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Chain rules for quantum Rényi entropies Frédéric Dupuis˚ arXiv:1410.5455v3 [quant-ph] 25 Feb 2015 Faculty of Informatics Masaryk University Brno, Czech Republic 26 February 2015 Abstract We present chain rules for a new definition of the quantum Rényi conditional entropy sometimes called the “sandwiched” Rényi conditional entropy. More precisely, we prove analogues of the equation HpAB|Cq “ HpA|BCq ` HpB|Cq, which holds as an identity for the von Neumann conditional entropy. In the case of the Rényi entropy, this relation no longer holds as an equality, but survives as an inequality of the form Hα pAB|Cq ě Hβ pA|BCq ` Hγ pB|Cq, where the parameters α, β, γ obey the β γ α “ β´1 ` γ´1 and pα ´ 1qpβ ´ 1qpγ ´ 1q ą 1; if pα ´ 1qpβ ´ 1qpγ ´ 1q ă 1, relation α´1 the direction of the inequality is reversed. 1 Introduction The Shannon entropy is one of the central concepts in information theory: it quantifies the amount of uncertainty contained in a random variable, and is used to characterize a wide range of information theoretical tasks. However, it is primarily useful for making asymptotic statements about problems involving repeated experiments, such as i.i.d. sources or channels. If one is interested not only in asymptotic rates, but in how quickly we can approach these rates with increasing block sizes, the natural quantities that arise are Rényi entropies. Likewise, in the case of “one-shot” problems, such as those arising in cryptographic settings, the (smooth) min- and max-entropies are usually the relevant quantities. (For instance, these quantities are relevant in scenarios such as privacy amplification [RK05; Ren05], decoupling [Dup09; DBWR14], and state merging [Ber08; DBWR14].) All of these information theoretical problems have a quantum counterpart, and it is therefore desirable to construct quantum versions of these entropic quantities. To do this systematically, one can use the fact that each of these entropies can be defined via a notion of divergence between two probability distributions; the entropy of a distribution p is then derived from the divergence between p and the uniform distribution. For example, the Shannon entropy can be defined in this way from the Kullback-Leibler divergence. One can then generalize the divergence quantities to quantum states and define quantum entropies in this way. However, these quantum counterparts to the divergences are not unique: the fact that quantum states can fail to commute leads to multiple non-equivalent definitions that all reduce to the classical version in the commutative case. In the case of ˚ Part of this work was performed while the author was at the Institute for Computer Science, Aarhus University, Aarhus, Denmark. 1 the Rényi divergence, this phenomenon has led to multiple non-equivalent definitions being in use. One of them [OP93] has been in use for several years and is defined as follows: let ρ and σ be two density operators. Then, their α-Rényi-divergence is given by D̃α pρ}σq :“ 1 Trrρα σ 1´α s log . α´1 Trrρs This definition has found many applications, most notably in hypothesis testing [ON00; Aud+07; Hay06], as well as for proving the fully quantum asymptotic equipartition property [TCR09; Tom12]. However, a new definition has recently been proposed in [ML+13; WWY14] which seems to possess better properties than the traditional definition in certain parameter regimes. It is defined as: ˆ ”´ 1´α 1´α ¯α ı˙ 1 1 Dα pρ}σq :“ log Tr σ 2α ρσ 2α . α´1 Trrρs (See Section 2.2 below for a more complete definition.) This new definition is sometimes called the “sandwiched” Rényi divergence, σ being the bread and ρ the meat in the expression above. The corresponding conditional Rényi entropy is then given by: Hα pA|Bqρ :“ ´ min Dα pρAB }idA σB q σB for a bipartite state ρAB . This new definition has already found a large number of applications: it characterizes the strong converse exponent in hypothesis testing [MO14a] (at least in some parameter regimes) and in classical-quantum channel coding [MO14b], and can be used to prove strong converses for a variety of information theoretical problems [WWY14; CMW14; TWW14]. Furthermore, because of this wide applicability, the fundamental properties of Dα and Hα are being investigated: a number of properties have been proven in [ML+13], including the data processing inequality for 1 ă α ď 2 and a duality property (see Fact 4 below, also independently proven in [Bei13]). The data processing inequality was also proven in [Bei13] for α ą 1 and in [FL13] for the full range α ě 12 . Moreover, in [AD13], the quantity was generalized to so-called α-zdivergences, and in [TBH14] the authors presented a duality property that involves both the new “sandwiched” definition and the traditional definition. The present work is in this vein. One of the most fundamental properties of the von Neumann entropy is the so-called chain rule: given a tripartite state ρABC , we can break down the conditional entropy HpAB|Cqρ into two parts: HpAB|Cqρ “ HpA|BCqρ ` HpB|Cqρ . While this rule no longer holds as an equality for the Rényi entropy, one can nonetheless hope to salvage it in the form of inequalities, as was done in [DBWR14; VDTR13] for the smooth min-/max-entropies. The result is the main theorem of this paper: Theorem 1. Let ρABC P DpA B Cq be a normalized density operator, and let α, β, γ P β γ α “ β´1 ` γ´1 . Then, we have that p 21 , 1q Y p1, 8q be such that α´1 Hα pAB|Cqρ ě Hβ pA|BCqρ ` Hγ pB|Cqρ (1) if pα ´ 1qpβ ´ 1qpγ ´ 1q ą 1, and Hα pAB|Cqρ ď Hβ pA|BCqρ ` Hγ pB|Cqρ (2) if pα ´ 1qpβ ´ 1qpγ ´ 1q ă 1. The proof is given in Section 3, where the two cases are split into Propositions 7 and 8, and where the statements proven are slightly more general. The proof is based on the Riesz-Thorin-type norm interpolation techniques developed in [Bei13], which are then applied to convenient expressions for the Rényi entropy. 2 2 Preliminaries 2.1 Notation In the table below, we summarize the notation used throughout the paper: Symbol A, B, C, . . . A, B, . . . LpA, Bq LpAq XAB XAÑB PospAq DpAq idA α1 , β 1 , γ 1 α̂, β̂, γ̂ OpAÑB p|iyA |iyB q X: XJ XA ě YA }X}p supp X Definition Quantum systems Hilbert spaces corresponding to systems A, B, . . . Set of linear operators from A to B LpA, Aq Operator in LpA Bq Operator in LpA, Bq Set of positive semidefinite operators on A Set of positive semidefinite operators on A with unit trace Identity operator on A α´1 β´1 γ´1 α , β , γ Dual parameter of α, β, γ: α1 ` α̂1 “ 2, etc. |jyB xi|A , for computational basis vectors |iyA , |jyB Adjoint of X. Transpose of X with respect to the computational basis. X ´ Y P PospAq. p 1 TrrpX : Xq 2 s p . Note that if p ă 1, this is not a norm. Support of X Note in particular the use of the shorthand α1 to denote α´1 α , i.e. the inverse of the Hölder conjugate of α; it will be used extensively. Using this shorthand, α P p 21 , 1q corresponds to α1 P p´1, 0q and α ą 1 corresponds to α1 P p0, 1q. Another convention used extensively throughout the paper is the implicit tensorization by the identity: if we have two operators XAB and YB , by XAB YB we mean XAB pidA YB q. We will also often omit subscripts when doing so should not create confusion. We will also make use of the operator-vector correspondence. We will endow every Hilbert space with its own computational basis, denoted by |iyA for the space A and so on, and we define the linear map OpAÑB : AB Ñ LpA, Bq by its action on the computational basis as follows: OpAÑB p|iyA |jyB q “ |jyB xi|A . 2.2 The quantum Rényi entropy We now present the definition of the quantum Rényi divergence, first defined in [ML+13; WWY14] and sometimes called the “sandwiched” Rényi divergence: Definition 2 (Quantum Rényi divergence). Let ρ, σ P PospAq, and let α P r 21 , 1q Y p1, 8q. Then, we define their Rényi α-divergence as ˆ „ˆ ˙α ˙ ´α1 ´α1 1 1 2 2 log Tr σ ρσ , Dα pρ}σq :“ α´1 Trrρs unless α ą 1 and supp ρ Ę supp σ, in which case Dα pρ}σq :“ 8. The Rényi entropy is then defined as follows: 3 Definition 3 (Quantum Rényi entropy). Let ρAB P PospA Bq and σB P DpBq, and let α P r 12 , 1q Y p1, 8q. Then, Hα pA|Bqρ|σ :“ ´Dα pρAB }idA σB q, and Hα pA|Bqρ :“ ´ inf ωB PDpBq Dα pρAB }idA ωB q. By taking the limit as α Ñ 1, we recover the von Neumann entropy; by taking the limit α Ñ 8, we get the min-entropy [Ren05]; by choosing α “ 12 , we get the max-entropy [KRS09]; and by choosing α “ 2 we get the collision entropy from [Ren05]. We also recall the duality property of quantum Rényi entropies proven in [ML+13, Theorem 9] and also independently in [Bei13, Theorem 9], which is the Rényi analogue of the duality between min- and max-entropy: Fact 4 (Duality of Rényi entropies). Let |ψyABC P A B C be a normalized pure state, and let ρABC :“ |ψyxψ|ABC . Then, we have that for any α P p 12 , 1q Y p1, 8q, Hα pA|Bqρ “ ´Hα̂ pA|Cqρ , where 1 α ` 1 α̂ “ 2. Note here that it is particularly convenient to rephrase the condition α1 ` the shorthand given in the table in Section 2.1: it corresponds to α1 “ ´α̂1 . 1 α̂ “ 2 using 3 Proof of the main result The first ingredient of the proof is the following generalization of Hadamard’s three-line theorem proven by Beigi in [Bei13] as part of his proof of the data processing inequality of the sandwiched Rényi divergence for α ą 1: Theorem 5 (Theorem 2 from [Bei13]). Let F : S Ñ LpAq, where S :“ tz P C : 0 ď Repzq ď 1u, be a bounded map that is holomorphic on the interior of S and continuous on the boundary. Let 0 ă θ ă 1 and define pθ by 1 1´θ θ “ ` . pθ p0 p1 For k “ 0, 1 define Mk “ sup }F pk ` itq}pk . tPR Then, we have }F pθq}pθ ď M01´θ M1θ . Note that in [Bei13], the theorem is stated for more general norms involving a positive operator σ (which can be chosen to be σ “ id to obtain this version) and adds a condition that p0 ď p1 which is not necessary, as can be seen by applying the theorem with F̄ pzq :“ F p1 ´ zq and θ̄ “ 1 ´ θ. The second ingredient is the following lemma, which gives a particularly useful expression for the Rényi entropy: 4 Lemma 6. Let |ψyABCD P A B C D be a normalized pure state, let XADÑBC “ OpADÑBC p|ψyq, and let α P p 21 , 1q Y p1, 8q, with α̂ such that α1 ` α̂1 “ 2, and let ρABCD “ |ψyxψ| and σC P DpCq. Then, Hα pAB|Cqρ|σ ›2 › α1 › α1 › ´α1 2 2 › › “ ´ log sup ›σC XτD › , τD PDpDq (3) 2 ›2 › › ´α1 › α1 2 Hα pB|Cqρ|σ “ ´ log ››σC X ›› , 2α › 21 › › α1 › α › Hα pA|BCqρ “ ´ log sup ›XτD 2 ›› . τD PDpDq (4) (5) 2α̂ Proof. We start by proving (3) from the represention in equation (19) from [ML+13]: # ´α1 ´1 α1 |ψy τD if α ă 1 1 log inf τD PDpDq xψ|idAB σC α Hα pAB|Cqρ|σ “ ´1 1 ´α1 α α1 log supτD PDpDq xψ|idAB σC τD |ψy if α ą 1, Putting the prefactor in the exponent, we get 1 1 1 α ´α τD |ψy α1 Hα pAB|Cqρ|σ “ ´ log sup xψ|idAB σC τD PDpDq ´α1 α1 for all α P r 21 , 1q Y p1, 8q. Now, consider the vector σC2 τD2 |ψy: by Lemma 11, we have that ˙ ˆ 1 1 1 1 ´α ´α α OpADÑBC σC2 τD2 |ψy J “ σC2 XτD α 2 and therefore xψ|idAB σC ´α1 1 α1 τD |ψy α1 › ˆ ´α1 ˙› 21 α1 › ›α J 2 2 “ ››OpADÑBC σC τD |ψy ›› 2 › › 21 1 1 ´α α › ›α J 2 › “ ››σC2 XτD › 2 by Lemma 12, from which (3) follows. We now derive (4) by expressing Hα pB|Cqρ|σ using (3): ´Hα pB|Cqρ|σ 2 “ sup τAD PDpADq “ sup τAD PDpADq › ›2 1 › α1 › ´α1 ›σ 2 XτAD α2 › › C › 2 ” ´α1 1 Tr X : σC XτAD α › › 11 1 › ›α “ ›X : σC ´α X › α › ›2 › ´α1 › α1 “ ››σC 2 X ›› , ı 11 α 2α where the third line follows from Lemma 10, and (4) follows. Finally, we prove (5) via 5 duality: Hα pA|BCqρ “ ´Hα̂ pA|Dqρ ›2 › › ´α̂1 › α̂1 › › 2 “ log inf ›XωD › ωD PDpDq 2α̂ › ´21 › › α1 › α “ log inf ››XωD 2 ›› ωD PDpDq 2α̂ › › 21 › α1 › α › “ ´ log sup ›XωD 2 ›› , ωD PDpDq 2α̂ where we used the fact that α̂1 “ ´α1 , and invoked Lemma 9 to get the third line. This concludes the proof. We are now ready to prove Theorem 1. We break it down into its two cases, with Proposition 7 below corresponding to Equation (1) and Proposition 8 further down corresponding to Equation (2). Note that we only prove the cases where α ą 1; the cases where α ă 1 then follow by applying the duality property (Fact 4) to all the terms. Proposition 7 (First case of Theorem 1). Let ρABC P DpA B Cq, and let α, β, γ ą 1 be such that α11 “ β11 ` γ11 . Then, we have that Hα pAB|Cqρ|σ ě Hβ pA|BCqρ ` Hγ pB|Cqρ|σ for any σC P DpCq. In particular, Hα pAB|Cqρ ě Hβ pA|BCqρ ` Hγ pB|Cqρ . Proof. Let |ψyABCD P A B C D be a purification of ρ, let XADÑBC “ OpADÑBC p|ψyq, and let τD P DpDq. We use Theorem 5 with the following choices: ´zγ 1 p1´zqβ 1 F pzq “ σC 2 XτD 2 1 1 1 “ “1´ p0 2β 2β̂ p1 “ 2γ θ“ α1 . γ1 Now, from these choices, we can show that 1 ´ θ “ α1 β1 : 1 ´ α1 {γ 1 1´θ “ 1 α α1 1 1 “ 1´ 1 α γ 1 “ 1. β 6 Furthermore, we can show that pθ “ 2: 1´θ θ 1 “ ` pθ p0 p1 α1 α1 ` 1 “ 2β̂β 1 2γ γ ˆ ˙ 1 α1 1 “ ` 2 β̂β 1 γ 1 γ ˙ ˆ 1 1 ` β1 1 ´ γ1 paq α ` “ 2 β1 γ1 ˙ ˆ 1 α 1 1 “ ` 1 1 2 β γ 1 “ , 2 where at line paq, we have used the fact that ´α1 1 γ “ 1 ´ γ 1 , and that α1 Finally, also note that F pθq “ σC2 XτD2 , and that › ´itγ1 › p1´itqβ 1 › › 2 2 › › M0 “ sup ›σC XτD › tPR ´itγ 1 due to the fact that σC 2 (6) › › β1 › › 2 › › “ ›XτD › 1 β̂ “ 2´ p0 p0 ´itβ 1 and τD 2 are unitary for all t P R. Likewise, › ´γ1 › › › M1 “ ››σC2 X ›› . p1 Hence, for any choice of normalized σ and τ , we have from Theorem 5 that › › › α1 › β1 › β1 α1 › › ´α1 › ›σ 2 Xτ 2 › ď ›Xτ 2 › D › › C › D› › ´γ1 › α11 › ›γ ›σ 2 X › . C › › › ›2 ›2 › β1 › β1 α1 › α1 › › ´α1 ›σ 2 Xτ 2 › ď ›Xτ 2 › D › › D› › C › ´γ1 › 21 › ›γ ›σ 2 X › , › C › 2β̂ 2 This leads to 2β̂ 2 2γ 2γ since α1 ą 0. Maximizing over τD on both sides yields: ›2 › α1 › α1 › ´α1 2 2 › › sup ›σC XτD › ď τD PDpDq Finally, using Lemma 6, we get that 2 › ›2 β1 › β1 › 2 › › sup ›XτD › τD PDpDq 2β̂ › ´γ1 › 21 ›γ › ›σ 2 X › . › › C Hα pAB|Cqρ|σ ě Hβ pA|BCqρ ` Hγ pB|Cqρ|σ , as advertised. 7 2γ 1 β “ 1 ` β1. We now turn to the second case of Theorem 1, corresponding to Equation (2), using essentially the same proof: Proposition 8 (Second case of Theorem 1). Let ρABC P DpA B Cq, and let α1 ą 0, and β 1 and γ 1 have opposite signs, and be such that α11 “ β11 ` γ11 . Then, we have that Hα pAB|Cqρ|σ ď Hβ pA|BCqρ ` Hγ pB|Cqρ|σ for every σC P DpCq. In particular, Hα pAB|Cqρ ď Hβ pA|BCqρ ` Hγ pB|Cqρ . Proof. Let |ψyABCD P A B C D be a purification of ρ, let XADÑBC “ OpADÑBC p|ψyq, and let τD P DpDq. We split the proof into two cases: first, we assume that β 1 ą 0. We use Theorem 5 with the following choices: ´α1 F pzq “ σC2 1 1 ´z α2βγ1 α1 XτD2 1 ´z α2 p0 “ 2 p1 “ 2γ θ“ ´β 1 γ1 β1 1 Now, note that F pθq “ XτD2 , and that 1 ´ θ “ αβ 1 and pθ “ 2β̂, as can be seen by a calculation very similar to Equation (6). Furthermore, we have that: › › › ´α1 ´it α1 γ1 α1 › α1 ´it › 2 2β 1 2 › XτD2 M0 “ sup ›σC › › › tPR 2 › › α1 › › ´α1 “ ››σC2 XτD2 ›› 2 ´itα1 γ 1 1 due to the fact that σC 2β ´itα1 are unitary for all t P R. Likewise, › ´γ1 › › › M1 “ ››σC2 X ›› . and τD 2 2γ Hence, for any choice of normalized σ and τ , we have that › β1 › α1 › α1 › ´α1 ď ››σC2 XτD2 ›› › ´γ1 › ´β1 1 › ›γ ›σ 2 X › . › C › ›2 ›2 › › β1 › β1 α1 › α1 › › ´α1 ›Xτ 2 › ď ›σ 2 Xτ 2 › D › › D› › C › ´γ1 › ´21 ›γ › ›σ 2 X › . C › › › › β1 › › ›Xτ 2 › › D› 2β̂ Now, since β 1 ą 0, this leads to: 2 2β̂ and therefore › ›2 β1 › β1 › sup ››XτD2 ›› ď τD PDpDq 2β̂ 2 2γ 2γ ›2 › α1 › α1 › ´α1 sup ››σC2 XτD2 ›› τD PDpDq 8 2 › ´γ1 › ´21 › ›γ ›σ 2 X › . › C › 2γ Using Lemma 6, we get that Hβ pA|BCqρ ě Hα pAB|Cqρ|σ ´ Hγ pB|Cqρ|σ , or, Hα pAB|Cqρ|σ ď Hβ pA|BCqρ ` Hγ pB|Cqρ|σ . We now turn to the case where β 1 ă 0 (and therefore γ 1 ą 0). We again use Theorem 5, but with these choices: 1 ´α1 `z α2 2 F pzq “ σC 1 1 α1 `z α2γβ1 2 XτC p0 “ 2 p1 “ 2β̂ θ“ ´γ 1 β1 ´γ 1 1 Now, note that F pθq “ σC2 X, that 1 ´ θ “ αγ 1 , and that pθ “ 2γ (see again Equation (6) for a similar calculation), and that › › 1 1› › ´α1 α1 α1 ´it α2γβ1 › › 2 ´it 2 2 XτD M0 “ sup ›σC › › tPR › 2 › › α1 › › ´α1 “ ››σC2 XτD2 ›› 2 ´itα1 β 1 2γ 1 ´itα1 are unitary for all t P R. Likewise, › › β1 › › M1 “ ››XτD2 ›› . and τD due to the fact that σC 2 2β̂ Hence, for any choice of normalized σ and τ , we have that › γ1 › α1 › α1 › ´α1 2 2 ď ››σC XτD ›› › › ´γ1 β1 › β1 › 2 ›Xτ › . › D› ›2 › ´γ1 › 21 › α1 › α1 ›γ › › ´α1 ›σ 2 X › ď ›σ 2 Xτ 2 › D › › › C › C › › ´2 β1 › β1 › ›Xτ 2 › . › D› › ´γ1 › › › ›σ 2 X › › C › 2γ Since γ 1 ą 0, this leads to 2 2 2γ 2β̂ 2β̂ Moving the rightmost term to the left-hand side and maximizing over τD on both sides, we get: › › ›2¸ ›2 › ´γ1 › 21 ˜ β1 › β1 α1 › α1 › › ´α1 ›γ › 2 2 2 2 ›σ X › sup ››XτD ›› ď sup ››σC XτD ›› . › › C 2γ τD PDpDq 2β̂ τD PDpDq Using Lemma 6, we get that Hγ pB|Cqρ|σ ` Hβ pA|BCqρ ě Hα pAB|Cqρ|σ . This concludes the proof. 9 2 Acknowledgments The author would like to thank Marco Tomamichel, Salman Beigi, Andreas Winter, Ciara Morgan, and Renato Renner for fruitful discussions and comments. The author acknowledges the support of the Czech Science Foundation GA CR project P202/12/1142 and the support of the EU FP7 under grant agreement no 323970 (RAQUEL). Part of this work was performed while the author was at Aarhus University, where he was supported by the Danish National Research Foundation and The National Science Foundation of China (under the grant 61061130540) for the Sino-Danish Center for the Theory of Interactive Computation, and also by the CFEM research centre (supported by the Danish Strategic Research Council). A Auxilliary lemmas Lemma 9. Let ρAB P DpA Bq. Then, we have that Hα pA|Bq “ ´ log inf σB PDpBq ›2 › ´α1 › α1 › ›Xσ 2 › , › B › 2α where X :“ OpABÑD p|ψyq for a purification |ψyABD of ρ. Proof. First, we write Hα pA|Bqρ|σ “ ´Dα pρAB }idA σB q „ˆ ´α1 ´α1 ˙α 1 log Tr σB2 ρσB2 “ 1´α › › › ´α1 ´α1 › ´1 2 2 “ 1 log ››σB ρσB ›› α α › ›1 1 ´α1 › α1 › ´α : 2 2 › › “ ´ log ›σB X XσB › α › ›2 ´α1 › α1 › 2 › › “ ´ log ›XσB › 2α where we have used the fact that ρAB “ X : X. Lemma 10. Let α P r 12 , 1q Y p1, 8q. Then, for any X P PospAq, we have that 1 }X}α “ sup TrrY α Xs Y PDpAq if α ą 1, and }X}α “ if α ă 1. (As usual, α1 “ α´1 α .) inf Y PDpAq 1 TrrY α Xs In particular, this means that 1 1 1 }X}αα1 “ sup TrrY α Xs α1 . Y PDpAq Proof. This is simply a reformulation of Lemma 11 in [ML+13]. 10 Lemma 11. Let |ψy P A B, and let XA P LpAq and YB P LpBq. Then, we have that J OpAÑB pXA YB |ψyq “ YB Opp|ψyqXA . Proof. This can be shown by a simple manipulation of indices; see for example [Wat11, Section 2.4]. Lemma 12. Let |ψy P A B. Then, xψ|ψy “ TrrOpAÑB p|ψyq: OpAÑB p|ψyqs and therefore, }|ψy} “ } OpAÑB p|ψyq}2 . Proof. Again, see [Wat11, Section 2.4]. References [AD13] Koenraad M.R. 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