Contraction Coefficients for Noisy Quantum Channels

Transcription

1 Contraction Coefficients for Noisy Quantum Channels Mary Beth Ruskai joint work with F. Hiai Daejeon February, M. B. Ruskai Contraction of Quantum Channels

2 Recall Relative Entropy Φ : M d M d Quantum channel, i.e., completely positiv, trace preserving (CPT) map Relative entropy H(ρ, γ) = Tr ( ρ log ρ ρ log γ ) ρ, γ 0 Tr ρ = Tr γ = 1 Basic property: Contracts under action of quantum channel, i.e., H [ Φ(ρ), Φ(γ) ] H(ρ, γ) Maximal contraction measure of noisiness of channel Φ H [ Φ(ρ), Φ(γ) ] η(φ) = sup ρ γ H(ρ, γ) Many other quantities contract compare contraction properties 2 M. B. Ruskai Contraction of Quantum Channels

3 Basic Notation for Operators on M d Φ denote adjoint wrt Hilbert-Schmidt inner prod P, Q = Tr P Q i.e., Tr [Φ(P)] Q = Tr P Φ(Q) Density matrices D {P M d : P 0, Tr P = 1} Tangent space {A = M d : A = A, Tr A = 0} Def. Left and Right mult are linear operators on M d L P (X ) = PX and R Q (X ) = XQ a) L P and R Q commute L P [R Q (X )] = PXQ = R Q [L P (X )] b) P = P L P, R P self-adjoint wrt H-S inner prod c) P, Q > 0 pos def L P, R P pos def, e.g Tr X PX 0 d) P, Q > 0 (L P ) 1 = L P 1, (R Q ) 1 = R Q 1 e) f (L P ) = L f (P), etc. log R Q = R log Q when Q > 0 3 M. B. Ruskai Contraction of Quantum Channels

4 Generalized Relative Entropy Petz (1986) defined quasi-entropy a.k.a. generalized realtive entropy, f -divergence G = {g : (0, ) R operator convex, g(1) = 0} H g (K, P, Q) Tr QK g ( L P R 1 ) Q (K Q) Thm: H g (K, P, Q) jointly convex in P, Q Thm: H g [ K, Φ(P), Φ(Q) ] Hg ( Φ(K), P, Q) g(x) G g(x) = x g(x 1 ) G H g (K, P, Q) = H g (K, Q, P) g = g g(x) = x log x H g (I, P, Q) = Tr P (log P log Q) g(x) = log x Hg (I, P, Q) = Tr Q(log Q log P) 4 M. B. Ruskai Contraction of Quantum Channels

5 Recover WYD Entropy g t (x) = g t (x) = x g t (x 1 ) = { 1 t(1 t) (x x t ) t 1. t (0, 2] x log x t = 1 { 1 t(1 t) (1 x t ) t 0 t [ 1, 1) log x t = 0 J t (K, P, Q) Tr QK ( g t LP R 1 ) Q (K Q) t [ 1, 2] 1 ( = Tr K PK Tr K P t KQ 1 t) t(1 t) J 1 (K, P, Q) = Tr KK P log P Tr K PK log Q J 0 (K, P, Q) = Tr K K Q log Q Tr KQK log P Recover both WYD entropy with linear term and H(P, Q), K = I 5 M. B. Ruskai Contraction of Quantum Channels

6 Riemannian metrics or Fisher information Henceforth consider only K = I and write H g (P, Q) a b H g (P + aa, P + bb, I ) = Tr A Ω κ P (B) = A, Ωκ P (B) a=b=0 2 for Tr A = Tr B = 0 in LHS, get pos quad form which is RHS with Ω κ P (X ) R 1 P κ( L P R 1 ) P X = L 1 P κ( R P L 1 ) P K = {κ : (0, ) (0, ) κ op convex, κ ( x 1) = xκ(x)} with κ(x) = g(x)+xg(x 1 ) (1 x) 2 = gsym(x) (1 x) 2 K relate H g and M κ P A, Ω κ P (B) Mκ P (A, B) is Riemmanian metric Ω P non-commutative multiplication by P 1 A, Ω κ P (A) quantum analogue of Fisher information 6 M. B. Ruskai Contraction of Quantum Channels

7 Examples g(x) g(x) κ(x) Ω ρ (X ) RelEnt x log x log x log x x ρ+ui X 1 ρ+ui du WY 4(1 x ) 4(x 4 x ) (1+ 4 ( x ) 2 LP + ) 2 (X ) R P x 1/2 x 1/2 x 1/2 ρ 1/2 X ρ 1/2 1 max 2 (x 1 1 1) 2 (x 2 1+x x) 2x ( 1 2 X ρ 1 + ρ 1 X ) min 2 1+x 2 L ρ+r ρ (X ) 7 M. B. Ruskai Contraction of Quantum Channels

8 Summarize G = {g : (0, ) R operator convex, g(1) = 0} G sym = {g : (0, ) R op convex, g(1) = 0, xg(x 1 ) = g(x)} g G g(x) = xg(x 1 ) G g sym = g(x) + g(x) G sym There is a 1-1 correspondence between a) sym rel entropy H gsym (P, Q) = H gsym (Q, P), g sym G sym b) op convex g G sym with g (1) = 2 c) op convex κ : (0, ) (0, ) with κ(x 1 ) = xκ(x), κ(1) = 1 d) monotone Riemannian metrics Any g G gives mono metric with κ(x) = g(x)+xg(x 1 ) (x 1) 2 = gsym(x) (x 1) 2 8 M. B. Ruskai Contraction of Quantum Channels

9 Geodesic distance Define geodesic distance for each κ K D κ (P, Q) inf ξ(t) 1 0 Tr ξ (t) Ω ξ(t) ξ (t)dt where ξ(t) smooth path in D with ξ(0) = P, ξ(1) = Q Remark: Restriction to Tr ξ(t) = 1 here typically gives larger distance than geodesic with ξ(t) path over all pos matrices. Trace distance P Q 1 = Tr P Q contracts under CPT maps 9 M. B. Ruskai Contraction of Quantum Channels

10 Contraction Theorems All of above decrease under quantum channels, i.e., for any CPT map Φ and for all P, Q D and Tr A = 0 Thm: H g [Φ(P), Φ(Q)] H g (P, Q) g G Thm: Tr Φ(A) Ω κ Φ(P) Φ(A) Tr A Ωκ P (A) κ K Thm: D κ [Φ(P), Φ(Q)] D κ (P, Q) Thm: Φ(P) Φ(Q) 1 P Q 1 k K Only need Φ pos κ(x) = g(x)+xg(x 1 ) (1 x) 2 = gsym(x) (1 x) 2 op conv and κ(x 1 ) = xκ(x) 10 M. B. Ruskai Contraction of Quantum Channels

11 Definition of Contraction Coefficients η g (Φ) RelEnt H g [Φ(P), Φ(Q)] sup P Q D H g (P, Q) η κ (Φ) Riem sup sup P D TrA=0 Φ(A) Ω k Φ(P) Φ(A) A Ω k P (A) η κ (Φ) geod D κ [Φ(P), Φ(Q)] sup P Q D D κ (P, Q) η Tr Φ(P) Φ(Q) 1 (Φ) sup P Q D P Q 1 Remark Classical: η geod (Φ) = η Riem (Φ) = η RelEnt (Φ) η Tr (Φ) Def. H g (p, q) for any g : (0, ) R convex with g(1) = 0. Only one Fisher information. 11 M. B. Ruskai Contraction of Quantum Channels

12 General Results Recall κ(x) = g(x)+xg(x 1 ) (1 x) 2 = gsym(x) (1 x) 2 Thm: η κ (Φ) geod = η κ (Φ) Riem ηg RelEnt sym (Φ) ηg RelEnt (Φ) 1 D κ (ρ, ρ + ɛa) = follows from Hiai-Petz lim = A, Ω ɛ 0 ɛ κ ρ(a other proofs straightforward Thm: Conj: Thm: ηκ Riem (Φ) η Tr (Φ) η Riem κ (Φ) η Tr (Φ) κ False in general η Riem κ (Φ) η Tr (Φ) for κ(x) = x 1/2 12 M. B. Ruskai Contraction of Quantum Channels

13 Integral Representations g(x) = g (1)(x 1) + c(x 1) 2 (x 1) 2 + dµ(s) 0 x + s H g (ρ, γ) = c Tr (ρ γ) 2 γ Tr (ρ γ) (ρ γ) dµ(s) L ρ + sr γ κ(x) = 1 0 ( 1 x + s + 1 ) 1 + s dm(s), sx dm(s) = 1 Tr A Ω κ P (A) = 0 ( 1 Tr A + L P + sr P 1 R P + sl P ) A 1+s 2 dm(s) Often suffices to prove bounds for Tr A 1 L ρ+sr γ (A) 13 M. B. Ruskai Contraction of Quantum Channels

14 Reformulation of η κ (Φ) Riem as eigenvalue problem Use HS inner product X, Y = Tr X Y and Φ denote adjoint Eigenvalue problem Φ Ω κ Φ(P) Φ(X ) = λω κ P (X ) X, Φ Ω κ Φ(P) Φ(X ) X, Ωκ P (X ) λ 1 X = P = (Ω κ P ) 1 (I ) e-vec to largest e-val 1 Φ(A) Ω κ Φ(P) By max-min principle λ 2 (Φ, P) = sup Φ(A) Tr A=0 A Ω κ P (A) ηκ Riem (Φ) = sup λ 2 (Φ, P) P D = sup sup P D Tr A=0 Φ(A) Ω κ Φ(P) Φ(A) A Ω κ P (A) Remark: (Ω κ P ) 1 Non-Comm. mult by P BUT Ω κ P 1 (Ω κ P ) 1 = Ω 1/κ(x 1 ) P 1 κ(x) K 1/κ(x 1 ) K 14 M. B. Ruskai Contraction of Quantum Channels

15 Eigenvalue (cont.) Equiv. Prob [ (Ω κ P ) 1 Φ Ω κ Φ(P)] Φ(X ) = λ2 (Φ, P)X Recall Υ positivity and trace preserving implies Υ(Y ) 1 Y 1 Apply Υ κ P (Ωκ P ) 1 Φ Ω κ Φ(P) get λ 2(Φ, P) X 1 Φ(X ) 1 ηκ Riem (Φ) = sup λ 2 (Φ, P) η Tr (Φ) P D But Υ κ P is positive Φ, P only for κ(x) = x 1/2 = 1/κ(x 1 ) Thm: κ(x) = x 1/2 important in mixing times of Markov chains η Riem x 1/2 (Φ) η Tr (Φ) Φ Conj: η Riem κ (Φ) η Tr (Φ) False for CQ qubit channel 15 M. B. Ruskai Contraction of Quantum Channels

16 Aside on partial order for κ K and CP of Ω κ P Def: κ 1 κ 2 if κ 1 (e t )/κ 2 (e t ) has positive Fourier transform Equiv cond: matrix with els κ 1(λ j /λ k ) κ 2 (λ j /λ k is pos semi-def Positive and C.P. equiv for Ω κ P because L P R 1 P (X ) is Schur product x jk λ j λ 1 k x jk in e-vec basis Thm: a) Ω κ P is C.P. κ(x) x 1/2 b) ( Ω κ P) 1 is C.P. x 1/2 κ(x) Υ κ P (Ωκ P ) 1 Φ Ω κ Φ(P) positive Φ, P only for κ(x) = x 1/2 Hiai, Kosaki, Petz & Ruskai: analyze partial order for many families using these conds 16 M. B. Ruskai Contraction of Quantum Channels

17 Summary Figure: Diagram of families in K parameterized to increase in order with the lower ball for K + and the upper K. The red curve describes the Heinz family kα H (0, 1 2 ) and k α H ( 1 2, 1); the blue curve the binomial family k α B ( 1, 1); the green curve the power difference family k α PD ( 2, 1). The left brown curve kt WYD with t [ 1 2, 2] and the right dotted WYD 4 brown curve the dual k t. Note crossings at 1+ and log x x) 2 x 1 17 M. B. Ruskai Contraction of Quantum Channels

18 Special Results for log Ω P (A) = 0 1 P + ui A 1 P + ui du Follows from ( ) 1 1(X ΩP ) = P t XP 1 t dt d dt Pt XP 1 t = P t log ( L P R 1 P 0 ) P 1 t Thm: η Riem κ (Φ) = η RelEnt g sym (Φ) η RelEnt g (Φ) = η RelEnt (Φ) g κ(x) = log x x 1 κ(x) = 1/κ(x 1 ) = x 1 x log x g sym (x) = (x 1) log x g(x) = x log x g(x) = log x 18 M. B. Ruskai Contraction of Quantum Channels

19 Qubit channels Bloch sphere representation ρ = 1 ] 2[ I + w σ = 1 2 [I + ] k w kσ k linear TP map Φ : [ I + w σ ] I + k (τ k + α k w k )σ k ( ) rep in Pauli basis τ 1 α τ 2 0 α 2 0 = τ T τ α 3 Unital τ = 0 positivity preserving iff α k 1 k CP cond (α 1 ± α 2 ) 2 (1 ± α 3 ) 2 but not needed non-unital CQ Φ : [ I + w σ ] I + ασ 1 + τw 3 σ 3 CP and positivity conditions coincide α 2 + τ M. B. Ruskai Contraction of Quantum Channels

20 Unital qubit results η κ (Φ) geod = η κ (Φ) Riem = η RelEnt g sym = T 2 = max αj 2 j (Φ) = η RelEnt (Φ) κ, g η Tr (Φ) = T = max α j j g Same as classical case. Proofs convert action of Φ to linear op T on R 3 and exploits fact that for Γ w pos lin op on R 3 1 Ty, ΓTw sup Ty y R 3 y, Γ 1 w y T y, Γ w T y = sup. y R 3 y, Γ Tw y 20 M. B. Ruskai Contraction of Quantum Channels

21 Non-unital CQ Qubit Results [ Φ : I + w σ] I + αw1 σ 1 + τσ 3 ( κ s (x) 1+s 2 η Riem κ s (Φ) = 1 x+s sx α 2 1 ( 1 s 1+s ) 2, 0 s 1 τ 2 ), 0 s 1 extreme points of K η Riem κ depends non-trivially on κ Thm: range of s, α, τ such that second inequality is strict in ηg RelEnt s (Φ α,τ ) η(g RelEnt s) sym (Φ α,τ ) > ηκ Riem s (Φ α,τ ) 21 M. B. Ruskai Contraction of Quantum Channels

22 Non-unital CQ Qubit Example (cont.) Φ : [ I + w σ ] I + ασ 1 + τw 3 σ 3 α 2 + τ 2 1 η Tr (Φ) = α, η Riem max (Φ) = α 2 1 τ 2, η Riem (Φ) ηriem ŴY (1+ x) 2 /4x (Φ) τ 2 α2 2(1 τ 2 ), η Riem x 1/2 (Φ) α 2 1 τ 2, η(log Riem α2 1+τ x)/(x 1) (Φ) log 2τ 1 τ, η Riem WY (Φ) η Riem 4/(1+ x) 2 (Φ) = η Riem min (Φ) = α 2. 2α τ 2, 22 M. B. Ruskai Contraction of Quantum Channels

23 Implications of CQ Results Recall C.P. condition α 2 + τ 2 1 Consistent: α 2 1 τ 2 Conj: (Ruskai) ηriem κ(x)=x 1/2 (Φ) η Tr (Φ) = α η Riem κ (Φ) η Tr (Φ) k K False α > 1 τ 2 η Riem max (Φ) = Conj: (Kastoryano-Temme) False η Riem x 1/2 (Φ) α2 1 τ 2 > α = ηtr (Φ) ηκ Riem (Φ) η Riem (Φ) k K κ(x)=x 1/2 For α 2 + τ 2 = 1, first ineq is tight so α2 1 τ η Riem (Φ) = 2 x 1/2 η Riem x 1/2 (Φ) = α < 1 = η Riem max (Φ) = α2 1 τ 2 = α α2 1 τ 2 η Riem (Φ) = α < 1 x 1/2 2 (1 + α) = τ 2 α2 2(1 τ 2 ) 23 M. B. Ruskai Contraction of Quantum Channels η Riem ŴY (Φ)

24 Remark on WYD relative entropy ( H WYD(t) (P, Q) = 1 t(1 t) Tr P Tr P t Q 1 t) = 1 t(1 t) (1 Tr P t Q 1 t) η RelEnt WYD(t) (Φ) = sup 1 Tr Φ(P) t Φ(Q) 1 t P,Q D 1 Tr P t Q 1 t [ ] H g K, Φ(P), Φ(Q) Question: Does contract depend on K in sup P,Q H g ( Φ(K), P, Q) ηwyd(t) RelEnt (Φ, K) sup P Q D Tr K Φ(P)K Tr K [Φ(P)] t K[Φ(Q)] 1 t Tr Φ(K) P Φ(K) Tr Φ(K) Pt Φ(K)Q1 t Ex: tensor prod Φ(ρ AB ) = Tr B ρ AB = ρ A, Φ(KA ) = K A I B 24 M. B. Ruskai Contraction of Quantum Channels

25 References D. Petz (1986) and (1996) and others given in refs below A. Lesniewski and M. B. Ruskai, Monotone Riemannian metrics and relative entropy on noncommutative probability spaces J. Math. Phys. 40 (1999), F. Hiai and D. Petz Riemannian metrics on positive definite matrices related to means Lin. Alg. Appl. 430, (2009). F. Hiai, M. Kosaski, D. Petz and M.B. Ruskai, Operator means associated with completely positive maps Lin. Alg. Appl. 439, (2013) (arxiv: ) F. Hiai and M.B. Ruskai, Contraction coefficients for noisy quantum channels J. Math. Phys (2016) 25 M. B. Ruskai Contraction of Quantum Channels