Universal recoverability in quantum information Mark M. Wilde - PowerPoint PPT Presentation

Universal recoverability in quantum information 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 mwilde@lsu.edu Based on arXiv:1505.04661, 1506.00981, 1509.07127, 1511.00267, 1601.01207, 1608.07569, 1610.01262 with Berta, Buscemi, Das, Dupuis, Junge, Lami, Lemm, Renner, Sutter, Winter QMATH 2016, Atlanta, Georgia, USA Mark M. Wilde (LSU) 1 / 28

Main message Entropy inequalities established in the 1970s are a mathematical consequence of the postulates of quantum physics They have a number of applications: for determining the ultimate limits on many physical processes (communication, thermodynamics, uncertainty relations, cloning) 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 We discuss progress in this direction Mark M. Wilde (LSU) 2 / 28

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 Operational interpretation (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 / 28

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 (among many) Lieb concavity theorem [L73] relative modular operator method (see, e.g., [NP04]) quantum Stein’s lemma [BS03] Mark M. Wilde (LSU) 4 / 28

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 P σ, N such that ρ = ( P σ, N ◦ N )( ρ ) , σ = ( P σ, N ◦ N )( σ ) This “Petz” recovery map has the following explicit form [HJPW04]: 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 P ( x | y ) is given by the Bayes theorem: P ( x | y ) q Y ( y ) = N ( y | x ) q X ( x ) where q Y ( y ) ≡ � x N ( y | x ) q X ( x ) Mark M. Wilde (LSU) 5 / 28

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] Mark M. Wilde (LSU) 6 / 28

One-shot measure of similarity for quantum states 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 . Mark M. Wilde (LSU) 7 / 28

New result of [Wil15, JSRWW15] Recoverability Theorem Let ρ and σ satisfy supp( ρ ) ⊆ supp( σ ) and let N be a channel. Then � ∞ � � �� ρ, P t / 2 D ( ρ � σ ) − D ( N ( ρ ) �N ( σ )) ≥ − dt p ( t ) log F σ, N ( N ( ρ )) , −∞ where p ( t ) is a distribution and P t σ, N is a rotated Petz recovery map: � � P t σ, N ( · ) ≡ U σ, t ◦ P σ, N ◦ U N ( σ ) , − t ( · ) , 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 R´ enyi generalization of a relative entropy difference and the Stein–Hirschman operator interpolation theorem Mark M. Wilde (LSU) 8 / 28

Universal Recovery Universal Recoverability Corollary Let ρ and σ satisfy supp( ρ ) ⊆ supp( σ ) and let N be a channel. Then D ( ρ � σ ) − D ( N ( ρ ) �N ( σ )) ≥ − log F ( ρ, R σ, N ( N ( ρ ))) , where � ∞ dt p ( t ) P t / 2 R σ, N ≡ σ, N −∞ (follows from concavity of logarithm and fidelity) Mark M. Wilde (LSU) 9 / 28

Universal Distribution 0 . 8 p ( t ) 0 . 6 0 . 4 0 . 2 0 − 3 − 2 − 1 0 1 2 3 t � � − 1 as a Figure: This plot depicts the probability density p ( t ) := π cosh( π t ) + 1 2 function of t ∈ R . We see that it is peaked around t = 0 which corresponds to the Petz recovery map. Mark M. Wilde (LSU) 10 / 28

R´ enyi generalizations of a relative entropy difference Definition from [BSW14, SBW14] � U σ − α ′ / 2 ρ 1 / 2 � � � ∆ α ( ρ, σ, N ) ≡ 2 � � N ( ρ ) − α ′ / 2 N ( σ ) α ′ / 2 ⊗ I E � α ′ log 2 α , � � 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, P σ, N ( N ( ρ ))) . Mark M. Wilde (LSU) 11 / 28

Stein–Hirschman operator interpolation theorem (setup) Let S ≡ { z ∈ C : 0 < Re { z } < 1 } , and let L ( H ) be the space of bounded linear operators acting on H . Let G : S → L ( H ) be an operator-valued function bounded on S , holomorphic on S , and continuous on the boundary ∂ S . 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 , ∞ ]. Mark M. Wilde (LSU) 12 / 28

Stein–Hirschman operator interp. theorem (statement) Then the following bound holds log � G ( θ ) � p θ ≤ � ∞ � � � � �� � G ( it ) � 1 − θ � G (1 + it ) � θ dt α θ ( t ) log + β θ ( t ) log , p 0 p 1 −∞ sin( πθ ) where α θ ( t ) ≡ 2(1 − θ ) [cosh( π t ) − cos( πθ )] , sin( πθ ) β θ ( t ) ≡ 2 θ [cosh( π t ) + cos( πθ )] , θ ց 0 β θ ( t ) = p ( t ) . lim Mark M. Wilde (LSU) 13 / 28

Proof of Recoverability Theorem Tune parameters � � [ N ( ρ )] z / 2 [ N ( σ )] − z / 2 ⊗ I E U σ z / 2 ρ 1 / 2 , Pick G ( z ) ≡ 2 p 0 = 2 , p 1 = 1 , θ ∈ (0 , 1) ⇒ p θ = 1 + θ Evaluate norms � � � U σ it / 2 ρ 1 / 2 � � ρ 1 / 2 � � � � � � N ( ρ ) it / 2 N ( σ ) − it / 2 ⊗ I E � G ( it ) � 2 = 2 ≤ 2 = 1 , � � � � � �� 1 / 2 ρ, P t / 2 � G (1 + it ) � 1 = σ, N ( N ( ρ )) . F Mark M. Wilde (LSU) 14 / 28

Proof of Recoverability Theorem (ctd.) Apply the Stein–Hirschman theorem � � � U σ θ/ 2 ρ 1 / 2 � � � [ N ( ρ )] θ/ 2 [ N ( σ )] − θ/ 2 ⊗ I E log � � 2 / (1+ θ ) � � θ/ 2 � � ∞ � ρ, ( P t / 2 ≤ dt β θ ( t ) log σ, N ◦ N )( ρ ) . F −∞ Final step Apply a minus sign, multiply both sides by 2 /θ , and take the limit as θ ց 0 to conclude. Mark M. Wilde (LSU) 15 / 28

Specializing to the Holevo Bound Specializing to the Holevo bound leads to a refinement. Given � � p X ( x ) | x �� x | X ⊗ ρ x � ϕ y | B ρ XB | ϕ y � B | y �� y | Y . ρ XB ≡ B , ω XY ≡ x y Then the following inequality holds � √ F ( ρ x B , E B ( ρ x I ( X ; B ) ρ − I ( X ; Y ) ω ≥ − 2 log p X ( x ) B )) , x where E B is an entanglement-breaking map of the form � ∞ ρ (1+ it ) / 2 | ϕ y �� ϕ y | B ρ (1 − it ) / 2 � B B E B ( · ) ≡ dt β 0 ( t ) � ϕ y | B ( · ) | ϕ y � B . � ϕ y | B ρ B | ϕ y � B −∞ y Mark M. Wilde (LSU) 16 / 28

Applying to Entropy Special case: Entropy gain (also called Entropy Production) Specializing to entropy gives the following bound for a unital quantum channel N : H ( N ( ρ )) − H ( ρ ) ≥ − log F ( ρ, N † ( N ( ρ ))) A different approach [BDW16] gives a stronger bound and applies to more general maps. For N a positive, subunital, trace-preserving map: H ( N ( ρ )) − H ( ρ ) ≥ D ( ρ �N † ( N ( ρ ))) ≥ 0 Mark M. Wilde (LSU) 17 / 28

Application to entropy uncertainty relations [BWW15] Let ρ ABE be a state for Alice, Bob, and Eve, and let X ≡ { P x A } and Z = { Q z A } be projection-valued measures for Alice’s system Define the post-measurement states: � | x �� x | X ⊗ σ x σ XBE ≡ where BE x σ x BE ≡ Tr A { ( P x A ⊗ I BE ) ρ ABE } � | z �� z | Z ⊗ ω z ω ZBE ≡ where BE z ω z BE ≡ Tr A { ( Q z A ⊗ I BE ) ρ ABE } Then H ( Z | E ) ω + H ( X | B ) σ x , z � P x A Q z A � 2 ≥ − log max ∞ − log F ( ρ AB , R XB → AB ( σ XB )) Mark M. Wilde (LSU) 18 / 28