily equivalent to the Schr¨ odinger representation. Indeed, letting (Hn )n≥0 be the orthonormal sequence of the Hermite polynomials in L2(IR; π−1/4 exp(−x2 /2)dx) the unitary operator U : l2 (IN ) → L2(IR; π−1/4 exp(−x2 /2)dx),

Uen = Hn

allows us to move from one representation to the other. p −−−−−−→

l2 (IN )   U y

l2 (IN )   yU

L2 (IR; π−1/4 exp(−x2 /2)dx) −−−−−−→ L2 (IR; π−1/4 exp(−x2 /2)dx) d −i dx We refer to Meyer’s book [69] for more details on this subject. The following is a “conversion table” Heisenberg representation p q N

Schroedinger representation d −i dx x 1 d2 2 − 2 +x −1 2 dx

8


Let ϕ be the state on B(L2 (IR; π−1/4 exp(−x2 /2)dx)) defined by the unit vector H0 (coinciding with the constant function 1) ϕ(a) = hH0, aH0i . Well-known facts allow us to compute easily the law of the random variables p, q. Notice first that the unitary groups generated by q and p are (exp(itq)v)(x) = exp(itx)v(x),

(exp(itp))v(x) = v(x + t).

Therefore the characteristic functions of p and q in the state ϕ are given by ϕ(exp(itq)) ϕ(exp(itp))

= =

hH0, exp(itq)H0 i = exp(−t2 /4), hH0, exp(itp)H0 i = exp(−t2 /4).

Thus p and q are two non-commuting random variables with gaussian distribution with mean 0 and variance 1/2. Moreover we have ϕ(N ) = hH0, N H0i = he0 , a∗ ae0 i = 0. Therefore, since N is a non negative random variable, it is almost surely 0 with respect to the probability law determined by the state ϕ.

1.4

Conditional expectation

Examples given in the previous section motivate the following Definition 1.6 Let (A, ϕ) be an algebraic probability space. For all element a of A we call the expectation, or mean value, of a in the state ϕ the number ϕ(a). For all a, b ∈ A we define the covariance ∗ ϕ (a − ϕ(a)) (b − ϕ(b)) and the variance

∗ ϕ (a − ϕ(a)) (a − ϕ(a)) .

Notice that here we defined an algebraic analogue of expectation, covariance and variance essentially for bounded random variables. In fact a is an element of A. We are now in a position to introduce the notion of conditional expectation. Definition 1.7 Let (A, ϕ) be an algebraic probability space and let A0 be a sub ∗-algebra of A with unit 1l. A conditional expectation is a linear map IE [ · | A0 ] : A 7→ A0 with the following properties: 1. for all a ∈ A such that a ≥ 0, we have IE [ a | A0 ] ≥ 0, 2. IE [ 1l | A0 ] = 1l, 3. for all a0 ∈ A0 and all a ∈ A we have IE [ a0 a | A0 ] = a0IE [ a | A0 ] ,

1.5. TOPOLOGIES ON ALGEBRAS

9 ∗

4. for all a ∈ A we have IE [ a∗ | A0 ] = (IE [ a | A0 ]) , 5. for all a ∈ A0 we have ϕ(a) = ϕ (IE [ a | A0 ]) . Example 1.3 The conditional expectations we shall use are of the following form. Let h, h1 be complex separable Hilbert spaces. Consider the Hilbert space k = h ⊗ h1 and consider the ∗-algebras A = B(k), A0 = B(h) ⊗ 1l. For all states ϕ on B(h) and ϕ1 on B(h1 ), consider the state ψ on A defined by ψ(a ⊗ a1 ) = ϕ(a)ϕ1 (a1 ) for all a ∈ B(h), a1 ∈ B(h1 ). Then the linear map IE [ · | A0 ] : A 7→ A0 ,

IE [ a ⊗ a1 | A0 ] = ϕ1 (a1) (a ⊗ 1l)

is a conditional expectation. Although our definition of conditional expectation in the ∗-algebraic language seems to be quite natural, it is too restrictive in quantum probability. In fact, contrary to the classical case, given a quantum probability space (A, ϕ) and a sub ∗-algebra let A0 of A with identity 1l, a conditional expectation IE [ · | A0 ] : A → A0 may not exists. A detailed discussion on conditional expectations in von Neumann algebras can be found in [73] and [77] by D. Petz. Definition 1.8 Let (A, ϕ) be an algebraic probability space. A filtration is a family (At] t≥0 of sub-∗-algebras of A such that As] ⊆ At] for all s ≤ t. Definition 1.9 Let (A, ϕ) be a quantum probability space and let (At] t≥0 be a filtration. Suppose that, for all t ≥ 0, there exists conditional expectations IE · | At] : A → At] . The family IE · | At] t≥0 is called projective if, for all s ≤ t, we have IE IE · | As] | At] = IE · | As] . We refer to the recent books of Meyer [69], Parthasarathy [74] and the references therein for a more detailed introduction to the language of quantum probability and other interesting examples.

1.5

Topologies on algebras

A von Neumann algebra (or W ∗ -algebra) could be defined intrinsically as a special C ∗algebra. However we will consider only “concrete” von Neumann algebras, that is sub-∗algebras of operators on some Hilbert space h closed under the weak or σ-weak or strong operator topology (see [21] pp.71-72). These topologies can be defined through the convergence of nets. Definition 1.10 Let (xα)α be a net in B(h) and let x ∈ B(h). We say that:

10

1. ALGEBRAIC PROBABILITY SPACES 1. (xα )α converges weakly to x if hv, xαui converges to hv, xui for every v, u ∈ h. P P 2. (xα )α converges σ-weakly to x if the sum nhvn , xαuni converges to the sum P nhvn , xuni 2 for every P pair2of sequences (vn )n, (un)n of elements of h such that the series n kunk and n kvn k converge. 3. (xα )α converges strongly to x if xαu converges to xu for every u ∈ h. P 4. (xα )α converges σ-strongly to x if the sum n k(xα −Px)unk2 converges to 0 for every sequence (un )n of elements of h such that the series n kunk2 converges.

Clearly (xα)α converges σ-weakly to x if and only if, for every trace class operator ρ in h, tr(xα ρ) converges to tr(xρ). The following facts will be frequently used: (a) the σ-weak topology is stronger than the weak topology and not comparable to the strong one, (b) the weak and σ-weak topology coincide on bounded subsets of B(h). Definition 1.11 A von Neumann algebra is a *-subalgebra of B(h) containing 1l closed in the weak (or by the bicommutant theorem [21] Th. 2.4.11 p.72 - σ-weak or strong or σ-strong) topology. We recall the following property of the cone of positive elements A+ of a von Neumann algebra A (see [21] Lemma 2.4.19 p.76). Proposition 1.12 Let A be a von Neumann algebra of operators acting on a Hilbert space h and let (xα ) be an increasing net in A+ with an upper bound in A+ . Then (xα)α has a least upper bound x = supα xα in A+ and the net converges σ-strongly to x. We define a useful class of positive functionals on a von Neumann algebra. Definition 1.13 Let A be a von Neumann algebra of operators acting on the Hilbert space h and let ω be a positive linear functional on A. We say that ω is normal if sup ω(xα ) = ω(sup xα ). α

α

The following fundamental properties of states on a von Neumann algebra are well-known (see [21] Theorem 2.4.21 p.76). Theorem 1.14 Let A be a von Neumann algebra of operators acting on a Hilbert space h and let ω be a state on A. The following conditions are equivalent: 1. ω is normal, 2. ω is σ-weakly continuous, 3. there exists a density matrix ρ (i.e. a positive trace-class operator on h with tr(ρ) = 1) such that ω(x) = tr(ρx). We shall use often a consequence of the equivalence of 1 and 2. Let us recall that a subset E of a Hilbert space h is called total in h if the linear manifold generated by E is dense in h. Proposition 1.15 Let A and B be von Neumann algebras, B acting on a Hilbert space h, and let T : A → B be a positive linear map. The following conditions are equivalent:

1.5. TOPOLOGIES ON ALGEBRAS

11

1. T is σ-weakly continuous (i.e. continuous with respect to the σ-weak topologies on A and B), 2. for every increasing net (xα)α in A+ with least upper bound x in A+ the increasing net (T xα )α in B+ converges σ-weakly to T x, 3. for every increasing net (xα )α in A+ with least upper bound x in A+ we have lim hu, (T xα)ui = sup hu, (T xα)ui = hu, (T x)ui α

α

for each u in a linear submanifold of h which is norm-dense in h, 4. for every increasing net (xα )α in A+ with least upper bound x in A+ we have lim hv, (T xα )ui = hv, (T x)ui α

(1.2)

for each v, u in a total subset of h. Proof. 1 implies 2. Indeed it suffices to note that the net (xα )α converges weakly to x by Proposition 1.12. 2 implies 3. Obvious since the linear functionals on B y → hu, yui with u ∈ h are σ-weakly continuous. 3 implies 4. We show first that 3 implies that the net (hu, (T xα)ui)α converges to hu, (T x)ui for each u ∈ h. Indeed, for every ε > 0, there exists uε in the dense subset such that ku − uεk < ε. The inequality T xα ≤ T x clearly implies kT xαk ≤ kT xk. We have then |hu, (T xα)ui − hu, (T x)ui| ≤

|hu − uε , (T xα)ui − hu − uε , (T x)ui|

+ + ≤

|huε, (T xα)(u − uε )i − huε, (T x)(u − uε)i| |huε, (T xα)uε i − huε , (T x)uεi| ku − uεk (kT xαk + kT xk) (kuk + kuεk)

+

|huε, (T xα)uε i − huε , (T x)uεi| .

Therefore we have lim |hu, (T xα)ui − hu, (T x)ui| ≤ 2εkT xk(2kuk + ε). α

Since ε is arbitrary, the net (hu, (T xα)ui)α converges to hu, (T x)ui for each u ∈ h. By the polarisation identity 1 X −k

v + ik u, (T xα)(v + ik u) i 4 3

hv, (T xα )ui =

k=0

it follows then that (1.2) holds for each v, u ∈ h. 4 implies 3. In fact 4 implies that (1.2) holds for each v, u in the dense subset of h linearly spanned by the total set. This linear span is obviously dense. 3 implies 2. Let (vn )n≥0, (un)n≥0 be two sequences of vectors in h such that the sequences (kvnk)n≥0, (kunk)n≥0 are square-summable. We must show that X X lim hvn , (T xα)uni = hvn , (T x)uni . α

n≥0

n≥0

To this end, for every ε > 0, take an integer ν such that X X kunk2 < ε, kvn k2 < ε. n>ν

n>ν

12


We have then

X X hv , (T x )u i − hv , (T x)u i n α n n n n≥0 n≥0 X ≤ (kT xαk + kT xk) kun k · kvn k n>ν

ν ν X X + hvn , (T xα)uni − hvn, (T x)uni . n=0

n=0

The first term, since kT xαk ≤ kT xk, can be estimated by ! X X 2 2 kT xk < 2εkT xk. kunk + kvn k n>ν

n>ν

Moreover, as we have shown in the proof that 3 implies 4, the property 3 implies the convergence of the net (hv, (T xα)ui)α to hv, (T x)ui for each v, u ∈ h. We have then X X lim hvn, (T xα)un i − hvn, (T x)uni ≤ 2εkT xk. α n≥0 n≥0 Since ε is arbitrary this shows that (1.2) holds. 2 implies 1. Let (xα )α be a net in A converging σ-weakly to x. For every pair (vn )n≥0 , (un)n≥0 of sequences of vectors in h such that (kvn k)n≥0, (kun k)n≥0 are square-summable let ω be the σ-weakly continuous functional on B X ω(y) = hvn , yuni . n≥0

By the complex polarisation identity ω can be written as a linear combination of four positive linear functionals ωk , k = 0, 1, 2, 3, with X

ωk (y) = (v + ik u), y(v + ik u) . n≥0

Therefore, in order to show that 2 implies 1, it suffices to prove that the net (ω(T xα ))α converges to ω(T x) for every positive linear functional ω of the above form. If, for such an ω, we have ω(T 1l) = 0, then we have also ω(T x) = 0 for each self-adjoint element x of A because −kxk1l ≤ x ≤ kxk1l and T is positive. Therefore ω(T x) = ω(T (x + x∗ ))/2 + iω(T (x − x)∗ /i)/2 = 0 for each x ∈ A. Thus there is nothing to prove. If ω(T 1l) > 0, then, letting ω e (y) = ω(T y)/ω(T 1l),

we define a state ω e on A. Condition 2 implies then that ω e is normal. Then, by virtue of Theorem 1.14, it is σ-weakly continuous so that lim ω(T xα) = lim ω(T 1l)e ω(T xα ) = ω(T 1l)e ω(T x) = ω(T x). α

α

This completes the proof. A map enjoying the property 3 is also called normal. We will often use the above equivalence to show that a positive map is continuous for the σ-weak topology. Moreover we will often use normal with the same meaning of σ-weak when no confusion can arise. We refer to the book [21] for more detailed results on von Neumann algebras.

2

Algebraic Markov Processes In this chapter we describe the abstract definition and the basic facts on algebraic Markov processes (see [5]). The main goal is to show that the fundamental definitions and properties of Markov processes are easily formulated in an algebraic language suitable for the study of Markov processes appearing in quantum theory. Moreover, we discuss in detail the notion of complete positivity which turns out to be the natural generalisation of positivity for commutative (classical) case and a non-commutative version of the Feynman-Kac formula which is the basic ingredient in the construction of Markov cocycles and processes.

2.1

Fundamental definitions

Definition 2.1 Let (A, ϕ) be an algebraic probability space and let B be a *-algebra. An algebraic stochastic process on A with values in B is a family u = (ut )t≥0 of algebraic random variables on A with values in B. The quantum stochastic process u is adapted with respect to a filtration At] t≥0 if ut(B) ⊆ At] for all t ≥ 0. In the remaining part of this chapter we fix the following framework: – an algebraic probability space (A, ϕ) with a filtration At] t≥0 and 1l ∈ A0] , – a projective family of conditional expectations IE · | At] t≥0, – an adapted process u with values in A0] , – a family of conditional expectations IE · | ut A0] , such that t≥0 IE IE[a | At] ] | ut A0] = IE a | ut A0] for every a in the ∗-algebra generated by us A0] with s ≥ t. Definition 2.2 An adapted algebraic stochastic process u is an algebraic Markov process, with respect to the filtration At] t≥0, if, for all s, t ≥ 0 and all X ∈ A0] , we have IE ut+s (X) | As] = IE ut+s (X) | us A0] .

(2.1)

An algebraic Markov process is covariant or homogeneous if, for all s, t ≥ 0 and all a ∈ A0] , we have ut IE us (a) | A0] = IE ut+s (a) | At] . (2.2)

13

14

2. ALGEBRAIC MARKOV PROCESSES

Example 2.1 To illustrate the above definition we show that an homogeneous classical Markov process can be considered as a covariant quantum Markov process. Let x = (xt)t≥0 be a classical (adapted) Markov process with values in a measurable space (E, E), initial law µ and transition probability function P : D × E × E → IR, where D = { (s, t) ∈ [0, +∞) | 0 ≤ s ≤ t }. Suppose that, for every (s, t) ∈ D, x ∈ E and every A ∈ E such that µ(A) = 0 we have P (s, t, x, A) = 0. We consider the canonical realization on the classical probability space (Ω, F, IP ) where Y E, F = ⊗t≥0E, Ω= t≥0

and IP is the probability measure on F defined by Z Z Z IE [f(xt1 , . . . , xtn )] = dµ(z) P (0, t1, z, dz1) P (t1, t2, z1 , dz2)... E E E Z ... P (tn−1, tn, zn−1, dzn)f(z1 , . . . , zn ) E

where 0 ≤ t1 < t2 < . . . < tn . Consider the filtration Ft] Y Ft] = E.

t≥0

given by

0≤s≤t

Consider a quantum probability space (A, ϕ) with a filtration At] A

=

t≥0

where

L∞ (Ω, F;C) I , At] = L∞ Ω, Ft];C I , ϕ(f(xt1 , . . . , xtn )) = IE [f(xt1 , . . ., xtn )] .

The classical process x defines a family of *-homomorphisms ut : A0] → At] ,

ut (f) = f(xt ).

Therefore the classical Markov property IE f(xt+s ) | Fs] = IE [ f(xt+s ) | xs ] for all s, t ≥ 0 and all f ∈ A0] is immediately translated, in the algebraic language, into the identity (2.1). The Markov process is time homogeneous if and only if Z Z f(y)P (s, t + s, x, dy) = f(y)P (0, t, x, dy) E

E

for all s, t ≥ 0, and all f ∈ A0] . Now (2.2) can be easily understood in view of the following correspondence table between the quantum and classical case quantum

IE ut+s(f) | At]

←→

IE us(f) | A0] ut IE us(f) | A0]

classical

←→

Z

←→

ZE

←→

ZE

f(y)P (t, t + s, xt, dy)

E

f(y)P (0, s, x, dy) f(y)P (0, s, xt , dy).

2.1. FUNDAMENTAL DEFINITIONS

15

Proposition 2.3 The following conditions are equivalent: 1. u is an homogeneous algebraic Markov process, 2. for all n ≥ 1 and all a1 , . . ., an ∈ A0] , 0 < t1 < . . . < tn, s ≥ 0 we have IE ut1+s (a1 ) . . . utn+s (an ) | As] = IE ut1 +s (a1 ) . . . utn +s (an ) | us(A0] ) , ut1 IE us(a1 ) | A0] = IE ut1 +s (a1 ) | At1] . Proof. Clearly it suffices to prove that, if u is an homogeneous quantum Markov process, then the first identity of condition 2 holds. We will consider the case n = 2 for simplicity. Using the Markov, covariance and *-homomorphism property of u we can show that the conditional expectation IE ut1+s (a1 )ut2 +s (a2 ) | As] is equal to

= = = = = =

IE ut1 +s (a1)IE ut2 +s (a2 ) | At1+s] | As] IE ut1 +s (a1)ut1 +s IE ut2 −t1 (a2 ) | A0] | As] IE ut1 +s a1IE ut2 −t1 (a2 ) | A0] | As] IE ut1 +s a1IE ut2 −t1 (a2 ) | A0] | us A0] IE ut1 +s (a1)ut1 +s IE ut2 −t1 (a2 ) | A0] | us(A0] ) IE ut1 +s (a1)IE ut2 +s (a2 ) | At1+s] | us(A0] ) IE ut1 +s (a1 )ut2+s (a2 ) | us(A0] ) .

This completes the proof. The following proposition shows that, as in classical probability, one can associate a semigroup to an algebraic Markov process. Proposition 2.4 Let u be a covariant algebraic Markov process. For all t ≥ 0 define the map Tt : A0] → A0] , (2.3) Tt(a) = IE ut (a) | A0] Then T = (Tt )t≥0 is a semigroup of operators in A0] with the following properties 1. for every integer n ≥ 1 and every family a1 , . . . , an, b1, . . . , bn of elements of A0] we have n X b∗p Tt(a∗p aq )bq ≥ 0 (2.4) p,q=1

for every t ≥ 0, 2. Tt (1l) = 1l for every t ≥ 0. Proof. T is a semigroup in fact, for every t, s ≥ 0 and a ∈ A0] ,we have, Tt+s(a) IE ut+s(a) | A0] (projectivity) = IE IE ut+s(a) | At] | A0] (covariance) = IE ut IE us(a) | A0] | A0] = Tt (Ts (a))

16


Clearly property 2 holds since 1l belongs to A0] and both ut and the conditional expectation preserve 1l. Finally for every integer n ≥ 1 and every family a1 , . . . , an, b1, . . . , bn of elements of A0] the left-hand side of (2.4) is equal to n X

b∗p IE ut(a∗p aq ) | A0] bq

p,q=1 n X

=

IE b∗p ut (a∗p )ut (aq )bq | A0]

p,q=1

= IE

"

n X

ut(ap )bp

!∗

p=1

n X

ut (aq )bq

q=1

!

A0]

#

which is clearly a positive operator because of property 1 of conditional expectation. This completes the proof. Property 1 of algebraic Markov semigroup is called complete positivity and plays a key role in our exposition. As we have seen in the proof of Proposition 2.4, it follows from complete positivity of conditional expectation. More general conditional expectations in quantum probability (see [3], [77]) are also completely positive. Therefore complete positivity is the proper generalization of positivity in the non-commutative framework. The next section will be devoted to a detailed study of this property and its connections with positivity.

2.2

Completely positive linear maps

Let A and B be two ∗-algebras with unit. We denote by 1l the unit of both since each time it will be clear from the context to which algebra it belongs. The obvious generalization of the notion of positivity for classical (sub) Markov operators is too weak when A and B are non commutative. Proposition 2.4 motivates the introduction of the following stronger notion of positivity. Definition 2.5 The linear map T : A → B is called: 1. n-positive if for every family a1 , . . . , an of elements of A and every family b1, . . . , bn of elements of B we have n X b∗p T (a∗p aq )bq ≥ 0, p,q=1

2. completely positive if it is n positive for every integer n ≥ 1. The following fact is obvious. Proposition 2.6 Let T : A → B be a ∗ -homomorphism. Then T is completely positive. In the remaining part of this section A and B will be C ∗ -algebras with unit 1l. Note that, for every integer n ≥ 1, the algebraic tensor product ∗-algebras A ⊗ Mn and B ⊗ Mn can be represented as the ∗-algebras of n × n matrices with entries in A and B respectively. Every element x of A ⊗ Mn can be written in the form X x= xij ⊗ Eij . (2.5) 1≤i,j≤n

2.2. COMPLETELY POSITIVE LINEAR MAPS

17

where Eij denotes the n × n matrix with all the entries equal to 0 except the ij-th which is equal to 1. Given a linear map T : A → B we can define linear maps Tn : A ⊗ Mn → B ⊗ Mn by Tn (a ⊗ Eij ) = T (a) ⊗ Eij

(2.6)

In order to give a useful condition equivalent to complete positivity by means of the maps Tn we prove a simple fact on positive elements of A ⊗ Mn in the case when A is a C ∗-algebra. Proposition 2.7 Let A be a C ∗ -algebra and let x be an element of A ⊗ Mn . The following conditions are equivalent: 1. x is positive, 2. x is a finite sum of matrices of the form X

a∗i aj ⊗ Eij

1≤i,j≤n

with a1, . . . , an ∈ A, 3. for all a1 , . . ., an ∈ A we have X

a∗i xij aj ≥ 0.

1≤i,j,≤n

Proof. 1 implies 2. In fact, since x is positive, it can be written in the form y∗ y with y ∈ A ⊗ Mn (see, for example, [21], Th. 2.2.10). Writing y in the form (2.5) we have x=

n X X

∗ y`i y`j ⊗ Eij .

`=1 1≤i,j≤n

2 obviously implies 3. 3 implies 1. By representing A as a sub-C ∗ -algebra of the algebra of all bounded operators on a Hilbert space H (see [21] Th. 2.1.10 p. 60) and decomposing H into cyclic orthogonal subspaces we may suppose that there exists a cyclic vector u for the representation. Condition 3 then implies then the inequality X hai u, xij aj ui ≥ 0. i,j

Therefore, since the vector u is cyclic, we have X hvi , xij vj i ≥ 0 i,j

for all v1 , . . ., vn ∈ H. This completes the proof. We are now in a position to prove the following Proposition 2.8 Let A, B be C ∗-algebras and let T : A → B be a linear map. The following conditions are equivalent: 1. T is completely positive,

18

2. ALGEBRAIC MARKOV PROCESSES 2. for every integer n ≥ 1 the map Tn defined by (2.6) is positive.

Proof. The P second condition implies the first by parts 2 and 3 of Proposition P 2.7. Since the operator x = i,j a∗i aj ⊗Eij in A⊗Mn is positive, we have that Tn (x) = i,j T (a∗i aj )⊗Eij is positive. P Conversely the first condition implies that i,j T (a∗i aj ) ⊗ Eij is positive. Therefore Tn is positive because of the equivalence of conditions 1 and 2 of Proposition 2.7. Proposition 2.9 Let T : A → B be a linear map where B is the C ∗ -algebra B(K) of all bounded operators on a Hilbert space K. Then T is completely positive if and only if for every integer n ≥ 1 and every a1 , . . ., an ∈ A, u1, . . . , un ∈ K X hui , T (a∗i aj )uj i ≥ 0. 1≤i,j≤n

Proof. Notice that the C ∗ -algebra B ⊗ Mn can be represented as the C ∗-algebra of all bounded operators on the n-fold direct sum K ⊕ . . . ⊕ K. Therefore the above condition is clearly equivalent to positivity of the map Tn on A ⊗ Mn for every integer n ≥ 1. The following counterexample, essentially due to W.B. Arveson [12], shows the existence of positive maps that are not completely positive. Let n ≥ 2 be an integer and let both A and B be the ∗-algebra Mn. Consider the positive linear map T : M n → Mn ,

T (a) = at

(2.7)

(at denoting the transpose matrix). We will prove that T is not 2-positive. Let b be the 2 × 2 matrix with entries in Mn having the matrix E11 as 11-th entry, E1n as 12-th entry, En1 as 21-th entry, Enn as 22-th entry. For example, when n = 2, we have     1 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 b=  , T2 (b) =  . 0 0 0 0 0 1 0 0 1 0 0 1 0 0 0 1 It is easy to check that the matrix b/2 is a self-adjoint projection. Thus it is positive. However the 2n × 2n real matrix Tn (b) is not positive because its elements do not satisfy the inequality |xij |2 ≤ xiixjj for i = n, j = n + 1. Other counterexamples of maps which are n-positive but not (n + 1)-positive for an arbitrary integer n can be found in [29]. We deduce now two useful properties of 2-positive maps. Proposition 2.10 Let A, B (B ⊆ B(K)) be C ∗ -algebras with unit and let T : A → B with be a 2-positive map. Then: 1. if T (1l) is invertible in B then for all a ∈ A we have the Schwarz inequality T (a∗ )(T (1l))−1 T (a) ≤ T (a∗ a),

(2.8)

2. for all a ∈ A we have the inequality T (a∗ )T (a) ≤ kT (1l)k T (a∗ a), 3. T is continuous and kT k = kT (1l)k .

(2.9)


19

Proof. The operator in B(K) ⊗ M2 ∗ T (a∗ a) a a a∗ T (a∗ ) 0 0 = T2 + T (a) T (1l) + ε1l a 1l 0 ε1l is positive for every ε > 0. Hence, for every u, v ∈ K we have hu, T (a∗a)ui + hu, T (a∗)vi + hv, T (a)ui + hv, (T (1l) + ε1l)vi ≥ 0. The operator T (1l) + ε1l has a bounded inverse. Taking v = −(T (1l) + ε1l)−1 T (a)u we have the inequality

u, T (a∗)(T (1l) + ε1l)−1T (a)u ≤ hu, T (a∗ a)ui for all u ∈ K. If T (1l) is invertible in B then, by letting ε tend to 0 we obtain the inequality (2.8). In any case, since 1l ≤ kT (1l) + ε1lk (T (1l) + ε1l)−1 , we find the inequality T (a∗ )T (a) ≤ kT (1l) + ε1lk T (a∗ )(T (1l) + ε1l)−1 T (a) ≤ kT (1l) + ε1lk T (a∗ a). Therefore, letting ε tend to 0, we obtain (2.9). 2 The inequality (2.9) and the property kx∗xk = kxk of C ∗ norms immediately yield

2 2 2 kT (a)k ≤ kT (1l)k · kT (a∗ a)k ≤ kT (1l)k · T (kak2 1l) = kT (1l)k · kak . This completes the proof. The following results show, in particular, that a positive linear map T : A → B is in fact completely positive when at least one of the C ∗-algebras A and B is commutative. We refer to [12], [82] for the proofs. Theorem 2.11 (Arveson) Let B be a commutative C ∗ -algebra. Then every positive linear map T : A → B is completely positive. Theorem 2.12 (Stinespring) Let A be a commutative C ∗ -algebra. Then every positive linear map T : A → B is completely positive. The following simple properties of completely positive maps turn out to be useful. Proposition 2.13 Let T : A → B and S : A → B be two completely positive linear maps. Then the map S + T completely positive. Proof. Obvious. Proposition 2.14 Let A1, A2 , A3 be C ∗ -algebras and T : A1 → A2 , S : A2 → A3 be two completely positive linear maps. Then the linear map S ◦T : A1 → A3 is completely positive. Proof. It suffices to notice that, for every integer n ≥ 1, the map (S ◦ T )n : A1 ⊗ Mn → A3 ⊗ Mn coincides with the composition Sn ◦ Tn . Proposition 2.15 Let K be a Hilbert space and let (Tm )m≥1 be a sequence of completely positive linear maps Tm : A → B(K). Suppose that, for every a ∈ A, the sequence (Tm (a))m≥1 converges weakly. Then the linear map T : A → B(K) defined by T (a) = lim Tm (a) m→∞

is completely positive.

20


Proof. By Proposition 2.9 it suffices to note that X X hui , T (a∗i aj )uj i = lim hui, Tm (a∗i aj )uj i ≥ 0, m→∞

1≤i,j≤n

1≤i,j≤n

for every integer n ≥ 1, every u1 , . . ., un ∈ K and every a1 , . . . , an ∈ A. W.F. Stinespring proved in [82] the following characterization of completely positive maps. Theorem 2.16 (Stinespring) Let B be a sub C ∗-algebra of the algebra of all bounded operators on a Hilbert space H and let A be a C ∗ -algebra with unit. A linear map T : A → B is completely positive if and only if it has the form T (a) = V ∗ π(a)V

(2.10)

where (π, K) is a representation of A on K for some Hilbert space K, and V is a bounded operator from H to K. Proof. Let T be a linear map of the form (2.10) and let (aij )1≤i,j≤n be a positive matrix in A ⊗ Mn. For all vectors (uj )1≤j≤n in H we have then X

hui, T (aij )uj i =

i,j

X

hV ui , π(aij )V uj i ≥ 0

i,j

because π is completely positive by Proposition 2.6. Conversely suppose that T is completely positive and consider the vector space A ⊗ H, the algebraic tensor product of A and H. On this space we define the bilinear form (·, ·) X (x, y) = hui , T (a∗i bj )vj i i,j

for x =

P

i

ai ⊗ ui and y =

P

j bj

⊗ vj in A ⊗ H. Since T is completely positive we have

(x, x) =

X

hui , T (a∗i aj )uj i ≥ 0

i,j

for all x ∈ A ⊗ H. Hence the bilinear form (·, ·) is positive. Consider the algebra homomorphism π0 defined on A with values in linear transformations in A ⊗ H ! X X π0(a) ai ⊗ ui = (aai ) ⊗ ui. i

i

Notice that, for all x, y as above, we have (x, π0(a)y) = (π0 (a∗ )x, y) . It follows that, for every x ∈ A ⊗ H, the linear map ω : A → C, I

ω(a) = (x, π0(a)x)

is a positive linear functional on A. Therefore we have (see [21] Prop. 2.3.11) 2

2

2

kπ0(a)xk = (x, π0(a∗ a)x) ≤ ka∗ ak ω(1l) = kak kxk .

(2.11)


21

Let N be the linear subspace of vectors x in A⊗H such that (x, x) = 0. Since N is invariant under π0(a) for every a ∈ A because of (2.11), we can consider the quotient pre-Hilbert space A ⊗ H/N defining the pre-scalar product on A ⊗ H/N by (x + N , x + N ) = (x, x) . Let K be the Hilbert space obtained by completion. By the above construction the ∗ homomorphism π0 extends to a representation π of A into B(K) such that π(a)(x + N ) = π0(a)x + N for a ∈ A and x ∈ A ⊗ H. Consider the linear operator V : H → K V u = 1l ⊗ u + N . This operator is bounded because of the inequality 2

2

kV uk = hu, T (1l)ui ≤ kT (1l)k kuk . A straightforward computation yields (2.10). Definition 2.17 A pair (π, V ) satisfying (2.10) is called a Stinespring representation of the completely positive map T . It is called a minimal Stinespring representation if the Hilbert space K coincides with the closure of the vector subspace generated by {π(a)V u | a ∈ A, u ∈ H} .

(2.12)

In other words a pair (π, V ) is a minimal Stinespring representation if the set (2.12) is total in K. Every completely positive map admits a minimal Stinespring representation. In fact, with the notation of the proof of Theorem 2.16, it suffices to consider as Hilbert space K the closure K1 of the vector space generated by (2.12). The restriction π1 of π to K1 also satisfies (2.10). The minimal Stinespring representation is unique in the following sense Proposition 2.18 Let π1 and π2 be two representations of A on Hilbert spaces K1 and K2 and let Vi : H → Ki be two bounded operators such that {πi (a)Vi u | a ∈ A, u ∈ H}, is total Ki for i = 1, 2 and such that T (a) = Vi∗ π(a)Vi for i = 1, 2. Then there exists a unitary map U : K1 → K2 such that U V1 = V2 ,

U π1(a) = π2(a)U

for all a ∈ A. Proof. Let U : K1 → K2 be the densely defined linear map defined by   n n X X U π1 (aj )V1 uj  = π2(aj )V2 uj j=1

j=1

(2.13)

22


for every integer n ≥ 1 and a1 , . . . , an ∈ A, u1, . . . , un ∈ H. A straightforward computation shows that (U π1(b)V1 v, U π1(a)V1 u)2 = hv, T (b∗ a)ui = (V1 v, π1(b∗ a)V1 u)1 , where (·, ·)j denotes the scalar product in Kj , for all a, b ∈ A and u, v ∈ H. Therefore U is an isometry and can be extended to K1 by an obvious density argument. In a similar way one can prove that also U ∗ : K2 → K1 is an isometry. Thus U is unitary. Finally, since U V1 u = U π1(1l)V1 u = π2(1l)V2 u = V2 u,

U π1 (a)V1 u = π2(a)V2 u

for every u ∈ H, (2.13) follows. We finish this section by proving K. Kraus’ characterisation [64] of σ-weakly continuous (i.e. normal) completely positive maps. Lemma 2.19 Let A and B be two von Neumann algebras of operators on Hilbert spaces H and K. A normal completely positive map T : A → B can be written in the form T (a) = V ∗ π(a)V where V is a bounded operator from K to a Hilbert space K1 and π is a normal representation of A in B(K1 ). Proof. Let (π, V ) be a minimal Stinespring representation of T with V : K → K1. We check that π is normal. Let (xα )α be a non-decreasing net of elements of A converging to x (x ∈ A) in the σ-weak topology. For all vectors u, v ∈ K and operators a, b ∈ A we have lim hπ(b)V v, π(xα)π(a)V ui α

= lim hV v, π(b∗ xαa)V ui α

= lim hv, T (b∗ xα a)ui α

= hv, T (b∗ xa)ui = hπ(b)V v, π(x)π(a)ui because T is normal. Thus π is normal by Proposition 1.15 4. Theorem 2.20 (Kraus) Let A be a von Neumann algebra of operators on a Hilbert space H and let K be another Hilbert space. A linear map T : A → B(K) is normal and completely positive if and only if it can be represented in the form T (a) =

∞ X

Vj∗ aVj

(2.14)

j=1

where (Vj )∞ j=1 are bounded operators from K to H such that the series strongly.

P

j

Vj∗ aVj converge

Proof. Clearly a completely positive map of the form (2.14) is normal. Indeed, for every non-decreasing net (xα )α of positive elements of A converging strongly to its least upper bound x we have X sup hu, T (xα)ui = sup hVj u, xαVj ui α

j

=

X

α

hVj u, xVj ui

j

=

hu, T (x)ui


23

for every u ∈ K. Conversely we can represent the normal completely positive map T in the form T (a) = V ∗ π(a)V with π normal as in Lemma 2.19. Therefore it suffices to establish (2.14) for the representation π. By decomposing K1 into cyclic orthogonal subspaces we can suppose that there exists a cyclic vector w for π(A) in K1. The state on A a → hw, π(a)wi is normal because π is normal. Hence (see, for example, [21] Th. 2.4.21 p.76) there exists a sequence (un)n≥1 of vectors in H such that X

2

kunk = 1,

hw, π(a)wi =

n≥1

X

hun , auni

n≥1

for all a ∈ A. Moreover, for every x ∈ A and n ≥ 1, we have 2

2

kxunk = hun , (x∗x)uni ≤ hw, π(x∗x)wi = kπ(x)wk . Therefore there exist contractions Vn : K1 → K such that Vn π(x)w = xun. For all x ∈ A we have also hπ(x)w, π(a)π(x)wi

=

hw, π(x∗ax)wi ∞ X huj , x∗axuj i

=

∞ X

hxuj , axuj i

=

j=1 ∞ X

hVj π(x)w, aVj π(x)wi

=

j=1 ∞ X

π(x)w, Vj∗ aVj π(x)w .

=

j=1

j=1

This completes the proof because w is cyclic for A. Remark. It is worth noticing here that Kraus’ representation (2.14) can also be written in the form T (a) = V ∗ (a ⊗ 1l)V where 1l denotes the identity operator in another Hilbert space H1 and V : K → H ⊗ H1 is a bounded operator. In fact it suffices to consider an orthonormal basis (ej ) in H1 and define V u = (Vj u) ⊗ ej . The notion of minimality for Kraus representations is obviously a special case of the analogue for Stinespring representation.

24


2.3

A quantum Feynman-Kac formula

In this section we outline a perturbation technique similar to the Feynman-Kac perturbation in classical probability. The abstract algebraic generalisation described by Accardi in [1] shows the importance of the notion of cocycle in the construction of algebraic Markov processes. We shall use the notation of Section 2.1. (A, ϕ) is an algebraic probability space with a filtration (At] )t≥0; conditional expectations IE · | At] will be denoted also by IEt] . Definition 2.21 A family (θt )t≥0 of ∗-homomorphisms of A is called a covariant shift if: 1. (semigroup property) θ0 (a) = a and θt (θs (a)) = θt+s (a) for all s, t ≥ 0 and all a ∈ A, 2. (left inverse) for all t ≥ 0 the map θt has a left inverse denoted θt∗ , i.e. for all a ∈ A, we have θt∗ (θt (a)) = a, 3. (covariance) for all s, t ≥ 0 and all a ∈ A we have θt IE θs (a) | A0] = IE θt+s (a) | At] . It is easy to check that the standard time shift of classical homogeneous Markov process is a covariant shift according to the above definition. In fact, with the notation of Example 2.1, is defined by θt (f(xt1 , . . . , xtn )) = f(xt+t1 , . . . , xt+tn ). Consider a family (jt)t≥0 of ∗-homomorphisms on A which are adapted in the sense that jt IE[ a | At] ] = IE[ jt (a) | At] ] (2.15) for every a ∈ A. We now try to find conditions in order that the algebraic process u on (A, ϕ) with values in A0] defined by ut (·) = jt (θt (·))

(2.16)

is a covariant algebraic Markov process. The following proposition gives a necessary condition. Proposition 2.22 Let (jt )t≥0 be a family of ∗-homomorphisms on A satisfying condition (2.15). Suppose that the algebraic process u defined by (2.16) is a covariant algebraic Markov process. Then, for all s, t ≥ 0 and all a ∈ θt+s (A0] ) we have IE0] [jt+s (a)] = IE0] [jt (θt (js (θt∗ (a))))] .

(2.17)

Proof. If u is a covariant quantum Markov process then, by Proposition 2.4, the family T of linear maps defined by (2.3) is a semigroup on A0] . Now, for all s, t ≥ 0 and all a ∈ A0] , denoting the conditional expectation with respect to As] by IEs] , we have Tt (Ts(a)) = IE0] jt θt IE0] [ js (θs (a))] covariance = IE0] jt IEt] [ θt (js (θs (a)))] j is adapted = IE0] IEt] [ jt (θt (js (θs (a))))] projectivity of IE = IE0] [ jt (θt (js (θs (a)))) ] inverse & semigroup = IE0] [ jt (θt ◦ js ◦ θt∗ ) (θt+s (a))] On the other hand Tt+s is defined by Tt+s (a) = IE0] [jt+s (θt+s (a))] for all a ∈ A0] . Therefore, since T is a semigroup, 2.17 holds.

2.3. A QUANTUM FEYNMAN-KAC FORMULA

25

Definition 2.23 A family (jt )t≥0 of ∗-homomorphisms on A satisfying (2.15) is called a Markov cocycle with respect to the covariant shift θ if, for all s, t ≥ 0 and all a ∈ A0] , we have jt+s (θt+s (a)) = jt (θt (js (θs (a)))) . (2.18) We prove now the fundamental result of [1]. Theorem 2.24 Let θ be a covariant shift and let (jt )t≥0 be a Markov cocycle with respect to θ. Then the algebraic process u defined by (2.16) is a covariant algebraic Markov process on (A, ϕ) with values in A0] . Proof. The proof of Proposition 2.22 shows that the maps (Tt)t≥0 are a semigroup. Let us show first that u is a quantum Markov process. For all a ∈ A0] and all s, t ≥ 0 we have IEs] [ut+s(a)] cocycle property projectivity of IE covariance of θ and IE

= IEs] [jt+s (θt+s (a))] = IEs] [js ((θs ◦ jt ◦ θs∗ ) θt+s (a))] = IEs] js IEs] [(θs ◦ jt ◦ θs∗ ) θt+s (a)] = IEs] (js ◦ θs ) IE0] [jt (θt (a)))] = IEs] [(js ◦ θs ) (Tt(a))]] = us (Tt(a))

Let us show now that u is covariant; for all a ∈ A0] and all s, t ≥ 0, we have ut IE0] [us(a)] = ut (Ts(a)) = IEt] [ut+s(a)] This completes the proof. Remark. The analogy with the classical Feynman-Kac formula becomes clear if we take a standard Brownian motion (wt)t≥0 with natural filtration (Ft )t≥0 Ft = σ {ws | 0 ≤ s ≤ t} as the classical stochastic process x in Example 2.1. Let Mt be the multiplicative functionals Z t Mt = exp − c(ws )ds 0

and let jt(a) = Mt a. Clearly jt : A → A is not an identity preserving homomorphism but it enjoys the cocycle property (2.18) with respect to the standard (classical) shift θ (see [2]). Proposition 2.25 Let H be a complex separable Hilbert space, let A be the *-algebra B(H) of all bounded operators on H and let At] t≥0 a filtration of sub-*-algebras of A. Consider a family (Vt )t≥0 of unitary operators on H such that Vt ∈ At] and, for all t ≥ 0, define the map jt : A → A, jt (a) = Vt aVt∗ (2.19) Suppose that, for all s, t ≥ 0, we have Vt+s = Vt θt (Vs ) Then j is a cocycle with respect to θ.

(2.20)

26


Proof. For all s, t ≥ 0 and all a ∈ A0] we have jt ◦ θt ◦ js ◦ θs (a)

= = =

Vt (θt (Vs θs (a)Vs∗ )) Vt∗ ∗ [Vt θt (Vs )] θt+s (a) [Vt θt (Vs )] jt+s (θt+s (a))

as required. The family of operators (Vt )t≥0 defining the cocycle j is called an operator cocycle or, when no confusion can arise, simply a cocycle.

3

Quantum Markov Semigroups Semigroups of completely positive operators are a fundamental tool in the theory of quantum Markov processes. Several results on this class of semigroups - mostly in the uniformly continuous case - are scattered in the literature. In this chapter we give a self-contained exposition of the basic results on uniformly continuous completely positive semigroups. Moreover we describe the construction of completely positive semigroups on B(h) that are continuous in the σ-weak topology and we give conditions - necessary and sufficient or simply sufficient - for the semigroup to be identity preserving (i.e. Markov).

3.1

Fundamental definitions and examples

In this section A will denote a W ∗ -algebra of operators acting on a Hilbert space H and A∗ will denote its predual. Definition 3.1 A quantum dynamical semigroup on A is a family T = (Tt)t≥0 of bounded operators on A with the following properties: 1. T0 (a) = a, for all a ∈ A, 2. Tt+s (a) = Tt (Ts(a)), for all s, t ≥ 0 and all a ∈ A, 3. Tt is completely positive for all t ≥ 0, 4. Tt is a σ-weakly continuous operator in A for all t ≥ 0, 5. for each a ∈ A, the map t → Tt(a) is continuous with respect to the σ-weak topology on A. Notice that, when 3 holds, then 4 is equivalent to normality of the maps Tt by Proposition 1.15. We introduce the usual definition of the infinitesimal generator of T . Definition 3.2 The infinitesimal generator the quantum dynamical semigroup T is the operator L whose domain D(L) is the vector space of elements a in A for which there exists an element b of A such that Tt (a) − a b = lim t→0 t in the σ-weak topology, and L(a) = b. 27

28

3. QUANTUM MARKOV SEMIGROUPS We now give some examples.

Example 3.1 Let A = B(H) and let (Pt)t≥0 be a strongly continuous semigroup on H. The family of linear operators Tt : A → A defined by Tt(a) = Pt∗ aPt is a quantum dynamical semigroup. In fact all the continuity properties of T follow from Kraus’ theorem and the strong continuity of P . Let G denote the infinitesimal generator of P . The infinitesimal generator L of T is given (formally if G is unbounded) by L(a) = G∗ a + aG. Notice that, if G is unbounded, then T is not necessarily strongly continuous. The following example shows that classical Markov semigroups can be viewed as a special class of quantum dynamical semigroups. However some care must be taken dealing with topologies. Example 3.2 Let (X, X ) be a measurable space and let µ be a σ-finite measure on X . Then L∞ (X,C; I dµ) is a commutative W ∗ -algebra of multiplication operators acting on the Hilbert space L2(X,C; I dµ). The predual is the Banach space L1 (X,C; I dµ). Consider a family (P (t, ·; ·))t≥0 of transition probabilities on X × X with the following properties: 1. for all x ∈ X, P (t, x; ·) is a probability measure on X , absolutely continuous with respect to µ for t > 0, and coinciding with the Dirac measure δx for t = 0, 2. for all measurable sets A ∈ X , P (t, ·; A) is a measurable essentially bounded function on X and the map t → P (t, ·; A) defined on [0, +∞[ with values in L∞ (X, X ; dµ) is σ-weakly continuous, 3. (Chapman-Kolmogorov equation) for all t, s ≥ 0, x ∈ X and A ∈ X we have Z P (t + s, x; A) = P (t, y; A)P (s, x; dy). X

Let (Tt)t≥0 be the family of operators on L∞ (X,C; I dµ) Z (Ttf)(x) = f(y)P (t, x; dy). X

Then T is a quantum dynamical semigroup in the W ∗ -algebra L∞ (X,C; I dµ). In fact property 1 of a quantum dynamical semigroup follows from P (0, x; dy) = δx (dy) and property 2 follows from the Chapman-Kolmogorov equation. Complete positivity follows from the fact that the W ∗ -algebra L∞ (X,C; I dµ) is commutative and that the linear operators Tt are positive. The continuity property 5 follows from the σ-weakly continuity of the map t → P (t, ·; A) for every A ∈ X . Indeed, writing f as the sum four positive functions and remembering that the σ-weak and weak topology coincide on bounded subsets of L∞ (X,C; I dµ), we can easily see that it suffices to show that, for every positive element f of L∞ (X,C; I dµ) and every g ∈ L2 (X,C; I dµ), the function Z Z Z 2 t → g(·), f(y)P (t, ·; dy)g(·) = |g(x)| dµ(x) f(y)P (t, x; dy) (3.1) X

X

X

3.1. FUNDAMENTAL DEFINITIONS AND EXAMPLES

29

is continuous. This is clearly the case whenever f is a simple function because of the σweakly continuity of the maps t → P (t, ·; A) with A ∈ X . Approximating f by an increasing (resp. decreasing) sequence of simple functions converging to f almost everywhere we see that (3.1) is lower (resp. upper) semicontinuous. Therefore the function (3.1) is continuous. In order to prove the continuity property 4, since Tt is positive, by Proposition 1.15 it suffices to check that it is normal. Thus we must show that, for every increasing net (fα ) in L1 (X,C; I dµ) converging σ-weakly to f in L1 (X,C; I dµ) and every g ∈ L1 (X,C; I dµ) non-negative we have Z Z Z Z sup g(x)dµ(x) fα (y)P (t, x; dy) = g(x)dµ(x) f(y)P (t, x; dy). (3.2) α

X

X

X

X

By [21] Lemma 2.4.19 p.76, the net (fα ) converges σ-strongly to f. Therefore, denoting the Radon-Nikodym derivative of P (t, x; dy) with respect to µ by p(t, x, ·), we have Z D E sup fα (y)P (t, x; dy) = sup (p(t, x, ·))1/2, fα(·)(p(t, x, ·))1/2 α α X D E = (p(t, x, ·))1/2, f(·)(p(t, x, ·))1/2 Z = f(y)P (t, x; dy) X

for every x ∈ X. It follows that the right-hand side is the least upper bound of the integrals of fα with respect to P (t, x; dy) and the same argument yields (3.2). This shows that (Tt)t≥0 is a quantum dynamical semigroup. It is worth noticing that a large class of classical Markov semigroups enjoys the so-called Feller property. This means, roughly speaking, that the semigroup can be restricted to a space of continuous functions and studied as a strongly continuous contractive semigroup. The original semigroup on L∞ (X, X ; µ) is then the unique σ-weakly continuous extension of this restriction. Clearly the quantum analogue of the Feller property is the following: there exists a C ∗ -algebra A0 that is dense in A in the σ-weak operator topology, invariant under the action of the quantum dynamical semigroup T , and such that the restriction of T to A0 is strongly continuous. We will not be concerned however with the problem of establishing whether some quantum dynamical semigroup enjoys this property. The following fact will be useful in the construction of another example of a quantum dynamical semigroup through a generalization of Example 3.1 Proposition 3.3 Let (X, X ) be a measurable space and let µ a finite measure on X . Let (U (t, x))t≥0,x∈X be a family of bounded operators on a Hilbert space H such that 1. for all x ∈ X the map t → U (t, x) is strongly continuous, 2. for all t ≥ 0 the map x → U (t, x) is strongly measurable, 3. for all t ≥ 0 there exists a positive function gt on X, integrable with respect to µ, such that sup kU (s, x)k ≤ gt(x) 0≤s≤t

Then the map Φ : [0, +∞[×B(H) → B(H) defined by the integral (in the σ-weak topology) Z Φ(t, a) = U (t, x)∗ aU (t, x)dµ(x) X

is σ-weakly continuous in both arguments and completely positive in the second.

30

3. QUANTUM MARKOV SEMIGROUPS

Proof. Let n ≥ 1 and a1 , . . . , an, b1, . . . , bn be elements of B(H). Since 2 Z X n ∗ ∗ bj Φ(t, aj ai )bi = aj U (t, x)bj dµ(x) ≥ 0 X j=1 i,j=1 n X

the map Φ(t, ·) is completely positive by Proposition 2.8. Note that Φ is uniformly bounded on sets of the form [0, t]×B(H) with t fixed. Therefore, since the σ-weak and weak topology coincide on bounded sets of B(H), in order to show that Φ is σ-weakly continuous in a it suffices to prove that, for every u ∈ H, the positive linear functional on B(H) Z a→ hU (t, x)u, aU (t, x)ui dµ(x) X

is σ-weakly continuous. To this end, by [21] Th. 2.4.21 p.76, it suffices to show that, for every increasing net (aα ) in B(H) with least upper bound a we have Z Z sup hU (t, x)u, aαU (t, x)ui dµ(x) = hU (t, x)u, aU (t, x)ui dµ(x). α

X

X

Since (aα ) converges strongly to a (see [21] Lemma 2.4.19) for all x ∈ X, we have sup hU (t, x)u, aαU (t, x)ui = hU (t, x)u, aU (t, x)ui . α

The conclusion then follows since the map on L1 (X, X , µ) Z g→ g(x)dµ(x) X

is a σ-weakly continuous functional. (This is in practice a monotone convergence theorem for increasing nets in L1 (X, X , µ)). Finally to show that Φ is σ-weakly continuous in t it suffices to use the inequality Z |hu, (Φ(t, a) − Φ(s, a))ui| ≤ 2c(r) kak kuk k(U (t, x) − U (s, x))uk dµ(x) X

for u ∈ H and t, s ∈ [0, r] (r > 0 fixed) where c(r) is a positive constant depending only on r and apply Lebesgue’s theorem. The following example is a generalization of Example 3.1 due to Parthasarathy ([74] Example 30.1 p. 258). However the semigroup he constructs there is not a quantum dynamical semigroup in the sense of his definition (Sect. 30 p.257) since it is not necessarily strongly continuous. We shall exhibit a counterexample. Example 3.3 Let (Bt )t≥0 be a classical Brownian motion on a filtered probability space (Ω, F, IP ) and let H be a complex separable Hilbert space. For every bounded self-adjoint operator L on H let us define the map Tt : B(H) → B(H) Z Tt(a) = exp(iLBt (ω))a exp(−iLBt (ω))dIP (ω) (3.3) Ω 2 Z √ √ 1 x = √ exp(ix tL)a exp(−ix tL) exp − dx 2 2π IR

3.1. FUNDAMENTAL DEFINITIONS AND EXAMPLES

31

Since (Bt )t≥0 has stationary independent increments it can be shown as in [74] Example√30.1 p. 258 that (Tt )t≥0 is a semigroup. Applying Proposition 3.3 with U (t, x) = exp(−ix tL) and µ equal to the standard gaussian measure on the Borel σ-algebra of IR it is easy to see that also the properties 3, 4 and 5 of Definition 3.1 are fulfilled. Hence T is a quantum dynamical semigroup. The infinitesimal generator L of T is given (at least formally if L is unbounded) by L(X) = −

1 2 L X − 2LXL + XL2 . 2

(3.4)

The quantum dynamical semigroup T admits restrictions to abelian subalgebras of B(H) which are classical Markov semigroups. Our argument here is only formal since the operator L is unbounded; we will rigorously deal with this example later. Let H be the Hilbert space L2(IR;C) I and let L be the first derivative with its natural domain. Denote by M (f) the multiplication operator by a bounded smooth function f M (f) : H → H,

(M (f)u)(x) = f(x)u(x).

A straightforward computation shows that L(M (f)) = M

1 00 . f 2

This means that the restriction of the semigroup T to the abelian C ∗ -algebra of bounded complex-valued continuous functions on the real line coincides with the classical Markov semigroup of Brownian motion. We give now the counterexample showing that T is not necessarily strongly continuous. Let us fix H = l2 (N) with canonical orthonormal basis (en )n≥0 and let N and S be the number operator and right shift operator    X  X D(N ) = u∈H n2 |un|2 < +∞ , Nu = nun en   n≥0 n≥0 X D(S) = H, Su = unen+1 , n≥0

where u =

P

n≥0 un en .

Clearly N 2 is the self-adjoint operator

 

 X  D(N 2 ) = u ∈ H n4 |un|2 < +∞ ,   n≥0

N u2 =

X

n2un en .

(3.5)

n≥0

Let L = N 2. We now prove that the quantum dynamical semigroup T is not strongly continuous on B(H). By virtue of (3.3), for every u ∈ H and x ∈ B(H) we have

= =

hu, (Tt(x) − x)ui 2 Z

1 y √ u, exp(iyN 2 )x exp(−iyN 2 ) − x u exp − dy 2t 2πt IR 2 Z D E √ √ 1 y √ u, exp(iy tN 2)x exp(−iy tN 2) − x u exp − dy 2 2π IR

32


Now let x be the right shift operator S and, for every n ≥ 1, let vn be the vector 1 vn = √ n

X

ek .

n 0, the function t → C 1/2P (t)u is differentiable and

2 D E d

1/2

C P (t)u = 2 0 we have Cε1/2P (t +

s)u =

Cε1/2P (t)u

+

Z

t+s

Cε1/2P (r)Gu dr t

for every s, t ≥ 0 and u ∈ R(λ; G)(D). Notice that, since u = R(λ; G)v with v ∈ D, then Gu = λR(λ; G)v − v belongs to D(C 1/2 ). Moreover, by Proposition 3.36, the function t → C 1/2P (t)Gu is continuous on [0, +∞[ for the norm k · k on h and satisfies the inequality kC 1/2P (t)Guk ≤ exp(bt)kC 1/2Guk. This allows us to let ε tend to 0 to get Z t+s C 1/2P (t + s)u = C 1/2P (t)u + C 1/2P (r)Gu dr. t

A straightforward computation shows that the difference

2

2

1/2

C P (t + s)u − C 1/2P (t)u

62


for s, t ≥ 0, can be written in the form

2 D E

1/2

C (P (t + s) − P (t)) u + 20

(n)

Proof. Let Rλ

be the sequence of positive linear maps considered in the proof of n>0

Theorem 3.25. (n) It suffices to show that, for all n ≥ 0, λ > b and u ∈ D(C 1/2 ), the operator Rλ (Cε ) satisfies

2 D E

(n) (λ − b) sup u, Rλ (Cε)u ≤ C 1/2u . (3.40) ε>0

The above inequality holds for n = 0 and u ∈ R(λ; G)(D). Indeed, since D is invariant under P (t), then P (t)R(λ; G)(D) = R(λ; G)(P (t)(D)) ⊆ R(λ; G)(D) is contained in D(C 1/2 ) and, integrating by parts, we have D E (0) λ u, Rλ (Cε)u Z ∞ = λ e−λt hP (t)u, CεP (t)ui dt Z0 ∞

2

≤ λ e−λt C 1/2P (t)u dt 0 Z ∞

2 D E

= C 1/2u + 20

This proves (3.40) for n = 0 and u ∈ R(λ; G)(D). Since R(λ; G)(D) is dense in D(C 1/2) for the norm k · kC , (3.40) holds also for u ∈ D(C 1/2 ).

3.6. SUFFICIENT CONDITIONS FOR CONSERVATIVITY

63

Suppose that (3.40) has been established for an integer n and every u ∈ D(C 1/2). Notice that for every u ∈ R(λ; G)(D) the vectors L` P (t)u belong to D(C 1/2 ). Thus, from the (n) second equation (3.17) (or from (3.27)) and the definition of Rλ , we have D E (n+1) u, Rλ (Cε)u ∞ Z ∞ D E X (n) = hu, Pλ(Cε)ui + e−λt L` P (t)u, Rλ (Cε )L` P (t)u dt `=1

0

1 X λ−b ∞

≤

hu, Pλ(Cε)ui +

Z

`=1

∞ 0

2

e−λt C 1/2L` P (t)u dt.

Using the inequality (3.37) and integrating by parts we obtain ∞ Z ∞

2 X

e−λt C 1/2L` P (t)u dt `=1

0

2 d

1/2 ≤ − C P (t)u dt e dt Z0 ∞

2

+b e−λt C 1/2P (t)u dt 0 Z ∞

2

1/2 2

= C u − (λ − b) e−λt C 1/2P (t)u dt 0

1/2 2 ≤ C u − (λ − b) hu, Pλ(Cε )ui . Z

∞

−λt

Therefore (3.40) for n+1 and u ∈ R(λ; G)(D)follows. Since R(λ; G)(D) is dense in D(C 1/2), this completes the proof. We can now prove the main results of this section. Theorem 3.39 Suppose that there exists an operator C satisfying hypothesis C such that hu, Fnui ≤ hu, Cui for each u ∈ D(C), n ≥ 1. Then the minimal quantum dynamical semigroup is Markov. Proof. Fix λ > b. Under the hypotheses of the theorem, for ε > 0, the bounded operators (Fn)ε and Cε satisfy the inequality (Fn)ε ≤ Cε (see, for example, [78] Chap. 8, Ex. 51, p.317). Applying Proposition 3.38, we obtain the estimate ∞ X

u, Qkλ(1l)u ≤

D E (min) lim inf sup u, Rλ ((Fn )ε)u

k=1

D E (min) sup u, Rλ (Cε )u

≤

n→∞ ε>0

ε>0

≤

2

(λ − b)−1 C 1/2u < +∞

for every u ∈ D(C 1/2). Therefore the minimal quantum dynamical semigroup is Markov because condition 2 of Theorem 3.28 is fulfilled. Remark. Notice that, in the above theorem, we did not assume that the quadratic form u → −2 0, the bounded operators Φε and Cε satisfy the inequality Φε ≤ Cε (see [78] Chap. 8, Ex. 51, p.317). Moreover, for u ∈ D(G), we have Z ∞

2

sup hu, Pλ(Φε )ui = e−λt Φ1/2P (t)u dt ε>0

0

=

∞ Z X `=1

∞ 2

e−λt kL` P (t)uk dt 0

= hu, Qλ(1l)ui . Therefore the increasing family of operators (Pλ (Φε ))ε>0 is uniformly bounded and, since D(G) is dense in h, it follows that it converges strongly to Qλ(1l) as ε goes to 0. The maps Qkλ being σ-weakly continuous, we have ∞ ∞ X X

u, Qk+1 u, Qk(Pλ (Φε ))u (1 l)u = sup λ ε>0

k=0

k=0

D E (min) = sup u, Rλ (Φε )u ε>0

by Theorem 3.25. From Proposition 3.38 we obtain the estimate ∞

2 D E X

(min) u, Qkλ(1l)u ≤ sup u, Rλ (Cε)u ≤ (λ − b)−1 C 1/2u k=1

ε>0

for every u ∈ D(C 1/2). Therefore the minimal quantum dynamical semigroup is Markov because condition 2 of Theorem 3.28 holds. It has been shown in [53] that condition C can be interpreted as an a priori estimate on the minimal quantum subMarkov semigroup associated with the operators G, L` . The following corollary gives an easier (but weaker) sufficient condition for conservativity. Corollary 3.41 Suppose that the hypothesis AA holds and there exists a self-adjoint operator C with domain coinciding with the domain of G and a core D for C with the following properties: a) L` (D) ⊆ D(C 1/2) for all ` ≥ 1,

3.6. SUFFICIENT CONDITIONS FOR CONSERVATIVITY

65

b) there exists a self-adjoint operator Φ such that −2 t + s.

Therefore we have the identity ρt (ρt+s f) = (σ−sf)1]0,t[ + (ρt+s f)1]t,t+s[ + f1]t+s,∞[ . By the tensor product factorization of Fock space and t-adaptedness of V (t) we write the right-hand side of (5.17) as

ve((σ−s g)1]0,t[ ), V (t)ue((σ−s f)1]0,t[ ) · e(g1]t+s,∞[ ), e(f1]t+s,∞[ )

· e((ρt+s g)1]t,t+s[ ), e((ρt+s f)1]t,t+s[ ) . Computing the second and third scalar product we write the right-hand side of (5.17) in the form

ve(g1]s,t+s[ ), Γ(σs)V (t)Γ(σ−s )ue(f1]s,t+s[ ) Z s Z ∞ · exp (¯ g f)(r)drf + (¯ gf)(r)dr 0

=

t+s

hve(g), θs (V (t))ue(f)i .

This proves the lemma. We recall now the notion of dual cocycle due to Journ´e (see [61] p. 294 and also [69] p. 174). Proposition 5.9 Let (V (t))t≥0 be a left (resp. right) cocycle. The family of operators (Ve (t))t≥0 defined by Ve (t) = Rt (V (t)∗ ) (5.18) is a left (resp. right) cocycle.

92

5. QUANTUM FLOWS

Proof. By virtue of the cocycle property, for every s, t ≥ 0 we have Ve (t + s)

= Rt+s (V (t + s)∗ ) = Rt+s (θs (V (t)∗ ) V (s)∗ ) = Rt+s (θs (V (t)∗ )) Rt+s (V (s)∗ ) .

Applying Rt+s to both sides of (5.16) we have Rt (V (t)∗ ) = Rt+s (θs (V (t)∗ )) . The same identity yields also Rt+s (V (s)∗ ) = Rt+s (Rs (Rs (V (s)∗ ))) = θt (Rs (V (s)∗ )) . This proves the lemma. Definition 5.10 The cocycle (Ve (t))t≥0 defined by (5.18) is called dual cocycle of the cocycle (V (t))t≥0 . We now study the relationship between a cocycle satisfying a left quantum stochastic differential equation and its dual. Lemma 5.11 Let t, s be two non-negative real numbers and let X(s) (resp. Y (t)) be an s-adapted (resp. t-adapted) bounded operator. For every u, v ∈ h and every f, g ∈ MS we have hY (t)ve(g), θt (Rs (X(s)))ue(f)i = hY (t)ve(ρt,t+s g), θt (X(s))ue(ρt,t+s f)i

(5.19)

where ρt,t+s is the time reversal operator on the interval [t, t + s] defined by f(2t + s − r) if t < r < t + s, (ρt,t+s f)(r) = f(r) otherwise. Proof. Since the homomorphisms θt and Rs are normal it suffices to prove the lemma for the factorised operators X(s) = X0 ⊗ Xs ,

Y (t) = Y0 ⊗ Yt

where X0 , Y0 are bounded operators in the initial space and Xs (resp. Yt ) is a bounded operator in the factor Γ(L2 (0, s)⊗k) (resp. Γ(L2 (0, t)⊗k)) of the Fock space Γ(L2 (IR+ )⊗k). In this case

the left-hand side of (5.19) can be written as the product of the factors hY0v, X0 ui, Yt e(g1[0,t] ), e(f1[0,t] ) and

e(g1[t,∞[ ), Γ(σt)Γ(ρs )Xs Γ(ρs )Γ(σ−t )e(f1[t,∞[ )

= e(ρs σ−t(g1[t,∞[ )), Xs e(ρs σ−t(f1[t,∞[ )) A simple computation shows that ρs σ−t(g1[t,∞[ ) = σ−t ρt,t+sg,

ρs σ−t(f1[t,∞[ ) = σ−tρt,t+s f.

Therefore we have

e(ρs σ−t (g1[t,∞[ )), Xs e(ρs σ−t(f1[t,∞[ )) = he(σ−t ρt,t+sg)), Xs e(σ−t ρt,t+s f)i

= e((ρt,t+s g)1[t,∞[ ), θt (Xs )e((ρt,t+s f)1[t,∞[ ) .

The product of this with Yt e(g1[0,t] ), e(f1[0,t] ) is equal to hYt ve(ρt,t+s g), θt (Xs )ue(ρt,t+s f)i . Multiplication by the initial space factor hY0 v, X0 ui then yields (5.19).

5.3. QUANTUM STOCHASTIC DIFFERENTIAL EQUATIONS

93

Proposition 5.12 Let L be an element of IB satisfying the inequality (5.15) and let (V (t))t≥0 be the unique (h, MS )-regular adapted contractive left cocycle (V (t))t≥0 satisfying (5.14). Then the dual cocycle (Ve (t))t≥0 satisfies the left quantum stochastic differential equation dVe (t) = Ve (t)dΛLb (t),

Ve (0) = 1l.

Proof. By virtue of Proposition 5.9, for every t, s ≥ 0 we have Ve (t + s) − Ve (t) = Ve (t)θt Ve (s) − 1l . Therefore, applying (5.19), we obtain D E ve(g), Ve (t + s) − Ve (t) ue(f) D E = Ve (t)∗ ve(g), θt (Rs (V (s)∗ − 1l)) ue(f) D E = Ve (t)∗ ve(ρt,t+s g), θt (V (s)∗ − 1l) ue(ρt,t+s f) The action of θt on a stochastic integral can be computed explicitly as follows Z s ∗ ∗ θt (V (s) − 1l) = θt dΛLb (r)V (r) =

0 t+s

Z

dΛLb (r)θt (V (r − t)∗ ) . t

Therefore, by the first fundamental formula of quantum stochastic calculus (5.6), we have D E ve(g), Ve (t + s) − Ve (t) ue(f) Z t+s ∗ ∗ e = V (t) ve(ρt,t+s g), dΛLb (r)θt (V (r − t) ) ue(ρt,t+s f) =

Z

t+s t

D

t

Ve (t)∗ v(e−∞ + g(2t + s − r))e(ρt,t+s g),

E Lb θt (V (r − t)∗ ) u(f(2t + s − r) + e+∞ )e(ρt,t+s f) dr.

Dividing by s and letting s tend to 0, for every t which is a continuity point for both g and f we have E E D d D ve(g), Ve (t)ue(f) = v(e−∞ + g(t))e(g), Ve (t)Lb u(f(t) + e+∞ )e(f) . dt This completes the proof.

5.3

Quantum stochastic differential equations

In this section we recall the main results on left quantum stochastic differential equations of the form (5.14) with a possibly unbounded operator L giving in particular necessary and sufficient conditions in order the operators V (t) to be isometries, coisometries or unitaries. These extend the results of [37] [50], [52] for special classes of operators L. Several results show that “good” operator cocycles satisfy a quantum stochastic differential equation (see, for example [7], [40], [57]). A full characterization (a quantum analogue

94

5. QUANTUM FLOWS

of the classical Stone’s theorem on strongly continuous unitary groups), however, is not available. As a first step we complete the study of the case when L is bounded. Proposition 5.13 Let L ∈ IB and let (V (t))t≥0 be the unique (h, MS )-regular adapted initial space bounded process solving the quantum stochastic differential equation dV (t) = V (t)dΛL(t),

V (0) = 1l.

(5.20)

Then: 1. the operators (V (t))t≥0 are contractions if and only if one of the following equivalent inequalities hold L + Lb + LLb F ≤ 0, F L + Lb + Lb L ≤ 0, (5.21) 2. the operators (V (t))t≥0 are isometries if and only if L + Lb + Lb L = 0,

(5.22)

3. the operators (V (t))t≥0 are coisometries if and only if L + Lb + LLb = 0,

(5.23)

4. the operators (V (t))t≥0 are unitary if and only if L + Lb + Lb L = 0,

L + Lb + LLb = 0.

(5.24)

Proof. Clearly the operators V (t) (t ≥ 0) are contractions if and only if the first inequality (5.21) holds by Corollary 5.7. The equivalence of the second inequality (5.21) and contractivity of the operators V (t) can be shown in the same way considering the dual cocycle Ve . In fact contractivity of Ve is equivalent to that of V and the dual cocycle satisfies a left quantum stochastic differential equation with Lb instead of L by Proposition 5.12. We prove then 2. For every v, u ∈ h and every f, g ∈ MS formula (5.6) yields D E Ve (t)∗ ve(g), Ve (t)∗ ue(f) = hve(g), ue(f)i Z t

+ Ve (s)∗ v(e−∞ + g(s))e(g), F (L + Lb + Lb L)Ve (s)∗ u(e−∞ + f(s))e(f) ds 0

Since Ve (t)∗ = Rt (V (t)) this shows that the operators (V (t))t≥0 are isometries if and only if (5.22) holds. The third statement can be proved in the same way considering the right cocycle (V (t)∗ )t≥0. The fourth statement follows immediately from 2 and 3. Remark. When L is the operator (5.5) we have  0 L3 + L∗1 + L∗1 L2 b b  L + L + L L = 0 L2 + L∗2 + L∗2 L2 0 0

 L4 + L∗4 + L∗1 L1 L1 + L∗3 + L∗2 L1  0

In this case, condition (5.22), (resp. (5.23)) is equivalent to the well-known condition for the solution of a quantum stochastic differential equation to be isometric (resp. coisometric) found in the seminal paper [58] by R.L. Hudson and K.R. Parthasarathy.


95

Note that, if the operators (Lk )4k=1 are bounded (for simplicity), the identity L4 + L∗4 + L∗1 L1 = 0 implies that 1 L4 + L∗1 L1 2 has the form iH where H is bounded self-adjoint operator. Therefore, the choice L1 = L2 = L3 = 0,

L4 = iH,

yields a group of unitary operators (V (t))t≥0 satisfying the Schroedinger equation dV (t) = iHV (t)dt. In this sense (5.20) is a generalization of the Schroedinger equation. The following propositions motivate partially the assumptions under which we shall study the quantum stochastic differential equation (5.20) with an unbounded operator L. Let us recall first the following (see, for example, [61] or [69] Ch. VI, Sect. 12). Proposition 5.14 Let (V (t))t≥0 be a contractive left cocycle in H. Then the family (P (t))t≥0 of operators on h defined by P (t) = IE0] [ V (t) ] (5.25) is a contraction semigroup in h. If the cocycle is strongly continuous then the semigroup (P (t))t≥0 is also. Proof. Since (θt )t≥0 is a covariant shift and V is a cocycle for every t, s ≥ 0 we have P (t + s)

= = =

IE0] [V (t + s)] IE0] [V (s)θs (V (t))] IE0] [V (s)IEs] [θs(V (t))]]

=

IE0] [V (s)θs (IE0] [V (t)])].

Notice that θs (P (t)) = P (t) because the shift (θt )t≥0 leaves invariant operators on the initial space. Hence we have P (t + s) = IE0] [V (s)]P (t) = P (s)P (t). Therefore (P (t))t≥0 is a semigroup. Moreover, for every u ∈ h, we have k(P (t + s) − P (t))uk =

sup

|hv, (P (t + s) − P (t))ui|

v∈h, kvk=1

=

sup

|hve(0), (V (t + s) − V (t))ue(0)i|

v∈h, kvk=1

≤ k(V (t + s) − V (t))ue(0)k . Strong continuity of V implies then the strong continuity of (P (t))t≥0. A similar argument shows that the semigroup is also contractive. From now on we shall always suppose that (P (t))t≥0 is strongly continuous. The infinitesimal generator G of P is defined as the set of u ∈ h such that the limit lim t−1 (P (t)u − u)

t→0+

exists in the strong (or equivalently weak) topology of h. The following natural hypothesis on the operator L that will be in force throughout the rest of this chapter. Hypothesis L

96

5. QUANTUM FLOWS 1. There exists an operator G which is the infinitesimal generator of a strongly continuous contraction semigroup in h and a core D for G such that the domain of the operator L contains the domain DS,G DS,G = { we−∞ + uf + ve+∞ | w ∈ h, u, v ∈ D, f ∈ k } , 2. for all u, v ∈ D, w ∈ h, we have Lwe−∞ = 0, E+∞ L(ve+∞ + uf) = 0 hw, Gvi = hwe−∞ , Lve+∞ i

(5.26)

3. for all x ∈ DS,G we have hF x, Lxi + hLx, F xi + hF Lx, Lxi ≤ 0.

(5.27)

Remark. The above hypothesis implies that the operators G and L`+∞ satisfy the hypothesis A introduced in Chapter 3. Indeed, for every u ∈ D, taking x = ue+∞ , (5.27) reads as ∞ X

` hGu, ui + L+∞ u, L`+∞ u + hu, Gui ≤ 0. `=1

Since D is a core for G, the operators L`+∞ can be extended to D(G) so that the above in equality holds also for u ∈ D(G). Remark. Since the coefficient L is unbounded but the solutions we shall construct are contractive the left quantum stochastic differential equation will be interpreted through quantum stochastic integrals defined on vectors of the form ue(f) with u ∈ D. We shall stress this fact by speaking of quantum stochastic differential equations in DS . ˆ can be approximated by a sequence of Under the hypothesis L an operator L in h ⊗ k bounded operators (Ln )n≥1 which are the “infinitesimal generators” of contractive cocycles. For every integer n ≥ 1, let R(n; G) be the bounded operator (n1l − G)−1. Note that the adjoint operator G∗ is the infinitesimal generator of the dual contraction semigroup and we have R(n; G∗ ) = R(n; G)∗ . Moreover the well-known properties of resolvent operators relations for all w ∈ h, v ∈ D(G) yield lim nR(n; G)w = w,

n→∞

lim nGR(n; G)v = Gv.

n→∞

ˆ satisfying the hypothesis L. For each n ≥ 1 Proposition 5.15 Let L be an operator in h⊗ k ˆ with domain DS defined by let In , Ln be the operators in h ⊗ k In = nR(n; G)E−∞ + E + nR(n; G)E+∞ ,

Ln = In∗ LIn .

(5.28)

Then the operator Ln has a bounded extension which is an element of IB satisfying the inequalities (5.21) and its uniform norm can estimated by √ kLn k ≤ 2 n + 3 n + 1 . Moreover, for all ξ ∈ DS , we have lim Ln ξ = Lξ.

n→∞

(5.29)


97

Proof. The operator Ln satisfies the inequality (5.27). Indeed it suffices to write (5.27) for vectors of the form In x with x ∈ DS,G (the vector In x belongs to the domain of L by the well-known properties of the operators R(n; G) and the remark after the hypothesis L) and use the commutation of the operator F with the operators In and In∗ . Moreover, because of (5.26), in order to prove that Ln is bounded, it suffices to estimate ˆ of the form PJ uj (fj + e+∞ ) with uj ∈ D, fj ∈ MS the norm of Ln ξ for vectors ξ in h⊗ k j=1 for all j. In this case we have kLn ξk ≤ kELn Eξk + kE−∞ Ln Eξk + kELn E+∞ ξk + kE−∞Ln E+∞ ξk

(5.30)

The norm of E−∞ Ln E+∞ can be estimated using the third identity (5.26). We have in fact, for each v, u ∈ D, hve−∞ , E−∞Ln E+∞ ue+∞ i = n2 hR(n; G)v, GR(n; G)ui Then, using the identity GR(n; G) = nR(n; G) − 1l and the contractivity of nR(n; G), we can estimate the norm of GR(n; G) by 2. Therefore we obtain the inequality |hve−∞ , E−∞Ln E+∞ ue+∞ i| ≤ 2n kvk · kuk which implies kE−∞ Ln E+∞ k ≤ 2n.

(5.31)

Taking a vector x ∈ DS in the range of E from the inequality (5.27) for Ln we obtain immediately the estimates kE + ELn Ek ≤ 1,

kELn Ek ≤ 2.

(5.32)

Consider now a vector x = ue+∞ with u ∈ D. The inequality (5.27) for Ln yields 2

2n2