Different quantum f-divergences and the reversibility of quantum operations

Different quantum f-divergences and the reversibility of quantum operations Fumio Hiai Tohoku University 2016, September (at Budapest) Joint work with Milán Mosonyi 1 1 F.H. and M. Mosonyi, Different quantum f-divergences and the reversibility of quantum operations, arxiv:1604.03089. Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 1 / 31

Plan Different quantum divergences - Standard f-divergences S f - Maximal f-divergences Ŝ f - Measured f-divergences S meas f - Various Rényi divergences Reversibility via quantum divergences - Reversibility via S f - Reversibility via Ŝ f - Reversibility via Rényi divergences Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 2 / 31

Different quantum divergences Notations H is a finite-dimensional Hilbert space. B(H) is the algebra of linear operators on H. B(H) + := {A B(H) : A 0}. B(H) ++ := {A B(H) : A > 0, i.e., positive invertible}. For A B(H) +, A 1 is the generalized inverse of A, and A 0 is the support projection of A. Tr is the usual trace functional on B(H). The Hilbert-Schmidt inner product is X, Y := Tr X Y, X, Y B(H). The left and the right multiplications of A B(H) are L A X := AX, R A X := XR, X B(H). If A, B B(H) +, then L A and R B are positive operators on B(H) with L A R B = R B L A. Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 3 / 31

Different quantum divergences Throughout the talk, assume that f : (0, ) R is an operator convex function. f admits the unique integral expression f(x) = f(1) + f (1)(x 1) + c(x 1) 2 (x 1) 2 + dµ f (s), x (0, ) x + s [0, ) with c 0 and a positive measure µ f on [0, ) satisfying [0, ) (1 + s) 1 dµ f (s) < +. Set f(0 + ) := lim x 0 f(x), in (, + ]. f(x) f ( ) := lim f (x) = lim x x x Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 4 / 31

Different quantum divergences Different quantum divergences: Standard f-divergences S f Definition For A, B B(H) ++ with spectral decompositions A = a Sp(A) ap a and B = b Sp(B) bq b, the (standard) f-divergence is S f (A B) := B 1/2, f(l A R B 1)B 1/2 = Tr B 1/2 f(l A R B 1)(B 1/2 ) with f(l A R B 1) = f(ab 1 )L Pa R Qb, a Sp(A) b Sp(B) which can be extended to general A, B B(H) + as S f (A B) := lim ε 0 S f (A + εi B + εi). The f-divergences are special cases of Petz s quasi-entropies. 2 2 D. Petz, Quasi-entropies for finite quantum systems, Rep. Math. Phys. 23 (1986), 57 65. Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 5 / 31

Properties of S f 3 4 Different quantum divergences Joint convexity For every A i, B i B(H) + and λ i 0, 1 i k, ( k k ) k S f λ i A i λ i B i λ i S f (A i B i ). i=1 i=1 Monotonicity or Data-processing inequality Assume that Φ : B(H) B(K) is a trace-preserving linear map such that the adjoint Φ : B(K) B(H) satisfies Φ (Y Y) Φ (Y )Φ (Y), Y B(K). For every A, B B(H) +, i=1 S f (Φ(A) Φ(B)) S f (A B). 3 A. Lesniewski and M.B. Ruskai, Monotone Riemannian metrics and relative entropy on noncommutative probability spaces, J. Math. Phys. 40 (1999), 5702 5724. 4 F.H., M. Mosonyi, D. Petz and C. Bény, Quantum f-divergences and error correction, Rev. Math. Phys. 23 (2011), 691 747. Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 6 / 31

Different quantum divergences Maximal f-divergences Ŝ f 5 6 Definition For A, B B(H) ++ define Ŝ f (A B) := Tr B f(b 1/2 AB 1/2 ) = Tr B 1/2 f(b 1/2 AB 1/2 )B 1/2, which can be extended to general A, B B(H) + as Ŝ f (A B) := lim ε 0 Ŝ f (A + εi B + εi). 5 D. Petz and M.B. Ruskai, Contraction of generalized relative entropy under stochastic mappings on matrices, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1 (1998), 83 89. 6 K. Matsumoto, A new quantum version of f-divergence, Preprint, 2014; arxiv:1311.4722. Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 7 / 31

Different quantum divergences Properties of Ŝ f Joint convexity Ŝ f (A B) is jointly convex in A, B B(H) +. More strongly, (A, B) B(H) ++ B(H) ++ B 1/2 f(b 1/2 AB 1/2 )B 1/2 is jointly operator convex. 7 Monotonicity For any trace-preserving positive linear map Φ : B(H) B(K) and A, B B(H) +, Ŝ f (Φ(A) Φ(B)) Ŝ f (A B). 7 E. Effros and F. Hansen, Non-commutative perspectives, Ann. Funct. Anal. 5 (2014), 74 79; arxiv:1309.7701. Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 8 / 31

Different quantum divergences Examples (1) Quadratic function When f(x) = x 2, S x 2(A B) = Tr A 2 B 1 = Ŝ x 2(A B). Hence, when f(x) = ax 2 + bx + c with a 0, S f (A B) = Ŝ f (A B) for all A, B B(H) +. (2) Log function When f(x) = x log x, the Umegaki relative entropy is S x log x (A B) = S(A B) := Tr A(log A log B) and the Belavkin-Staszewski relative entropy is Ŝ x log x (A B) = S BS (A B) := Tr A log(a 1/2 B 1 A 1/2 ). Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 9 / 31

Different quantum divergences (3) Power functions For α [ 1, 2] \ {0, 1} consider f (α) (x) := s(α)x α 1 if α [ 1, 0) (1, 2],, where s(α) := 1 if α (0, 1). Then S f (α)(a B) = s(α)tr A α B 1 α, Ŝ f (α)(a B) = s(α)tr B 1/2 (B 1/2 AB 1/2 ) α B 1/2. When 0 < α < 1, we rewrite Ŝ f (α)(a B) = Tr B # α A, where # α denotes the weighted geometric mean. Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 10 / 31

Inequality Different quantum divergences For every A, B B(H) +, Strict inequality S f (A B) Ŝ f (A B). In fact, Ŝ f is maximal among monotone quantum f-divergences (due to Matsumoto 5 ). Strict inequality If f(0 + ) < + and supp µ f 2(dim H) 2, then S f (A B) < Ŝ f (A B) for every non-commuting A, B B(H) + with A 0 B 0. Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 11 / 31

Different quantum divergences Particular cases Assume that A, B B(H) + are non-commuting with A 0 B 0. For f(x) = x log x, S(A B) < S BS (A B). S S BS was first shown in 8. For f(x) = x α with 0 < α < 1, Tr B # α A < Tr B 1 α A α, first proved in 9. For f(x) = x α with 1 < α < 2, Note For α > 2, Tr B 1 α A α < Tr B 1/2 (B 1/2 AB 1/2 ) α B 1/2. Tr B 1 α A α > Tr B 1/2 (B 1/2 AB 1/2 ) α B 1/2. 8 F.H. and D. Petz, The proper formula for relative entropy and its asymptotics in quantum probability, Comm. Math. Phys. 143 (1991), 99 11. 9 F.H., Equality cases in matrix norm inequalities of Golden-Thompson type, Linear and Multilinear Algebra 36 (1994), 239 249. Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 12 / 31

Different quantum divergences Measured f-divergences S meas f A measurement (or POVM) on H is M = (M x ) x X, where X a finite set, M x B(H) +, which gives a CPTP map M : B(H) C X, M(A) := (Tr AM x )δ x. Definition For A, B B(H) + define x X M x = I, S meas (A B) := sup { S f f (M(A) M(B)) : M measurement on H }, S pr (A B) := sup{ S f f (M(A) M(B)) : M projective measurement x X on H }. Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 13 / 31

Different quantum divergences By definition, S meas (Φ(A) Φ(B)) S meas (A B) holds, and S meas f f f minimal among monotone quantum f-divergences. Consequently, S pr f all coincide if AB = BA. (A B) Smeas(A B) S f (A B) Ŝ f (A B); f Note S pr = S meas is known for many f, while it is unknown whether f f for all f. = S meas f If f(0 + ) < + and supp µ f Sp(L A R B 1) + dim H, then S pr f S pr f (A B) < S f(a B) for every non-commuting A, B B(H) + with A 0 B 0. Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 14 / 31 is

Definition For α, z > 0 with α 1, Different quantum divergences Various Rényi divergences The (conventional) Rényi divergence is D α (A B) := 1 α 1 log Tr Aα B 1 α Tr A. The sandwiched Rényi divergence 10 is D 1 α (A B) := α 1 1 α Tr (B 2α AB 1 α 2α ) α log. Tr A 10 M. Müller-Lennert, F. Dupuis, O. Szehr, S. Fehr and M. Tomamichel, On quantum Rényi entropies: A new generalization and some properties, J. Math. Phys. 54 (2013), 122203. Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 15 / 31

Different quantum divergences Definition (Cont.) The α-z-rényi divergence 11 12 is D α,z (A B) := 1 Tr (B log α 1 Note D α = D α,1, D α = D α,α and 1 α 2z A α 1 α z B 2z ) z. Tr A D (A B) = 2 log F(A, B) + 2 log Tr A, 1/2 where F(A, B) := Tr A 1/2 B 1/2 is the fidelity, lim α 1 D α (A B) = S(A B) := Tr A(log A log B), lim α D α (A B) = D max(a B) := inf{γ : A e γ B}. 11 V. Jaksic, Y. Ogata, Y. Pautrat and C.-A. Pillet, Entropic fluctuations in quantum statistical mechanics. An Introduction, in: Quantum Theory from Small to Large Scales, August 2010, in: Lecture Notes of the Les Houches Summer School, vol. 95, Oxford University Press, 2012. 12 K.M.R. Audenaert and N. Datta, α-z-rényi relative entropies, J. Math. Phys. 56 (2015), 022202. Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 16 / 31

Different quantum divergences Monotonicity or Data-processing inequality For each (α, z) in the blue region below, Tr (B 1 α 2z concave for α < 1 and jointly convex for α > 1, and the monotonicity D α,z (Φ(A) Φ(B)) D α,z (A B) holds for any TPCP map Φ. A α z B 1 α 2z ) z is jointly z 1 DPI is conjectured for the red region. 0 1 2 α Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 17 / 31

Different quantum divergences Monotonicity or Data-processing inequality (Cont.) - 0 < α < 1 and z max{α, 1 α} (F.H., 2013), - 1 < α 2 and z = 1 (Ando, 1979), - α > 1 and z = α (Müller-Lennert et al. 10, Frank and Lieb 13, Beigi 14, Wilde, Winter and Yang 15 ) - 1 < α 2 and z = α/2 (Carlen, Frank and Lieb 16 ). 13 R.L. Frank and E.H. Lieb, Monotonicity of a relative Rényi entropy, J. Math. Phys. 54 (2013), 122201. 14 S. Beigi, Sandwiched Rényi divergence satisfies data processing inequality, J. Math. Phys. 54 (2013), 122202. 15 M.M. Wilde, A. Winter and D. Yang, Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Rényi relative entropy, Comm. Math. Phys. 331 (2014), 593 622. 16 E.A. Carlen, R.L. Frank and E.H. Lieb, Some operator and trace function convexity theorems, Linear Algebra Appl. 490 (2016), 174 185. Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 18 / 31

Different quantum divergences Proposition For any A, B B(H) +, when α (0, 1/2], D α (A B) Dpr(A B) = Dmeas(A B) D α (A B). When α [1/2, + ), D pr α α (A B) = Dmeas(A B) D α (A B) D α(a B). α α In particular, D pr (A B) = Dmeas (A B) = 1/2 1/2 D (A B) 1/2 = 2 log F(A, B) + 2 log Tr A. Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 19 / 31

Reversibility via quantum divergences Reversibility via quantum divergences Reversibility problem Let D be a quantum divergence and A, B B(H) +. Let Φ : B(H) B(K) be a stochastic map (typically, a TPCP map). If D(Φ(A), Φ(B)) = D(A, B) < +, then, is there a reverse stochastic map Ψ : B(K) B(H) such that Ψ(Φ(A)) = A and Ψ(Φ(B)) = B? Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 20 / 31

Reversibility via quantum divergences A very classical counterpart (Halmos-Savage, 1949; Kullback-Leibler, 1951) Let (Ω, A) be a measurable space, B a σ-subalgebra of A, and µ, ν be probability measures on A with µ ν. Let E : L (Ω, A, ν) L (Ω, B, ν) be the conditional expectation w.r.t. ν. Then B is sufficient for µ, ν (equivalently, E is reversible for µ, ν) the Radon-Nykodym derivative dµ dν d(µ B ) d(ν B ) = dµ dν S(µ B ν B ) = S(µ ν). is B-measurable, i.e., Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 21 / 31

Reversibility via quantum divergences We assume: Reversibility via S f f(0 + ) < +, so f extends to an operator convex function on [0, ). Φ : B(H) B(K) is a 2-positive trace-preserving map. A, B B(H) + with A 0 B 0. Define a unital 2-positive map Φ B : B(H) B(K) as Φ B (X) := Φ(B) 1/2 Φ(B 1/2 XB 1/2 )Φ(B) 1/2, X B(H), whose adjoint Φ B : B(K) B(H) is Φ B (Y) = B1/2 Φ (Φ(B) 1/2 YΦ(B) 1/2 )B 1/2, Y B(K). Note that Φ (Φ(B)) = B. B Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 22 / 31

Reversibility via quantum divergences Combining our previous paper (2011) 4 and Jenčová (2012) 17, Theorem The following conditions are equivalent: (i) Φ is reversible for A, B. (ii) Φ (Φ(A)) = A. B (Φ (Φ(B)) = B is automatic.) B (iii) S f (Φ(A) Φ(B)) = S f (A B) for some f satisfying supp µ f Sp(L A R B 1) Sp(L Φ(A) R Φ(B) 1). (iv) S f (Φ(A) Φ(B)) = S f (A B) for all f. (v) Φ (Φ(B) 1/2 Φ(A)Φ(B) 1/2 ) = B 1/2 AB 1/2. (B 1/2 AB 1/2 is regarded as a kind of non-commutative Radon-Nikodym derivative of A w.r.t. B.) (vi) B 1/2 AB 1/2 F Φ Φ B, the fixed-point subalgebra of Φ Φ B. 17 A. Jenčová, Reversibility conditions for quantum operations, Rev. Math. Phys. 24 (2012), 1250016, 26 pp. Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 23 / 31

Reversibility via quantum divergences Reversibility via Ŝ f The multiplicative domain of Φ B is M ΦB := {X B(H) : Φ B (XY) = Φ B (X)Φ B (Y), Φ B (YX) = Φ B (Y)Φ B (X), Y B(H)} = {X B(H) : Φ B (X X) = Φ B (X) Φ B (X), Φ B (XX ) = Φ B (X)Φ B (X) }. Associated with an operator monotone function h 0 on (0, ) with h(1) = 1, the operator mean σ h (in the sense of Kubo-Ando, 1980) is A σ h B := A 1/2 h(a 1/2 BA 1/2 )A 1/2, A, B B(H) ++. We say that σ h is non-linear if h is non-linear. Note Φ(A σ B) Φ(A) σ Φ(B) holds for any operator mean σ and any general positive map Φ (essentially due to Ando, 1979). Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 24 / 31

Reversibility via quantum divergences The next theorem gives equivalent conditions for equality case of Ŝ f under Φ. The implication (a) (g) was shown by Matsumoto 6. Theorem With the same assumption on Φ, A, B as above, the following conditions are equivalent: (a) Ŝ f (Φ(A) Φ(B)) = Ŝ f (A B) for some non-linear f. (b) Ŝ f (Φ(A) Φ(B)) = Ŝ f (A B) for all f. (c) Tr Φ(A) 2 Φ(B) 1 = Tr A 2 B 1. (d) Φ(A) σ Φ(B) = Φ(A σ B) for some non-linear operator mean σ. (e) Φ(A) σ Φ(B) = Φ(A σ B) for all operator means σ. (f) Φ(A)Φ(B) 1 Φ(A) = Φ(AB 1 A). (g) B 1/2 AB 1/2 M ΦB. Conditions (a) (g) are usually strictly weaker than (i) (vi) of the previous theorem. Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 25 / 31

Reversibility via quantum divergences Reversibility via sandwiched Rényi divergences Jenčová 18 recently proved Theorem Let Φ : B(H) B(K) be a 2-positive trace-preserving map, 1 < α <, and A, B B(H) + with A 0 B 0. Then Φ is reversible for A, B if and only if D α (Φ(A) Φ(B)) = D α (A B). 18 A. Jenčová, Characterization of sufficient quantum channels by a Rényi relative entropy, arxiv:1604.02831. Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 26 / 31

Reversibility via quantum divergences Another recent result 19 is Theorem Let Φ : B(H) B(K) be a CPTP map, α [1/2, ) \ {1}, γ := 1 α 2α, and A, B B(H) + with A 0 B 0 if α > 1. Then if and only if D α (Φ(A) Φ(B)) = D α (A B) B γ (B γ AB γ ) α 1 B γ = Φ ( Φ(B) γ [ Φ(B) γ Φ(A)Φ(B) γ ] α 1 Φ(B) γ). 19 F. Leditzky, C. Rouzé and N. Datta, Date processing for the sandwiched Rényi divergence: a condition for equality, arxiv:1604.02119. Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 27 / 31

Theorem Reversibility via quantum divergences Reversibility via α-z-rényi divergence Assume that Φ : B(H) B(H) is a 2-positive bistochastic map and D α,z (Φ(A) Φ(B)) = D α,z (A B). Then Φ is reversible for A, B if one of the following is satisfied: - α z 1, Φ(B) = B and A 0 B 0 (without 2-positivity), - α < z 1, Φ(B) = B and B 0 A 0, - 0 < 1 α z 1, Φ(A) = A and B 0 A 0 (without 2-positivity), - 0 < 1 α < z 1, Φ(A) = A and A 0 B 0, - α z max{1, α/2}, Φ(B) = B and A 0 B 0 (without 2-positivity), - α > 1, z 1, z > α 1, Φ(A) = A and B 0 = I. Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 28 / 31

Reversibility via quantum divergences Corollary Assume that Φ : B(H) B(H) is 2-positive bistochastic map and D α (Φ(A) Φ(B)) = D α (A B). Then Φ is reversible for A, B if each of the following is satisfied: - Φ(B) = B and A 0 B 0 (for arbitrary α (0, ) \ {1} without 2-positivity), - 1/2 α < 1, Φ(A) = A and B 0 A 0 (without 2-positivity), - 1/2 < α < 1, Φ(A) = A and A 0 B 0. Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 29 / 31

Reversibility via quantum divergences Note When α = z = 1/2 (the case of fidelity), F(Φ(A), Φ(B)) = F(A, B) implies the reversibility of Φ for A, B when Φ(B) = B and A 0 B 0 (or Φ(A) = A and B 0 A 0 ). [This is not true without the assumption Φ(B) = B (or Φ(A) = A) 20.] The reversibility via D max does not hold even when Φ(B) = B and A 0 = B 0. 20 M. Mosonyi and T. Ogawa, Quantum hypothesis testing and the operational interpretation of the quantum Rényi relative entropies, Comm. Math. phys. 334 (2015), 1617 1648. Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 30 / 31

Reversibility via quantum divergences Thank you! Fumio Hiai (Tohoku University) Quantum f-divergences and reversibility 2016, September (at Budapest) 31 / 31