Recoverability in quantum information theory

Transcription

1 Recoverability in quantum information theory Mark M. Wilde Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana, USA Beyond i.i.d., July 5-10, 2015, Banff, Alberta, Canada Mark M. Wilde (LSU) 1 / 23

2 Main message Entropy inequalities established in the 1970s are a mathematical consequence of the postulates of quantum physics They are helpful in determining the ultimate limits on many physical processes (communication, thermodynamics, uncertainty relations) Many of these entropy inequalities are equivalent to each other, so we can say that together they constitute a fundamental law of quantum information theory There has been recent interest in refining these inequalities, trying to understand how well one can attempt to reverse an irreversible physical process This poster presentation discusses progress in this direction Mark M. Wilde (LSU) 2 / 23

3 Background entropies Umegaki relative entropy [Ume62] The quantum relative entropy is a measure of dissimilarity between two quantum states. Defined for state ρ and positive semi-definite σ as D(ρ σ) Tr{ρ[log ρ log σ]} whenever supp(ρ) supp(σ) and + otherwise Physical interpretation with quantum Stein s lemma [HP91, NO00] Given are n quantum systems, all of which are prepared in either the state ρ or σ. With a constraint of ε (0, 1) on the Type I error of misidentifying ρ, then the optimal error exponent for the Type II error of misidentifying σ is D(ρ σ). Mark M. Wilde (LSU) 3 / 23

4 Background entropies Relative entropy as mother entropy Many important entropies can be written in terms of relative entropy: H(A) ρ D(ρ A I A ) (entropy) H(A B) ρ D(ρ AB I A ρ B ) (conditional entropy) I (A; B) ρ D(ρ AB ρ A ρ B ) (mutual information) I (A; B C) ρ D(ρ ABC exp{log ρ AC + log ρ BC log ρ C }) (cond. MI) Equivalences H(A B) ρ = H(AB) ρ H(B) ρ I (A; B) ρ = H(A) ρ + H(B) ρ H(AB) ρ I (A; B) ρ = H(B) ρ H(B A) ρ I (A; B C) ρ = H(AC) ρ + H(BC) ρ H(ABC) ρ H(C) ρ I (A; B C) ρ = H(B C) ρ H(B AC) ρ Mark M. Wilde (LSU) 4 / 23

5 Fundamental law of quantum information theory Monotonicity of quantum relative entropy [Lin75, Uhl77] Let ρ be a state, let σ be positive semi-definite, and let N be a quantum channel. Then D(ρ σ) D(N (ρ) N (σ)) Distinguishability does not increase under a physical process Characterizes a fundamental irreversibility in any physical process Proof approaches Lieb concavity theorem [L73] relative modular operator method (see, e.g., [NP04]) quantum Stein s lemma [BS03] Mark M. Wilde (LSU) 5 / 23

6 Strong subadditivity Strong subadditivity [LR73] Let ρ ABC be a tripartite quantum state. Then I (A; B C) ρ 0 Equivalent statements (by definition) Entropy sum of two individual systems is larger than entropy sum of their union and intersection: H(AC) ρ + H(BC) ρ H(ABC) ρ + H(C) ρ Conditional entropy does not decrease under the loss of system A: H(B C) ρ H(B AC) ρ Mark M. Wilde (LSU) 6 / 23

7 Equality conditions [Pet86, Pet88] When does equality in monotonicity of relative entropy hold? D(ρ σ) = D(N (ρ) N (σ)) iff a recovery map R P σ,n such that ρ = (R P σ,n N )(ρ), σ = (R P σ,n N )(σ) This Petz recovery map has the following explicit form [HJPW04]: R P σ,n (ω) σ 1/2 N ( (N (σ)) 1/2 ω(n (σ)) 1/2) σ 1/2 Classical case: Distributions p X and q X and a channel N (y x). Then the Petz recovery map R P (x y) is given by the Bayes theorem: where q Y (y) x N (y x)q X (x) R P (x y)q Y (y) = N (y x)q X (x) Mark M. Wilde (LSU) 7 / 23

8 More on Petz recovery map Linear, completely positive by inspection and trace non-increasing because Tr{R P σ,n (ω)} = Tr{σ 1/2 N ( (N (σ)) 1/2 ω(n (σ)) 1/2) σ 1/2 } = Tr{σN ( (N (σ)) 1/2 ω(n (σ)) 1/2) } = Tr{N (σ)(n (σ)) 1/2 ω(n (σ)) 1/2 } Tr{ω} For N (σ) positive definite, the map perfectly recovers σ from N (σ): R P σ,n (N (σ)) = σ 1/2 N ( (N (σ)) 1/2 N (σ)(n (σ)) 1/2) σ 1/2 = σ 1/2 N (I ) σ 1/2 = σ Mark M. Wilde (LSU) 8 / 23

9 Functoriality Normalization [LW14] For identity channel, the Petz recovery map is the identity map: R P σ,id = id. If there s no noise, then no need to recover Tensorial [LW14] Given a tensor-product state and channel, then the Petz recovery map is a tensor product: R P σ 1 σ 2,N 1 N 2 = R P σ 1,N 1 R P σ 2,N 2. Individual action suffices for pretty good recovery of individual states Composition [LW14] Given N 2 N 1, then R P σ,n 2 N 1 = R P σ,n 1 R P N 1 (σ),n 2. To recover pretty well overall, recover pretty well from the last noise first and the first noise last Mark M. Wilde (LSU) 9 / 23

10 Petz recovery map for strong subadditivity Strong subadditivity is a special case of monotonicity of relative entropy with ρ = ω ABC, σ = ω AC I B, and N = Tr A Then N ( ) = ( ) I A and Petz recovery map is R P C AC (τ C ) = ω 1/2 AC ( ) ω 1/2 C τ C ω 1/2 C I A ω 1/2 AC Interpretation: If system A is lost but H(B C) ω = H(B AC) ω, then one can recover the full state on ABC by performing the Petz recovery map on system C of ω BC, i.e., ω ABC = R P C AC (ω BC ) Mark M. Wilde (LSU) 10 / 23

11 Approximate case Approximate case would be useful for applications Approximate case for monotonicity of relative entropy What can we say when D(ρ σ) D(N (ρ) N (σ)) = ε? Does there exist a CPTP map R that recovers σ perfectly from N (σ) while recovering ρ from N (ρ) approximately? [WL12] Approximate case for strong subadditivity What can we say when H(B C) ω H(B AC) ω = ε? Is ω ABC approximately recoverable from ω BC by performing a recovery map on system C alone? [WL12] Mark M. Wilde (LSU) 11 / 23

12 Other measures of similarity for quantum states Trace distance Trace distance between ρ and σ is ρ σ 1 where A 1 = Tr{ A A}. Has a one-shot operational interpretation as the bias in success probability when distinguishing ρ and σ with an optimal quantum measurement. Fidelity [Uhl76] Fidelity between ρ and σ is F (ρ, σ) ρ σ 2 1. Has a one-shot operational interpretation as the probability with which a purification of ρ could pass a test for being a purification of σ. Bures distance [Bur69] Bures distance between ρ and σ is D B (ρ, σ) = 2 ( 1 ) F (ρ, σ). Mark M. Wilde (LSU) 12 / 23

13 Breakthrough result of [FR14] Remainder term for strong subadditivity [FR14] unitary channels U C and V AC such that ( ) H(B C) ω H(B AC) ω log F ω ABC, (V AC R P C AC U C )(ω BC ) Nothing known from [FR14] about these unitaries! However, can conclude that I (A; B C) is small iff ω ABC is approximately recoverable from system C alone after the loss of system A. Remainder term for monotonicity of relative entropy [BLW14] unitary channels U and V such that D(ρ σ) D(N (ρ) N (σ)) log F Again, nothing known from [BLW14] about U and V. ( ) ρ, (V R P σ,n U)(N (ρ)) Mark M. Wilde (LSU) 13 / 23

14 New result of [Wil15] New Theorem: Let ρ and σ be such that supp(ρ) supp(σ) and let N be a quantum channel. Then the following inequality holds [ ( ) ] D (ρ σ) D (N (ρ) N (σ)) log sup F ρ, R P,t σ,n (N (ρ)), t R where R P,t σ,n is the following rotated Petz recovery map: R P,t σ,n ( ) ( U σ,t R P σ,n U N (σ), t ) ( ), R P σ,n is the Petz recovery map, and U σ,t and U N (σ), t are defined from U ω,t ( ) ω it ( ) ω it, with ω a positive semi-definite operator. Two tools for proof: Rényi generalization of a relative entropy difference and the Hadamard three-line theorem Mark M. Wilde (LSU) 14 / 23

15 Rényi generalizations of a relative entropy difference Definition from [BSW14, SBW14] α (ρ, σ, N ) 2 α log ( N (ρ) α /2 N (σ) α /2 I E ) Uσ α /2 ρ 1/2 2α, where α (0, 1) (1, ), α (α 1)/α, and U S BE is an isometric extension of N. Important properties lim α (ρ, σ, N ) = D (ρ σ) D (N (ρ) N (σ)). α 1 ( ) 1/2 (ρ, σ, N ) = log F ρ, R P σ,n (N (ρ)). Mark M. Wilde (LSU) 15 / 23

16 Hadamard three-line theorem Let S {z C : 0 Re {z} 1}, and let L (H) be the space of bounded linear operators acting on a Hilbert space H. Let G : S L (H) be a bounded map that is holomorphic on the interior of S and continuous on the boundary. Let θ (0, 1) and define p θ by 1 = 1 θ + θ, p θ p 0 p 1 where p 0, p 1 [1, ]. For k = 0, 1 define M k = sup G (k + it) pk. t R Then G (θ) pθ M 1 θ 0 M θ 1. Mark M. Wilde (LSU) 16 / 23

17 Three (or so) line proof Pick t R G (z) ([N (ρ)] z/2 [N (σ)] z/2 I E ) Uσ z/2 ρ 1/2, p 0 = 2, p 1 = 1, θ (0, 1) p θ = θ Then (N ) M 0 = sup (ρ) it/2 N (σ) it/2 I E Uσ it ρ 1/2 2 ρ 1/2 2 = 1, M 1 = sup G (1 + it) 1 = t R [ sup F t R ( ρ, R P,t σ,n (N (ρ)) ) ] 1/2. Apply the three-line theorem to conclude that [ ( ) ] G (θ) 2/(1+θ) sup F ρ, R P,t θ/2 σ,n (N (ρ)). t R Take a negative logarithm and the limit as θ 0 to conclude. Mark M. Wilde (LSU) 17 / 23

18 SSA refinement as a special case Let ρ ABC be a density operator acting on a finite-dimensional Hilbert space H A H B H C. Then the following inequality holds [ ( I (A; B C) ρ log sup F ρ ABC, R P,t C AC (ρ BC )) ], t R where R P,t C AC is the following rotated Petz recovery map: R P,t C AC ( ) (U ρac,t R P C AC U ρ C, t ) ( ), the Petz recovery map R P C AC is defined as [ ] R P C AC ( ) ρ1/2 AC ρ 1/2 C ( ) ρ 1/2 C I A ρ 1/2 AC, and the partial isometric maps U ρac,t and U ρc, t are defined as before. Mark M. Wilde (LSU) 18 / 23

19 Conclusions The result of [FR14] already had a number of important implications in quantum information theory. The new result in [Wil15] applies to relative entropy differences, has a brief proof, and improves our understanding of the input and output unitaries (but see [SFR15] for the special case of SSA) By building on [SFR15, Wil15], we can now generalize these results: there is a universal recovery map which depends only on σ and N and has the form [SRWW15]: X for some probability measure µ. µ(dt) R P,t σ,n (X ) It is still conjectured that the recovery map can be the Petz recovery map alone (not a rotated Petz map). Mark M. Wilde (LSU) 19 / 23

20 References I [BLW14] Mario Berta, Marius Lemm, and Mark M. Wilde. Monotonicity of quantum relative entropy and recoverability. December arxiv: [BS03] Igor Bjelakovic and Rainer Siegmund-Schultze. Quantum Stein s lemma revisited, inequalities for quantum entropies, and a concavity theorem of Lieb. July arxiv:quant-ph/ [BSW14] Mario Berta, Kaushik Seshadreesan, and Mark M. Wilde. Rényi generalizations of the conditional quantum mutual information. March arxiv: [Bur69] Donald Bures. An extension of Kakutani s theorem on infinite product measures to the tensor product of semifinite w -algebras. Transactions of the American Mathematical Society, 135: , January [FR14] Omar Fawzi and Renato Renner. Quantum conditional mutual information and approximate Markov chains. October arxiv: [HJPW04] Patrick Hayden, Richard Jozsa, Denes Petz, and Andreas Winter. Structure of states which satisfy strong subadditivity of quantum entropy with equality. Communications in Mathematical Physics, 246(2): , April arxiv:quant-ph/ Mark M. Wilde (LSU) 20 / 23

21 References II [HP91] Fumio Hiai and Denes Petz. The proper formula for relative entropy and its asymptotics in quantum probability. Communications in Mathematical Physics, 143(1):99 114, December [Lin75] Göran Lindblad. Completely positive maps and entropy inequalities. Communications in Mathematical Physics, 40(2): , June [L73] Elliott H. Lieb. Convex Trace Functions and the Wigner-Yanase-Dyson Conjecture. Advances in Mathematics, 11(3), , December [LR73] Elliott H. Lieb and Mary Beth Ruskai. Proof of the strong subadditivity of quantum-mechanical entropy. Journal of Mathematical Physics, 14(12): , December [LW14] Ke Li and Andreas Winter. Squashed entanglement, k-extendibility, quantum Markov chains, and recovery maps. October arxiv: [NO00] Hirsohi Nagaoka and Tomohiro Ogawa. Strong converse and Stein s lemma in quantum hypothesis testing. IEEE Transactions on Information Theory, 46(7): , November arxiv:quant-ph/ Mark M. Wilde (LSU) 21 / 23

22 References III [NP04] Michael A. Nielsen and Denes Petz. A simple proof of the strong subadditivity inequality. arxiv:quant-ph/ [Pet86] Denes Petz. Sufficient subalgebras and the relative entropy of states of a von Neumann algebra. Communications in Mathematical Physics, 105(1): , March [Pet88] Denes Petz. Sufficiency of channels over von Neumann algebras. Quarterly Journal of Mathematics, 39(1):97 108, [SBW14] Kaushik P. Seshadreesan, Mario Berta, and Mark M. Wilde. Rényi squashed entanglement, discord, and relative entropy differences. October arxiv: [SFR15] David Sutter, Omar Fawzi, and Renato Renner. Universal recovery map for approximate Markov chains. April arxiv: [SRWW15] David Sutter, Renato Renner, Mark M. Wilde, and Andreas Winter. Universal recovery from a decrease of quantum relative entropy. June arxiv:150?.?????. Mark M. Wilde (LSU) 22 / 23

23 References IV [Uhl76] Armin Uhlmann. The transition probability in the state space of a *-algebra. Reports on Mathematical Physics, 9(2): , [Uhl77] Armin Uhlmann. Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory. Communications in Mathematical Physics, 54(1):21 32, [Ume62] Hisaharu Umegaki. Conditional expectations in an operator algebra IV (entropy and information). Kodai Mathematical Seminar Reports, 14(2):59 85, [Wil15] Mark M. Wilde. Recoverability in quantum information theory. May arxiv: [WL12] Andreas Winter and Ke Li. A stronger subadditivity relation? csajw/stronger subadditivity.pdf, Mark M. Wilde (LSU) 23 / 23