Jan 9, 2014 - testing, and classical data compression with quantum side information. .... Here (2) comes from the data processing inequality for collision ...
Quantum Achievability Proof via Collision Relative Entropy Salman Beigi1 , Amin Gohari1,2 1
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran 2
Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran
arXiv:1312.3822v2 [quant-ph] 9 Jan 2014
Abstract In this paper, we provide a simple framework for deriving one-shot achievable bounds for some problems in quantum information theory. Our framework is based on the joint convexity of the exponential of the collision relative entropy, and is a (partial) quantum generalization of the technique of Yassaee et al. (2013) from classical information theory. Based on this framework, we derive one-shot achievable bounds for the problems of communication over classical-quantum channels, quantum hypothesis testing, and classical data compression with quantum side information. We argue that our one-shot achievable bounds are strong enough to give the asymptotic achievable rates of these problems even up to the second order.
1
Introduction
Yassaee et al. [1] have recently proposed a general technique for proving upper bounds on the probability of error for classical network information theory problems in the one-shot case. By one-shot we mean a setup whose goal is to find a strategy for (say) transmitting one out of M messages in a single use of the network with (average or maximal) probability of error being as small as possible. This is unlike the traditional setup where the focus is on asymptotic rates and vanishing error probability. Some salient features of the method of [1] are as follows: (1) it provides one-shot results whose form resembles (in a systematic way) that of asymptotic results; (2) it fundamentally differs from the traditional methods where first various error events are identified, and then union bound (and packing or covering lemmas on individual error events) is used. Instead, the technique uses Jensen’s inequality only and is able to consider the effect of all the error events at once (hence yielding stronger results). Further, since one-shot versions of mutual packing and covering lemma are not known to exist, this technique outperforms the traditional ones; (3) the technique is able to recover the second-order asymptotics (dispersion) of the point-to-point channel capacity and provides new finite blocklength achievability results for many other network problems. The technique of [1] can roughly be explained as follows. For the problem of point-to-point channel capacity, for example, this technique (as usual) uses a random coding to encode the message. However, to decode the message at the receiver’s side, it does not use the maximum likelihood decision rule. Instead, the technique uses a decoder called stochastic likelihood coder (SLC) in [1]. Putting in the terminology of quantum information theory, this decoder is nothing but the transpose channel. Then to analyze this scenario, expectation value of the probability of successful decoding is bounded from below using Jensen’s inequality. This simple technique is general enough to be applied to several problems in classical network information theory, and interestingly gives a tight asymptotic bound up to the second order for the pointto-point channel capacity problem. Our contributions: Motivated by the appealing features of the technique of [1], we are interested to see if it extends to quantum information theory. In this paper we provide a partial generalization that in particular can be applied to the problems of communication over classical-quantum (c-q) channels, quantum hypothesis testing, and classical data compression with quantum side information. To analyze these problems, we observe that the probability of successful decoding can be written in terms of collision relative entropy [2]. Then to find a lower bound on the expectation value of the probability of successful decoding (over the random choice of codewords) we use the joint convexity of the exponential of collision relative entropy and apply Jenssen’s inequality. This, for instance, gives a lower bound on the one-shot capacity of c-q channels in terms of collision relative entropy. To compute the asymptotics of this bound we prove a lower bound on the collision relative entropy in terms of another quantity called information spectrum relative entropy whose asymptotics, up to the second order, has been computed by Tomamichel and Hayashi [3]. This gives a simple proof for the achievability parts of the second order asymptotics of the problems of the capacity of c-q channels recently computed by Tomamichel and Tan [4], quantum hypothesis testing previously computed
1
by Tomamichel and Hayashi [3] and Li [5], and classical data compression with quantum side information previously derived by Tomamichel and Hayashi [3]. Related works: A one-shot achievable bound (and a converse) for c-q channel coding was also derived by Wang and Renner [6] (see also [7]). This one-shot achievability result has very recently been used by Tomamichel and Tan [4] to compute the dispersion of c-q channels. This one-shot achievable bound is proved using the Hayashi-Nagaoka operator inequality [7, Lemma 2]. To prove our results however, we do not use this inequality. Quantum hypothesis testing and data compression with quantum side information in the one-shot case have also been studied in [8] and [9] respectively.
1.1
Notation
Throughout this paper quantum and classical systems are denoted by uppercase letters B, X etc. We save X to denote classical systems whose alphabet set is denoted by calligraphic letter X . So a density matrix P ρX over X is determined by its diagonal elements that are indexed by elements x ∈ X , i.e., ρX = x∈X pX (x)|xihx|, where pX is a distribution over X . In this paper we consider samples from this distribution whose outcomes are elements of X . However, to distinguish elements x ∈ X with samples drawn from pX we denote the latter by mathtt lowercase x. As a result letting ρXB to be a density matrix over the joint system XB, X ρXB = pX (x)|xihx| ⊗ ρx , x∈X
it is important to differentiate between ρx and ρx ; while the former is the state of B when the subsystem X is in the state x, the latter is a random density matrix, taking the value of ρx when x = x. Needless to say, ρX is the marginal density matrix over X. For a positive semi-definite U we use U −1 for the inverse of U restricted to supp(U ) which is the span of eigenvectors of U with non-zero eigenvalues. Thus U U −1 = U −1 U is equal to the projection on supp(U ) and not necessarily I, the identity operator. All logarithms and exponential functions in the paper are in base 2.
2
Collision relative entropy and information spectrum
For positive semi-definite (not necessarily normalized) ρ and σ, the collision relative entropy or collision divergence is defined by 2 − 41 − 14 , (1) D2 (ρkσ) = log tr σ ρσ if supp(ρ) ⊆ supp(σ), and by D2 (ρkσ) = ∞ otherwise. This quantity (as a conditional entropy) was first introduced in [2, Definition 5.3.1] and has found applications in quantum information theory [10, 11, 12]. Collision divergence is a member of the family of quantum (sandwiched) R´enyi divergences defined in [13, 14] (and before that in talks [15] and [16]), and further studied in [17, 18]. Some of the known properties of D2 (·k·) are as follows: Theorem 1. [13, 14] Collision relative entropy satisfies the following properties: • (Positivity) For (normalized) density matrices ρ, σ we have D2 (ρkσ) ≥ 0. • (Data processing) For any quantum channel N we have D2 (ρkσ) ≥ D2 (N (ρ)kN (σ)). • (Joint convexity) exp D2 (·k·) is jointly convex, i.e., for posisive semi-definite matrices ρx , σx and probability distribution p(x) we have X p(x) exp D2 (ρx kσx ) ≥ exp D2 (ρkσ), x
where ρ =
P
x p(x)ρx
and σ =
P
x
p(x)σx .
• (Monotonicity in σ) If σ ′ ≥ σ then D2 (ρkσ ′ ) ≤ D2 (ρkσ).
2
The other important quantity that we use in this paper, is the information spectrum relative entropy introduced by Tomamichel and Hayashi [3]. We need the following notation to define this relative entropy. For a hermitian matrix U let ΠU≥0 be the orthogonal projection onto the union of eigenspaces of U with non-negative eigenvalues. Also for two hermitian operators U, V let ΠU≥V = ΠU−V ≥0 . Now for two density matrices σ, ρ and ǫ > 0, define the information spectrum relative entropy by Dsǫ (ρkσ) := sup{R| tr ρΠρ≤2R σ ≤ ǫ}.
Information spectrum relative entropy is used in [3] to compute the second order asymptotics of some information processing tasks including quantum hypothesis testing and source coding with quantum side information. The reason for introducing Dsǫ (·k·) is that its second order asymptotics can be computed in terms of relative entropy and relative entropy variance. To express this result we first need the following definition: Definition 2. [3, 5] The information variance or the relative entropy variance V (ρkσ) is defined by V (ρkσ) := tr ρ(log ρ − log σ − D(ρkσ))2 . Here D(ρkσ) is the relative entropy D(σkρ) := tr(σ(log σ − log ρ)).
√ Theorem 3. [3] For every two density matrices ρ, σ, fixed 0 < ǫ < 1 and δ proportional to 1/ n we have p Dsǫ±δ (ρ⊗n kσ ⊗n ) = nD(ρkσ) + nV (ρkσ)Φ−1 (ǫ) + O(1), Ru 1 2 where Φ is the cumulative distribution of the standard normal distribution Φ(u) := −∞ √12π e− 2 t dt, and Φ−1 (ǫ) = sup{u| Φ(u) ≤ ǫ}. In this paper we derive our achievability bounds in terms of collision relative entropy. In the following theorem we prove a lower bound on the collision relative entropy in terms of information spectrum relative entropy. We will use this lower bound to compute the asymptotics of our achievable bounds. Theorem 4. For every 0 < ǫ, λ < 1 and density matrices ρ, σ we have −1 exp D2 (ρkλρ + (1 − λ)σ) ≥ (1 − ǫ) λ + (1 − λ) exp − Dsǫ (ρkσ) .
Proof. Let Π = Πρ≤2R σ and Π′ = I − Π = Πρ>2R σ where R is a real number to be determined. Define the pinching map N (ρ) := ΠρΠ + Π′ ρΠ′ . Then the following chain of inequalities hold. exp D2 (ρkλρ + (1 − λ)σ) ≥ exp D2 (N (ρ)kλN (ρ) + (1 − λ)N (σ)) ≥ exp D2 (Π′ ρΠ′ kλΠ′ ρΠ′ + (1 − λ)Π′ σΠ′ )
≥ exp D2 (Π′ ρΠ′ kλΠ′ ρΠ′ + (1 − λ)2−R Π′ ρΠ′ ) −1 = λ + (1 − λ)2−R tr(Π′ ρ).
(2) (3) (4)
Here (2) comes from the data processing inequality for collision relative entropy. For (3) observe that since Π and Π′ are orthogonal to each other, exp D2 (N (ρ)kλN (ρ) + (1 − λ)N (σ)) is a summation of two nonnegative terms. For (4) we use the last property of collision relative entropy stated in Theorem 1 and that by the definition of Π′ we have Π′ (ρ − 2R σ)Π′ ≥ 0. Now the result follows by letting R = Dsǫ (ρkσ) and using the fact that by the definition of information spectrum relative entropy we have tr(Π′ ρ) ≥ 1 − ǫ.
3
A one-shot achievable bound for c-q channels
Consider a c-q channel with input X and output B which maps x ∈ X to ρx . Take an arbitrary distribution pX (x) on the input. This distribution induces a density matrix over the joint system XB: X ρXB = pX (x)|xihx| ⊗ ρx . (5) x∈X
As usual, to communicate a classical message m ∈ {1, . . . , M } over the channel, we encode the message m with xm . In other words, we generate the random codebook C = {x1 , · · · , xM } where the elements xm are drawn independently from pX (x). So if the message to be communicated is m, the output of the channel is ρxm . As mentioned above, Yassaee et al. [1] use SLC to decode the message. Putting in the terminology of quantum information theory and applying it to c-q channels, this decoder is nothing but the pretty good 3
measurement. So we assume that to decode the message the receiver applies the pretty good measurement corresponding to signal states {ρx1 , . . . , ρxM } with POVM elements X − 21 X − 21 ρx i ρx m . (6) ρx i Em = i
i
There is nothing new in quantum information theory up to this point. The error analysis of this encoder/decoder however contains the main idea of this work, and as in [1] is based on Jensen’s inequality. In the following we find that the probability of successful decoding can be written in term of collision relative entropy and then we use joint convexity.
Theorem 5. The expected value of the probability of successful decoding of the pretty good measurement (6) corresponding to a randomly generated codebook according to distribution pX is bounded by
1 1 −1
ρX ⊗ ρB , ρXB + 1 − EC Pr[succ] ≥ M exp D2 ρXB M M
where ρXB is defined in (5).
For a classical channel, the above bound reduces to Theorem 1 of [1], although it is not expressed in terms of collision relative entropy in [1]. 1 1 Remark 6. One might guess that replacing M ρXB + 1 − M ρX ⊗ ρB by ρX ⊗ ρB may also result in a valid lower bound. This is incorrect since this would imply that the rate D2 (ρXB kρX ⊗ρB ) > D1 (ρXB kρX ⊗ρB ) = I(X; B)ρ is asymptotically achievable, which we know is not the case. Proof. Let us define σUXB =
M 1 X |mihm| ⊗ |xm ihxm | ⊗ ρxm , M m=1
and let σUX and σB its corresponding marginal density matrices. Note that σUXB is a random density matrix. A direct computation verifies that the probability of successful decoding is equal to M 1 X Pr[succ] = tr (Em ρxm ) M m=1 ! M X −1/4 2 1 X X −1/4 tr ρx i ρx m ρx i = M m=1 i i
=
1 exp D2 (σUXB kσUX ⊗ σB ). M
Here the last step is our key observation; The success probability of pretty good measurement (transpose channel in general) can be written in terms of collision relative entropy. Now using the data processing inequality and joint convexity properties of Theorem 1, the expected value of the probability of correct decoding with respect to randomly chosen codewords, is lower bounded by 1 EC exp D2 (σUXB kσUX ⊗ σB ) M 1 EC exp D2 (σXB kσX ⊗ σB ) ≥ M 1 exp D2 EC σXB kEC σX ⊗ σB . ≥ M Computing EC σXB and EC σX ⊗ σB gives the desired result: 1 X EC σXB = EC |xm ihxm | ⊗ ρxm = EC |x1 ihx1 | ⊗ ρx1 = ρXB . M m 1 X EC σX ⊗ σB = EC 2 |xm ihxm | ⊗ ρxm′ M ′ EC Pr[succ] =
m,m
1 X 1 X = EC 2 |xm ihxm | ⊗ ρxm′ |xm ihxm | ⊗ ρxm + EC 2 M m M ′ m6=m
1 1 ρX ⊗ ρB . ρXB + 1 − = M M 4
Combining the above theorem with Theorem 4 we obtain the following one-shot lower bound: Corollary 7. For every distribution pX and ǫ > δ > 0, it is possible to transmit one out of M messages using a single use of the c-q channel, with the average probability of error being at most ǫ, provided that ǫ−δ δ exp Ds (ρXB kρX ⊗ ρB ) + 1 . M= 1−ǫ
3.1
Second order asymptotics of c-q channels
For a channel x 7→ ρx let M ∗ (n, ǫ) be the maximum size of a codebook with codewords of length n whose average probability of successful decoding (under the optimial decoding algorithm) is at least 1 − ǫ. Then by the HSW theorem log M ∗ (n, ǫ) (for small ǫ > 0) is roughly equal to nC where C is the Holevo information of the channel given by C = max I(X; B)ρ . pX
(7)
Here I(X; B)ρ denotes the mutual information corresponding to the joint state (5). Our goal in the second ∗ order analysis is to find a more accurate estimate of √ log M (n, ǫ). Based on the∗ method of types, √ it is already √ ∗ shown by Winter [19] that log M (n, ǫ) = nC+O( n). So we may write log M (n, ǫ) = nC+ nf (ǫ)+o( n). By computing the second order asymptotics we mean finding f (ǫ). The second order asymptotics of c-c channels was first computed by Strassen [20]. Under a mild condition on the channel, he showed that for every 0 < ǫ < 1/2, √ (8) log M ∗ (n, ǫ) = nC + nV Φ−1 (ǫ) + o(log n). Here V is a parameter of the channel (independent of ǫ) which following [21] we call channel dispersion. Channel dispersion has recently been studied further by Polyanskiy et al. [21] and Hayashi [22]. For c-q channels, the second order asymptotics has been computed very recently by Tomamichel and Tan [4]. Here we re-derive the achievability part of their result. Theorem 8. For every c-q channel x 7→ ρx and 0 < ǫ < 1/2 we have √ log M ∗ (n, ǫ) ≥ nC + nV Φ−1 (ǫ) + O(log n), where C is the channel capacity given by (7), and V is the channel dispersion given by V = min V (ρXB kρX ⊗ ρB ), pX ∈P
where ρXB is defined in (5), P is the set of capacity achieving input distributions, i.e. the set of distributions pX that achieve the optimal value in (7), and V (·k·) is given in Definition 2. Proof. By Corollary 7, for every distribution pX and ǫ > δ > 0 we have ǫ−δ ⊗n ⊗n M ∗ (n, ǫ) ≥ Dsδ (ρ⊗n kρ ⊗ ρ ) + log . XB X B 1−ǫ √ Letting δ = ǫ − 1/ n, using Theorem 3, and optimizing over pX gives the desired result.
4
Quantum hypothesis testing
Suppose that a physical system is randomly prepared in one of the two states ρ, σ, which are called the hypotheses. To distinguish which hypothesis is the case we apply a POVM measurement {Fρ , Fσ } on the system. Such a measurement may cause an error in detecting the right hypothesis, and the goal of the hypothesis testing problem is to find the smallest possible probability of such an error. Indeed there are two types of error: Type I error is defined to be the probability of mis-detecting the hypothesis when the system is prepared in state ρ, and Type II error is defined similarly when the system is prepared in state σ. Then we have Pr[type I error] = tr(ρFσ ) and Pr[type II error] = tr(σFρ ). In the asymmetric hypothesis testing problem we assume that the cost associated to type II error is much higher than the cost corresponding to type I error. So the probability of type II error should be minimized, while we only put a bound the probability of type I error. Quantum Stein’s lemma [23, 24] 5
(n)
(n)
states that for every ǫ > 0, there is a POVM {Fρ , Fσ } to distinguish the hypotheses ρ⊗n and σ ⊗n such that Pr[type I error] ≤ ǫ and Pr[type II error] ≤ exp − nD(ρkσ) + o(n) . Moreover, D(ρkσ) is the optimal such error exponent. Also the one-shot hypothesis testing problem has been studied in [8]. Here we first prove a one-shot bound for the quantum hypothesis problem and then compute the asymptotics of our bound.
Theorem 9. For every ǫ > 0 there is a POVM measurement {Fρ , Fσ } for the one-shot hypothesis testing problem such that Pr[type I error] ≤ 2ǫ and Pr[type II error] ≤ exp − Dsǫ (ρkσ) − log ǫ .
Proof. Consider the POVM with elements
Fρ = (ρ + M σ)−1/2 ρ(ρ + M σ)−1/2
Fσ = (M −1 ρ + σ)−1/2 σ(M −1 ρ + σ)−1/2 ,
and
where M is a positive real number to be determined. Observe that the choice of this POVM is motivated by the proof of Theorem 5. The probability of type I error is equal to 1 − Pr[type I error] = tr (ρFρ ) = exp D2 (ρkρ + M σ). Then using Theorem 4 we have −1 . 1 − Pr[type I error] ≥ (1 − ǫ) 1 + M exp − Dsǫ (ρkσ)
The probability of type II error, 1 − Pr[type II error] = tr(σFσ ), is equal to
1 − Pr[type II error] = exp D2 (σkM −1 ρ + σ) M −1 1 = (1 + M −1 )−1 exp D2 σ ρ + σ 1 + M −1 1 + M −1 −1 −1 ≥ (1 + M ) ,
where in the last line we use the positivity of collision relative entropy (see Theorem 1). Letting M = exp Dsǫ (ρkσ) + log ǫ gives the desired result.
The second order asymptotics of quantum hypothesis testing has been found independently by [3] and [5]. The achievability part of their result is a simple consequence of Theorem 9 and Theorem 3.
Theorem 10. For every ǫ > 0 and n there exists a POVM to distinguish ρ⊗n and σ ⊗n such that Pr[type I error] ≤ ǫ and that p √ − log Pr[type II error] = nD(ρkσ) + nV (ρkσ)Φ−1 (ǫ) + o( n).
5
Data compression with quantum side information
Let ρXB =
X x
pX (x)|xihx| ⊗ ρx ,
be a c-q state. Suppose that Alice receives x with probability pX (x), in which case Bob receives system B in state ρx . The goal of Alice is to transmit x to Bob by sending a message m. To this end, they may fix an encoding hash function h : X → {1, . . . , M }. Then Alice after receiving x sends m = h(x) to Bob. Having m, Bob chooses a measurement {Fxm | x ∈ X , h(x) = m}, that depends on m and applies it to ρx to decode x. The probability of successful decoding is equal to X Pr[succ] = pX (x)1{m=h(x)} tr(ρx Fxm ), x,m
where 1{m=h(x)} is equal to 1 if h(x) = m and is equal to 0 otherwise. This problem is called classical source coding (or data compression) with quantum side information, also known as the c-q Slepian-Wolf problem. Needless to say, the goal is to minimize M while keeping the average probability of successful decoding close to one. It is shown in [25] that the optimal asymptotic rate of communication required to achieve this goal is the conditional entropy H(X|B)ρ . A one-shot achievable and converse bound for this problem is derived in [9] in terms of smooth conditional max-entropy. Moreover, the second order asymptotics of this problem has been computed in [3]. Here we prove a one-shot achievable bound for this problem. 6
Theorem 11. For every ǫ > 0, there is an encoding map h : X → {1, . . . , M } and decoding measurements {Fxm | x ∈ X , h(x) = m} such that the probability of correct decoding is bounded by
1−ǫ 1 1
. ρXB + IX ⊗ ρB ≥ Pr[succ] ≥ exp D2 ρXB 1 − M M 1 + M −1 exp − Dsǫ (ρXB kIX ⊗ ρB )
Proof. Suppose that Alice’s encoding map h : X → {1, . . . , M } is chosen randomly, i.e., h(x) is chosen uniformly at random and independent of other h(x′ ), x′ 6= x. As before we use h(x) (in mathtt format) to denote this random function. Let the decoding POVM of Bob when receiving m ∈ {1, . . . , M } be {Fxm | x ∈ X , h(x) = m} where X −1/2 X −1/2 Fxm = pX (x) pX (x′ )ρx′ 1{h(x′ )=m} ρx pX (x′ )ρx′ 1{h(x′ )=m} . (9) x′
x′
Then the expectation value of the average probability of correct decoding is equal to X X Eh Pr[succ] = Eh pX (x)1{h(x)=m} tr (ρx Fxm ) = M Eh pX (x)1{h(x)=1} tr ρx Fx1 . x,m
x
This can be written in terms of collision relative entropy as Eh Pr[succ] = M Eh exp D2 (σXB kτXB ) where X X X σXB = p(x)1{h(x)=1} |xihx| ⊗ ρx , and τXB = 1{h(x)=1} |xihx| ⊗ p(x′ )1{h(x′ )=1} ρx′ . x
x
x′
Then using the joint convexity of exp D2 (·k·) we arrive at Eh Pr[succ] ≥ M exp D2 (Eh σXB kEh τXB ). We 1 ρXB , and have Eh σXB = M X Eh τXB = Eh p(x′ )1{h(x)=1} 1{h(x′ )=1} |xihx| ⊗ ρx′ x,x′
=
1 1 p(x′ ) 1{x=x′ } + 2 1{x6=x′ } |xihx| ⊗ ρx′ M M ′
X x,x
= Finally, using Theorem 4 we obtain
1 1 1 − 2 ρXB + 2 IX ⊗ ρB . M M M
1 1 1 1 ρXB − 2 ρXB + 2 IX ⊗ ρB Eh Pr[succ] ≥ M exp D2 M M M M
1 1 = exp D2 ρXB 1 − ρXB + IX ⊗ ρB M M −1 −1 −1 ≥ (1 − ǫ) 1 − M + M exp − Dsǫ (ρXB kIX ⊗ ρB ) −1 ≥ (1 − ǫ) 1 + M −1 exp − Dsǫ (ρXB kIX ⊗ ρB ) .
Corollary 12. For every ǫ > δ > 0 there is a protocol for classical data compression with quantum side information with M = ⌈exp − Dsδ (ρXB kIX ⊗ ρB ) − log(ǫ − δ) ⌉, whose probability of error is at most ǫ.
Remark 13. Again using Theorem 3 the achievability part of the asymptotic analysis of [3] for the c-q Slepian-Wolf problem can be derived from the above corollary.
6
Conclusion
We proposed a partial quantum extension of the technique of [1] and applied it to three basic information theoretic problems in the quantum case. In our generalization we noticed that some of the expressions of [1] can be written in terms of collision relative entropy, and used the joint convexity of its exponential. A full generalization of the technique of [1] to more complicated scenarios such as channels with state, and quantum Marton coding requires new tools to meet the challenges of dealing with non-commuting operators as well as proving (joint) operator convexity of certain functions to apply Jensen’s inequality. Acknowledgements. SB was in part supported by National Elites Foundation and by a grant from IPM (No. 91810409). 7
References [1] M. H. Yassaee, M. R. Aref, A. Gohari, A Technique for Deriving One-Shot Achievability Results in Network Information Theory, IEEE Symposium on Information Theory, 2013. [2] R. Renner, Security of Quantum Key Distribution, PhD thesis, ETH Zurich, Dec. 2005. arXiv: quantph/0512258. [3] M. Tomamichel and M. Hayashi, A Hierarchy of Information Quantities for Finite Block Length Analysis of Quantum Tasks, IEEE Trans. Inf. Theory 59 (11), 7693-7710 (2013). [4] M. Tomamichel and V. Y. F. Tan, Second-Order Asymptotics of Classical-Quantum Channels, arXiv:1308.6503 [5] K. Li. Second Order Asymptotics for Quantum Hypothesis Testing. Aug. 2012. arXiv:1208.1400 [6] L. Wang and R. Renner. One-Shot Classical-Quantum Capacity and Hypothesis Testing. Phys. Rev. Lett., 108 (20), May 2012. [7] M. Hayashi and H. Nagaoka. General Formulas for Capacity of Classical-Quantum Channels. IEEE Trans. IT , 49 (7):1753-1768, July 2003. [8] M. Hayashi, Optimal sequence of quantum measurements in the sense of Stein’s lemma in quantum hypothesis testing, Journal of Physics A: Mathematical and General, 35 (50), 10759-10773, 2002. [9] J. M. Renes and R. Renner, One-Shot Classical Data Compression With Quantum Side Information and the Distillation of Common Randomness or Secret Keys, IEEE Trans. Inf. Theory 58, 1985 (2012). [10] M. Berta, P. J. Coles, and S. Wehner, An equality between entanglement and uncertainty, arXiv:1302.5902 (2013). [11] F. Dupuis, O. Fawzi, and S. Wehner, Entanglement sampling and applications, arXiv: 1305.1316 (2013). [12] F. Dupuis, M. Berta, J. Wullschleger, and R. Renner, One-shot decoupling, arXiv:1012.6044 (2010). [13] M. M¨ uller-Lennert, F. Dupuis, O. Szehr, S. Fehr, and M. Tomamichel, On quantum R´enyi entropies: a new definition and some properties, arXiv:1306.3142 (2013). [14] M. Wilde, A. Winter, and D. Yang, Strong converse for the classical capacity of entanglement-breaking channels, arXiv:1306.1586 (2013). [15] M. Tomamichel, Smooth entropies–a tutorial: With focus on applications in cryptography, Tutorial at QCRYPT 2012, slides available at http://2012.qcrypt.net/docs/slides/Marco.pdf (2012). [16] S. Fehr, On the conditional R´enyi entropy, Lecture at the Beyond IID Workshop at the University of Cambridge, January 2013. [17] R. L. Frank, E. H. Lieb, Monotonicity of a relative R´enyi entropy, arXiv:1306.5358 (2013). [18] S. Beigi, Sandwiched R´enyi Divergence Satisfies Data Processing Inequality, arXiv:1306.5920 (2013). [19] A. Winter. Coding Theorem and Strong Converse for Quantum Channels. IEEE Trans. IT, 45(7):24812485, 1999. [20] V. Strassen. Asymptotische Absch¨ atzungen in Shannons Informationstheorie. In Trans. Third Prague Conf. Inf. Theory, 689-723, Prague, 1962. [21] Y. Polyanskiy, H. V. Poor, and S. Verd´ u, Channel Coding Rate in the Finite Blocklength Regime. IEEE Trans. on Inf. Theory, 56(5):2307-2359 (2010). [22] M. Hayashi, Information Spectrum Approach to Second-Order Coding Rate in Channel Coding. IEEE Trans. Inf. Theory, 55(11):4947-4966 (2009). [23] F. Hiai and D. Petz, The Proper Formula for Relative Entropy and its Asymptotics in Quantum Probability, Comm. Math. Phys., vol. 143, no. 1, pp. 99-114, Dec. 1991. [24] T. Ogawa and H. Nagaoka. Strong Converse and Stein’s Lemma in the Quantum Hypothesis Testing. IEEE Trans. Inf. Theory. 46, 2428 (2000). [25] I. Devetak and A. Winter, Classical data compression with quantum side information, Phys. Rev. A 68, 042301 (2003).
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