Quantum coarse-grained entropy and thermodynamics ˇ anek,1, ∗ J. M. Deutsch,2 and Anthony Aguirre1 Dominik Safr´ 1
SCIPP and Department of Physics, University of California, Santa Cruz, CA 95064, USA 2 Department of Physics, University of California, Santa Cruz, CA 95064, USA (Dated: August 1, 2017)
arXiv:1707.09722v1 [quant-ph] 31 Jul 2017
We extend classical coarse-grained entropy, commonly used in many branches of physics, to the quantum realm. Classical coarse graining partitions phase space, which is difficult to extend to quantum mechanics. However by performing two non-commuting kinds of measurement, first to gather coarse-grained positional information, and then energy, our prescription leads to results in accord with the thermodynamic entropy for equilibrium systems and appears to extend well to nonequilibrium situations. This should have many applications, particularly to experiments on isolated quantum systems.
Entropy, and its increase, are crucial concepts applied across an array of physical theories and systems. Yet entropy has many distinct proposed definitions [1] with major open questions as to which if any of these definitions apply and when. Consider a closed physical system – be it isolated in a laboratory or “the whole Universe” – undergoing Hamiltonian evolution. Classically, thermal entropy and its increase are generally treated through coarse-graining of phase space. A system can have time-evolving coarsegrained quantities, such as order parameters, energy, and currents – as is the case in fluid dynamics or phase transitions [2, 3]. An entropy measure can then naturally and generically rise if it attributes higher entropy to coarsegrained states of greater phase-space volume; this works even if the Gibbs entropy is conserved, or zero. In the quantum case, the standard von Neumann entropy would be constant (and zero for a pure state), in close correspondence to the classical Gibbs entropy. Yet despite its utility, the corresponding procedure of coarsegraining has been hard to formulate in quantum mechanics due to the lack of commutation of conjugate degrees of freedom. This problem is particularly severe when discussing coarse-grained entropy, where phase space volume is a crucial concept. Instead, quantum mechanical definitions of equilibrium entropy have been developed, such as diagonal entropy [4–7], entropy of an observable [8–10], or the entanglement entropy [11–14], that can give rise to the thermodynamic entropy even in pure states [7, 15–17]. However, their relation to the coarse-graining used in classical systems is obscure or lacking, and they can behave oddly in certain cases – for example entanglement entropy is zero if computed for a partition of a system that separates the state into a product state, while being nonzero for many equally-reasonable partitions. In this letter, we argue that we can, in fact, define a coarse-grained quantum mechanics in a satisfactory and surprisingly simple way, by focusing explicitly on the outcomes probabilities for measurements that may be coarse-grained and may be performed in a noncommuting sequence. For this reason we call our formu-
lation Observational entropy. We identify two special cases of this idea that are particularly interesting. The first such entropy can be understood as uncertainty in outcomes of two consecutive measurements, first in coarse-grained position and then in energy. The second entropy describes a set of local quantities trying to equilibrate with each other. Both of these entropies converge to thermodynamic entropy and extend well to non-equilibrium situations, making them suitable candidates for description of isolated quantum systems. More generally, this definition of entropy, like classical Boltzmann entropy, can describe quantum systems becoming disordered within a chosen coarsegrained description – the quantum mechanical equivalent of “Spilling coffee on the table.” Observational entropy elucidates the dynamics of a variety of quantum thermodynamic systems and may shed light on thorny questions such as the entropy of black holes and horizons in general, or the arrow of time in the Universe as a whole. Experimentally, it could have applications in cold atoms, where experiments on isolated quantum systems are now becoming feasible [18, 19]. We start by considering making a single observation on a quantum system characterized by a density matrix ρˆ. In analogy to classical physics, we define measurements of the system that partition it into coarse-grained macrostates. We do this through a set of trace-preserving projectors {Pˆi }i , indexed by i, acting on a Hilbert space H. For example, given a system with N indistinguishable particles, we can coarse grain them into p bins, each of width δ. We wish to make observations that will give us the bin that every particle is in. To do this, we denote ⃗ = (x(1) , . . . , x(N ) ), where each the particle positions by x element can take one of the equidistant values x1 , . . . , xp . ⃗, we define a set of coarse-grained projectors With i → x (δ) (δ) CX = {Pˆx⃗ }x⃗ , where Pˆx⃗ = ∑ ∣x ˜⃗⟩⟨x ˜⃗∣
(1)
⃗∈Cx⃗ x ˜
and Cx⃗ represents a hypercube of dimension N and width δ = xj+1 − xj , that represents the possible particle posi-
2 tions in a single macrostate. Our coarse-graining CX then represents measurements that can be done that will characterize the system positional macrostate at a scale of δ. Performing the above coarse-grained measurement does not give the precise position of the particles. After the measurement, if the particles were confined to a lattice, a further measurement could be done that would ⃗ precisely. In more genergive the positional basis states x ality, after performing a coarse-grained measurement defined by a set of projectors {Pˆi }, the number of possible outcomes of a second measurement that would determine the basis state of the system is tr[Pˆi ] and so with no more information, we would then assign equal weights to these different outcomes. The probability of finding the system in a particular subspace Hi of the total Hilbert space is equal to pi = tr[Pˆi ρˆ]. Therefore the probability of finding the system in any of the basis states is pi /tr[Pˆi ]. This allows us to define the Observational entropy for coarse-graining C as the Shannon entropy of these probabilities, pi . (2) SO(C) (ˆ ρ) ≡ − ∑ pi ln tr[ Pˆi ] i The idea of coarse-grained projections is mentioned very early on by von Neumann [20] with an expression similar to this for the particular case of coarse-grained energies, that he attributes to Eugene Wigner. It is mentioned later by Werhl in generality [21], in connection with developing a quantum mechanical master equation. By itself, it does not partition phase space sufficiently to define the equivalent of a coarse-grained classical entropy capable of sensible time dependent evolution. However it has a number of interesting properties that we studied and are briefly discussed below. It is straightforward to show that a state that is completely within one of the subspaces Hi has SO(C) (ˆ ρ) = ln dim Hi , which aligns with the coarse-graining interpretation that we gave: because the basis state is not measured to more precision than the one given by coarsegraining C, this entropy represents the inability of such measurements to acquire more accurate information even if the state of ρˆ is known to more precision. The degree to which coarse-grained measurements specify a system can be made more precise by considering two coarse-grainings C1 and C2 , and saying that C2 is “finer than” C1 , denoted by writing C1 ↪ C2 , if projectors in C1 can always be written as the sum of projectors in C2 . In this case, it can be proven that SO(C1 ) (ˆ ρ) ≥ SO(C2 ) (ˆ ρ). This intuitively means that entropies will be larger, the more that the system is coarse-grained. We now summarize some interesting properties of the Observational entropy. The detailed definitions and proofs will be published elsewhere [22]. • There are general bounds that can be proven for it:
where SVN is the von Neumann entropy. • The Observational entropy is extensive. Consider a composite of m sub-systems characterized together by a separable state ρˆ = ρˆ(1) ⊗ ⋯ ⊗ ρˆ(m) . If we impose a coarse-graining C = C (1) ⊗ ⋅ ⋅ ⋅ ⊗ C (m) = {Pˆi1 ⊗ ⋅ ⋅ ⋅ ⊗ Pˆim }i1 ,...,im , which coarse grains the different subsystems separately, then m
SO(C) (ˆ ρ) = ∑ SO(C (k) ) (ˆ ρ(k) ) .
• If the coarse-graining C is composed of projectors that commute with the Hamiltonian, the Observational entropy SO(C) (ˆ ρt ) (of the time-evolving density matrix ρˆt ), does not vary in time. • Otherwise, for a large class of nonequilibrium initial states, the Observational entropy increases. A provable result is that starting with an initial state that is contained in one of the subspaces Hi the Observational entropy increases or remains the same, at least for a short time. This definition of entropy can partition Hilbert space using a single set of coarse-grainings C that correspond to a set of measurements, for example of position, that one can perform on the system. But to get a useful generalization of coarse-grained classical entropies, we should consider a second set of measurements corresponding, for example, to coarse-graining in energy. Indeed our classical notion of a coarse-grained phase space requires consideration of two types of measurements, for example position and momentum, that in the quantum mechanical case do not commute. Therefore we need to generalize the above definition of entropy to allow for the series of measurements of non-commuting projectors. We will find that this leads to a surprisingly simple prescription for coarsed-grained but fully quantum mechanical entropy. For simplicity, consider two different coarse-grainings C1 = {Pˆi1 }i1 and C2 = {Pˆi2 }i2 that may not commute. First a measurement of projectors in C1 is performed, then measurements in projectors in C2 . The probability of such an outcome is pi1 i2 = tr[Pˆi2 Pˆi1 ρˆPˆi1 Pˆi2 ] = tr[Pˆi2 Pˆi1 ρˆPˆi1 ]. The denominator in our definition of SO(C) was tr[Pˆi ] and now will be generalized to Vi1 ,i2 ≡ tr[Pˆi2 Pˆi1 ]. This has an interpretation as a volume in Hilbert space because it is always positive and ∑i1 ,i2 Vi1 ,i2 = dim H. This can be generalized further [23] to give Definition 1. Let (C1 , . . . , Cn ) be an ordered set of coarse-grainings. We define the Observational entropy with coarse-grainings (C1 , . . . , Cn ) as SO(C1 ,...,Cn ) (ˆ ρ) ≡ − ∑ pi1 ,...,in ln i1 ,...,in
SVN (ˆ ρ) ≤ SO(C) (ˆ ρ) ≤ ln dimH,
(3)
(4)
k=1
pi1 ,...,in , ˆ tr[Pin ⋯Pˆi1 ⋯Pˆin ] (5)
3 where the sum goes over elements such that pi1 ,...,in ≡ tr[Pˆin ⋯Pˆi1 ρˆPˆi1 ⋯Pˆin ] ≠ 0.
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We can generalize the notion of finer coarse-grainings to multiple coarse-grainings and prove the following theorem.
SVN (ˆ ρ) ≤ SO(C1 ,...,Cn ) (ˆ ρ) ≤ ln dimH.
(6)
SVN (ˆ ρ) = SO(C1 ,...,Cn ) (ˆ ρ) if and only if Cρˆ ↪ (C1 , . . . , Cn ). SO (ˆ ρ) = ln dimH if and only if ∀i1 , . . . , in , pi1 ,...,in = tr[Pˆin ⋯Pˆi1 ⋯Pˆin ] . dimH
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Theorem 1. For any ordered set of coarse-grainings (C1 , . . . , Cn ) and any density matrix ρˆ,
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Cρˆ consists of projectors from the spectral decomposition of ρˆ. Cρˆ ↪ (C1 , . . . , Cn ) means that the projectors in Cρˆ can be always be written as the sum of Pˆi1 ⋯Pˆin . With general Definition 1 in mind, it is possible to consider many possible kinds of Observational entropies. It is not obvious that any of these have any relation to thermodynamic entropy, but we now describe two versions that do bear a close connection. In Eq. (1) we introduced coarse-graining in position space with p number of bins. Consider these and “finegrained” energy projectors CE = {PˆE }E ,
PˆE = ∣E⟩⟨E∣.
(7)
We construct entropy SxE ≡ SO(CX ,CE ) ,
(8)
which means that we first measure the coarse-grained position of the system, and then its energy. The second quantity employs a different coarsegraining and is similar in spirit but mathematically distinct. We start by considering Hilbert space divided into two parts H(1) and H(2) , the joint system being ˆ can then be sepaH = H(1) ⊗ H(2) . The Hamiltonian H rated into three terms ˆ =H ˆ (1) ⊗ Iˆ + Iˆ ⊗ H ˆ (2) + H ˆ (int) , H
(9)
ˆ (1) and H ˆ (2) are the Hamiltonians that describe where H internal interactions in the first and second systems reˆ (int) is an interaction term. For large spectively, and H subsystems and local interactions, the magnitude of this term is expected to small and hence we have introduced a parameter to indicate this. Consider a coarse-graining that projects in the eigenstates of the local Hamiltonians ˆ (1) and H ˆ (2) . We call this the factorized Observational H entropy (FOE). It can be more formally written as SF ≡ SO(C
ˆ (1) ⊗CH ˆ (2) H
ρ). ) (ˆ
(10)
In other words, we construct basis states for H that are the (tensor) products of eigenstates of the local Hamilˆ (1) and H ˆ (2) , and form projectors from these tonians H
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FIG. 1. Time evolution of SxE (line) and the factorized Observational entropy SF (dotted line) starting in a pure thermal state. After t = 30, the right wall is expanded to double the system size and the system continues to evolve. The dotted straight lines represent thermodynamic entropies of the canonical ensemble.
states. This can be easily generalized to an arbitrary number m of local Hamiltonians, rather than two. We will see below that the FOE and SxE have similar behavior but are quite distinct mathematically and differ in their physical interpretation. The FOE is more theoretically tractable and SxE has an elegant interpretation in terms of measurement and may prove to be quite accessible experimentally. Let us start with a numerical analysis of these quantities. We consider a one dimensional lattice model of spinless fermions, with both nearest-neighbor (NN) and nextnearest-neighbor (NNN) hopping and interactions [24]. We also choose NN interactions of V and NNN of V ′ . ̵ = V = t = 1. For generic systems we We always take h choose the well-studied case U ′ = t′ = 0.96, and also try U ′ = t′ = 0 to investigate the integrable case [16, 17, 24]. We employ hard wall boundary conditions for our numerical experiments so that we can study the expansion of a gas from a smaller to a larger box. First we investigate the dynamics of the two Observational entropies SxE and SF , both of which are coarsegrained into four subsystems (p = m = 4) of length 4 in the full system of length L = 16. The graph of the evolution is shown in Fig. 1. We start with a system with N = 4 particles confined to a box of size L = 8. The system starts in what can be described as a “pure thermal state.” It is a superposition of all energy eigenstates, each eigenstate having a random complex amplitude drawn from a distribution with a variance given by the Gibbs distribution at inverse temperature β = 1. For t < 30 the system is in equilibrium. At t = 30, we suddenly enlarge the box to size L = 16 and compute the
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microcanonical eigenstate superpose DOS
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microcanonical eigenstate superpose DOS
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FIG. 2. The top three curves show observational entropy SxE for a microcanonical state, a random superposition of neighboring energy eigenstates, and energy eigenstates, from top to bottom. The lowest curve is the logarithm of the density of states, as described in the text.
FIG. 3. The top three curves show SF for a microcanonical state, a random superposition of neighboring energy eigenstates, and energy eigenstates, from top to bottom. The lowest curve is the same as in Fig. 2 describing the entropy computed from the density of states.
continued evolution. Both entropies increase rapidly but smoothly, until they reach equilibrium. The dashed lines represent the values obtained for the entropy using the canonical distribution. Because of finite-size effects, this differs from the computed values of the SxE and SF by approximately 10%. We also performed similar analyses for systems with other initial conditions such as starting in eigenstates of the smaller box, and also analyzed the integrable case [22]. As expected, the integrable case shows substantially larger fluctuations. To investigate behavior of these two entropies in more detail, we also plot SxE and SF as functions of energy for various equilibrium states as shown in Figs 2 and 3 respectively; this is particularly relevant for studying the long-time limit. Both entropies are coarse-grained into 4 subsystems (p = m = 4) of the full generic (nonintegrable) system of size L = 20, and computed for energy eigenstates, random superposed pure states, and microcanonical mixed states. The random superposed pure states were obtained by superposing k = 30 neighboring energy eigenstates with complex amplitudes drawn uniformly from the unit disk, then normalizing. The microcanonical states were obtained by adding together the density matrices of k = 30 neighboring energy eigenstates with equal weights. Because of significant finite size effects, we eschew using the canonical ensemble for comparison, and instead focus on the microcanoncial ensemble given by the density of states ρ(E); we plot SDOS ≡ ln(ρ(E)∆E).[25] SDOS gives an entropy that, up to an unimportant additive constant, is equivalent to the thermodynamic entropy given by the canonical ensemble [26]. The results for the two quantities are quite similar;
the same is true for the time-dependent analysis shown in Fig. 1. This is not a coincidence. It can be shown that in the → 0 limit in Eq. (9), SxE and FOE are the same, and there are strong arguments that the quantities are very closely tied for finite (see [22]). As shown on Fig. 1, both SxE and FOE approximate the thermodynamic entropy in the long-time limit. Figures 2 and 3 provide even more compelling evidence for this convergence, as follows. It is possible to prove that up to order , FOE of a canonical state is equivalent to the canonical entropy [22]. Because of significant finite size effects, we focused on the microcanoncial ensemble instead, which is equivalent to the canonical in the thermodynamic limit [26]. Curves for both SxE and FOE approximate the microcanonical entropy computed from the density of states – in fact, they are almost parallel to each other. The differences of order O(1) are unimportant in the thermodynamic limit. The superposed states have random phases, meaning that they describe the state of a typical wave-function at some time far in the future, which provides additional support to the claim that in the long time limit, and for generic systems, these two Observational entropies converge to the thermodynamic entropy. Convergence of SxE and FOE to the thermodynamic entropy can be also shown analytically [22] for generic (i.e., non-integrable) systems of large size, by using connections between non-integrable systems and random matrix theory. These results show that both Observational entropies, in the form of SxE and FOE, extend the idea of classical Boltzmann entropy to quantum mechanical systems. It is worthwhile briefly comparing the above approach
5 with a well-known entropy used for closed quantum systems. The entanglement entropy of a bipartite division of a system into subsystems A and B is also closely related to the thermodynamic entropy in equilibrium [15– 17]. But it is a distinct quantity that is fundamentally different from SxE or FOE. For example, if the state is a product state of A and B, then the entanglement entropy is zero, but SxE is not. We expect that the thermodynamic entropy of the complete system should still be large, and thus for a non-equilibrium situation, the entanglement entropy cannot give us a sensible measure, at least in this case, for the thermodynamic entropy. On the other hand SxE is largely unaffected by this lack of entanglement for short ranged systems. It is surprising that we can adequately describe the thermodynamic entropy of ρˆ through the application of the non-commuting measurements of position and energy (SxE ), or by coarse-graining in local energy states (FOE). It is also intriguing that these quantities are quantitatively so similar both at equilibrium and in their detailed time-dependence. This kind of entropy may play a useful role in experiments, for example on cold atoms, in which these kinds of measurements and coarse-grainings are possible. Consider a situation where many copies of a system are prepared and then evolve for a time t. It is hard to measure the entanglement entropy between two subsystems at time t directly [27], and to compute it one needs to know the full density matrix for at least one of the subsystems, which can only be accomplished through a very large set of measurements. On the other hand, to obtain SxE , we first determine the coarse-grained position of particles. For indistinguishable particles, this is equivalent to measuring the coarse-grained density, which is frequently performed in cold atom experiments [28]. After measuring this density, the state energy is observed. If the resolution of the apparatus is not fine-grained enough to get individual eigenstates, an Observational entropy with finite energy coarse-graining can still be calculated theoretically, and compared with experimental data. We have argued through both analytical and numerical work that it is indeed possible to extend coarse grained entropy to quantum mechanics, and shown that for a variety of initial states and for non-integrable systems, this entropy generically rises, approaching the correct thermodynamic value. It is easily understood in terms of performing subsequent measurements, has the mathematical properties expected of entropy, and has close ties to experimental techniques. Thus Observational entropy is a very promising candidate for understanding the nonequilibrium evolution of entropy, and the second law of thermodynamics, in closed quantum systems. This research was supported by the Foundational
Questions Institute (FQXi.org), of which AA is Associate Director, and by the Faggin Presidential Chair Fund.
∗
[email protected] [1] R. Balian, in Poincar´e Seminar 2003 (Springer, 2004) pp. 119–144. [2] S. K. Ma, Statistical Mechanics (World Scientific, 1985). [3] J. Sethna, Statistical mechanics: entropy, order parameters, and complexity, Vol. 14 (Oxford University Press, 2006). [4] R. C. Tolman, The principles of statistical mechanics (Courier Corporation, 1938). [5] D. Ter Haar and W. Band, Am. J. Phys. 22, 641 (1954). [6] E. T. Jaynes, Phys. Rev. 108, 171 (1957). [7] A. Polkovnikov, Ann. Phys. 326, 486 (2011). [8] R. S. Ingarden and K. Urbanik, Acta Physica Polonica 21, 281 (1962). [9] M. Grabowski and P. Staszewski, Rep. Math. Phys. 11, 233 (1977). [10] F. Anz` a and V. Vedral, Sci. Rep. 7 (2017). [11] L. Bombelli, R. K. Koul, J. Lee, and R. D. Sorkin, Phys. Rev. D 34, 373 (1986). [12] M. Srednicki, Phys. Rev. Lett. 71, 666 (1993). [13] C. H. Bennett, H. J. Bernstein, S. Popescu, and B. Schumacher, Phys. Rev. A 53, 2046 (1996). [14] C. H. Bennet, Phys. Scrip 76, 210 (1998). [15] J. M. Deutsch, New J. Phys. 12, 075021 (2010). [16] J. M. Deutsch, H. Li, and A. Sharma, Phys. Rev. E 87, 042135 (2013). [17] L. F. Santos, A. Polkovnikov, and M. Rigol, Phys. Rev. E 86, 010102 (2012). [18] A. M. Kaufman, M. E. Tai, A. Lukin, M. Rispoli, R. Schittko, P. M. Preiss, and M. Greiner, Science 353, 794 (2016). [19] S. Trotzky, Y.-A. Chen, A. Flesch, I. P. McCulloch, U. Schollw¨ ock, J. Eisert, and I. Bloch, Nature Phys. 8, 325 (2012). [20] J. von Neumann, Eur. Phys. J. H 35, 201 (2010). [21] A. Wehrl, Rev. Mod. Phys. 50, 221 (1978). ˇ anek, J. M. Deutsch, and A. Aguirre, To be pub[22] D. Safr´ lished (2017). [23] This is not a unique generalization, but is selected for its desirable properties; see [22]. [24] L. F. Santos and M. Rigol, Phys. Rev. E 81, 036206 (2010). [25] To give it the right units, √ we have multiplied ρ by a typical energy ∆E = σ(E)/ N , where σ computes the standard deviation and N is the number of particles. This choice of ∆E is rather arbitrary as it is unimportant in the thermodynamic limit. [26] D. Ruelle, Statistical Mechanics: Rigorous Results (World Scientific, 1999). [27] M. Rispoli, P. Preiss, E. Tai, A. Lukin, R. Schittko, A. Kaufman, R. Ma, R. Islam, and M. Greiner, in APS Division of Atomic, Molecular and Optical Physics Meeting Abstracts (2016). [28] K. Levin, A. Fetter, and D. Stamper-Kurn, Ultracold Bosonic and Fermionic Gases, Vol. 5 (Elsevier, 2012).
