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On the Quantum Rényi Relative Entropies and Their On the quantum R´enyirelative entropies and their use 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 [email protected] On Sabbatical with Stanford Institute for Theoretical Physics, Stanford University, Stanford, California 94305 Mark M. Wilde (LSU) 1 / 68 Motivation Entropy and relative entropy are at the core of information theory R´enyientropy and R´enyirelative entropy are fruitful generalizations of this concept, having many applications When generalizing to the quantum case, there is not a unique way to generalize R´enyirelative entropy In this talk, I'll review three quantum generalizations of R´enyirelative entropy and their information-processing properties I'll also discuss their applications in quantum information theory Mark M. Wilde (LSU) 2 / 68 Entropy One of the most famous formulas in all of science [Sha48]: X H(pX ) := pX (x) log2 pX (x) − x Figure: Screen capture from the movie Arrival (2016) Equal to the optimal limit of classical data compression [Sha48] Mark M. Wilde (LSU) 3 / 68 Relative entropy Relative entropy [KL51] generalizes entropy Let pX ( ) be a probability distribution, and let qX be a measure.2 PThenX X pX (x) D(p q ) := p (x) log X X X 2 q (x) k x X Distinguishability measure with interpretation in hypothesis testing Mark M. Wilde (LSU) 4 / 68 Special cases of relative entropy Entropy is a special case of relative entropy: H(pX ) = D(pX 1); − k where 1 is the vector of all ones. Mutual information is a special case of relative entropy: I (X ; Y ) := D(pXY pX pY ) = inf D(pXY pX qY ) k ⊗ qY 2P(Y) k ⊗ Mark M. Wilde (LSU) 5 / 68 Data-processing inequality Let NY jX be a classical channel and define qY := NY jX (qX ) where X qY (y) = N (y x)qX (x): Y jX j x2X Then the data-processing inequality for relative entropy is D(pX qX ) D(N (pX ) N (qX )) k ≥ Y jX k Y jX Mark M. Wilde (LSU) 6 / 68 R´enyirelative entropy R´enyirelative entropy is a generalization of relative entropy with parameter α (0; 1) (1; ) [R´en61]: 2 [ 1 ! 1 X α 1−α Dα(pX qX ) := log pX (x) qX (x) k α 1 2 − x2X Mark M. Wilde (LSU) 7 / 68 Special cases of R´enyirelative entropy R´enyientropy [R´en61]is a special case: ! 1 X α Hα(pX ) := log pX (x) = Dα(pX 1) 1 α 2 − k − x2X R´enyimutual information [Csi95] defined as Iα(X ; Y ) := inf Dα(pXY pX qY ) qY 2P(Y) k ⊗ Mark M. Wilde (LSU) 8 / 68 Properties of R´enyirelative entropy Converges to relative entropy: D(pX qX ) = lim Dα(pX qX ) k α!1 k Data-processing inequality for all α (0; 1) (1; ): 2 [ 1 Dα(pX qX ) Dα(N (pX ) N (qX )) k ≥ Y jX k Y jX Additivity: for probability distributions pX1 and pX2 , measures qX1 and qX , and for all α (0; 1) (1; ): 2 2 [ 1 Dα(pX pX qX qX ) = Dα(pX qX ) + Dα(pX qX ) 1 ⊗ 2 k 1 ⊗ 2 1 k 1 2 k 2 Ordering: For α > β > 0 Dα(pX qX ) Dβ(pX qX ) k ≥ k Mark M. Wilde (LSU) 9 / 68 Quantum states A quantum state ρA is a positive semi-definite, unit trace operator (i.e., Hermitian matrix with all eigenvalues non-negative and summing to one) Subscript notation indicates ρA is a state of a quantum system A Also called density operator or density matrix ( A) denotes set of density operators acting on a Hilbert space A D H H Classical probability distributions are a special case in which density operator is diagonal Mark M. Wilde (LSU) 10 / 68 Quantum channels A quantum channel A!B is a completely positive, trace-preserving map, which evolves systemN A to system B Complete positivity and trace preservation are equivalent to Choi state N Φ := (idR A!B )(ΦRA) RB ⊗N N 1 being positive semi-definite and TrB [Φ ] = IR , where R A and RB dR ' maximally entangled state ΦRA is defined as 1 X ΦRA := i j R i j A; dR j ih j ⊗ j ih j i;j A classical channel is a special case in which the Choi state is a diagonal density operator. Mark M. Wilde (LSU) 11 / 68 Quantum relative entropy Quantum relative entropy of a state ρ and a positive semi-definite operator σ is defined as [Ume62] D(ρ σ) := Tr[ρ(log ρ log σ)] k 2 − 2 Standard definition with operational meaning [HP91, ON00] Mark M. Wilde (LSU) 12 / 68 Special cases of quantum relative entropy Quantum entropy [vN27] is a special case of relative entropy: H(ρ) := Tr[ρ log ρ] = D(ρ I ) − 2 − k Quantum mutual information [Str65] is a special case of relative entropy: I (A; B)ρ := D(ρAB ρA ρB ) = inf D(ρAB ρA σB ) k ⊗ σB 2D(HB ) k ⊗ Mark M. Wilde (LSU) 13 / 68 Data-processing inequality for quantum relative entropy Let ρ be a state, σ a positive semi-definite operator, and a quantum channel. Then [Lin75] N D(ρ σ) D( (ρ) (σ)) k ≥ N kN Mark M. Wilde (LSU) 14 / 68 Petz{R´enyirelative entropy There are at least two meaningful ways to generalize the classical R´enyirelative entropy to the quantum case. Let us begin with the Petz{R´enyirelative entropy. Petz{R´enyirelative entropy [Pet86] defined for α (0; 1) (1; ): 2 [ 1 1 α 1−α Dα(ρ σ) := log Tr[ρ σ ] k α 1 2 − Mark M. Wilde (LSU) 15 / 68 Properties of Petz{R´enyirelative entropy Converges to quantum relative entropy in limit α 1: ! D(ρ σ) = lim Dα(ρ σ) k α!1 k Data-processing inequality [Pet86] for all α (0; 1) (1; 2]: 2 [ Dα(ρ σ) Dα( (ρ) (σ)) k ≥ N kN Additivity: For states ρ1 and ρ2, and positive semi-definite operators σ1 and σ2: Dα(ρ1 ρ2 σ1 σ2) = Dα(ρ1 σ1) + Dα(ρ2 σ2) ⊗ k ⊗ k k Ordering [MH11]: For α > β > 0 Dα(ρ σ) Dβ(ρ σ) k ≥ k Mark M. Wilde (LSU) 16 / 68 Sandwiched R´enyirelative entropy Different quantum generalization of classical R´enyirelative entropy: Sandwiched R´enyirelative entropy [MLDS+13, WWY14] defined for all α (0; 1) (1; ): 2 [ 1 1 (1−α)=2α (1−α)=2α α Deα(ρ σ) := log Tr[(σ ρσ ) ] k α 1 2 − Mark M. Wilde (LSU) 17 / 68 Properties of sandwiched R´enyirelative entropy Converges to quantum relative entropy in limit α 1 [MLDS+13, WWY14]: ! D(ρ σ) = lim Deα(ρ σ) k α!1 k Data-processing inequality for all α [1=2; 1) (1; ) [FL13]: 2 [ 1 Deα(ρ σ) Deα( (ρ) (σ)) k ≥ N kN Additivity: For states ρ1 and ρ2, and positive semi-definite operators σ1 and σ2: Deα(ρ1 ρ2 σ1 σ2) = Deα(ρ1 σ1) + Deα(ρ2 σ2) ⊗ k ⊗ k k Ordering [MLDS+13]: For α > β > 0, Deα(ρ σ) Deβ(ρ σ) k ≥ k Mark M. Wilde (LSU) 18 / 68 Quantum hypothesis testing [Hel67, Hel69, Hol72, Hol73] Quantum system prepared in the state ρ or σ and objective is to figure out which one was prepared Make a quantum measurement Λ; I Λ to figure out which was prepared f − g Assign outcome Λ to \guess ρ", and the outcome I Λ to \guess σ" − Probability of committing a Type I error (\false alarm"): Tr[(I Λ)ρ] − Probability of committing a Type II error (\missed detection"): Tr[Λσ] Mark M. Wilde (LSU) 19 / 68 Hypothesis testing relative entropy [BD10, BD11, WR12] Minimize Type II error probability subject to a constraint on Type I error probability: " DH (ρ σ) := log2 inf Tr[Λσ] : Tr[(I Λ)ρ] "; Λ I k − Λ≥0f − ≤ ≤ g More generally can define this quantity when σ is positive semi-definite Obeys data-processing inequality (operational argument): D" (ρ σ) D" ( (ρ) (σ)) H k ≥ H N kN Mark M. Wilde (LSU) 20 / 68 Quantum Stein's lemma [HP91, ON00] Optimal achievable rate for hypothesis testing: 1 " ⊗n ⊗n E(ρ, σ) := inf lim inf DH (ρ σ ) "2(0;1) n!1 n k Optimal strong converse rate for hypothesis testing: 1 " ⊗n ⊗n Ee(ρ, σ) := sup lim sup DH (ρ σ ) "2(0;1) n!1 n k Always have E(ρ, σ) Ee(ρ, σ) ≤ Quantum Stein's lemma: E(ρ, σ) = Ee(ρ, σ) = D(ρ σ) k Mark M. Wilde (LSU) 21 / 68 Relating quantum hypothesis testing and relative entropy Let ρ be a state and σ a positive semi-definite operator Lower bound [Hay07, AMV12, QWW18]: For " (0; 1], and α (0; 1): 2 2 " α 1 D (ρ σ) Dα(ρ σ) + log H k ≥ k α 1 2 " − Upper bound [CMW16]: For " [0; 1), and α (1; ): 2 2 1 " α 1 D (ρ σ) Deα(ρ σ) + log H k ≤ k α 1 2 1 " − − Mark M. Wilde (LSU) 22 / 68 Application of lower bound Apply lower bound and additivity to find for all " (0; 1] and α (0; 1) that 2 2 1 " ⊗n ⊗n 1 ⊗n ⊗n α 1 D (ρ σ ) Dα(ρ σ ) + log n H k ≥ n k n(α 1) 2 " − α 1 = Dα(ρ σ) + log k n(α 1) 2 " − Take n limit to find for all α (0; 1) that ! 1 2 1 " ⊗n ⊗n lim inf D (ρ σ ) Dα(ρ σ) n!1 n H k ≥ k Since lower bound holds for all α (0; 1), conclude that 2 1 " ⊗n ⊗n lim inf DH (ρ σ ) sup Dα(ρ σ) = D(ρ σ) n!1 n k ≥ α2(0;1) k k Mark M.
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