Capacity Estimates of TRO Channels

Capacity Estimates of TRO Channels arxiv: 1509.07294 and arxiv: 1609.08594 Li Gao University of Illinois at Urbana-Champaign QIP 2017, Microsoft Research, Seattle Joint work with Marius Junge and Nicholas LaRacuente Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 1 / 23

Quantum capacity of a channel Quantum channel (cptpm) N : B(H A ) B(H B ) Stinespring dilation: N : ρ A tr E (V ρ A V ) Complementary channel: N E : ρ A tr B (V ρ A V ) Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 2 / 23

Quantum capacity of a channel Quantum channel (cptpm) N : B(H A ) B(H B ) Stinespring dilation: N : ρ A tr E (V ρ A V ) Complementary channel: N E : ρ A tr B (V ρ A V ) How many qubits per use of N can one transmit over it? Coherent information: Q (1) (N ) = sup ρ H(B) σ H(E) σ σ BE = V ρ A V and H(B) σ = tr(σ B log σ B ). Quantum capacity [Lloyd-Shor-Devetak, 97]: Q(N ) = lim k 1 k Q(1) (N k ) N, Q (1) (N ) < Q(N ) [DiVincenzo-Shor-Smolin, 98] Regularization (the limit) is necessary but hard to compute. Qubit depolarizing channel D λ (ρ) = (1 λ)ρ + λ 1 2. Q(D λ ) in general is open. Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 2 / 23

Degradable channels N is degradable if M s.t. N E = M N [Devetak-Shor, 05] shows that if N is degradable then Q (1) (N ) = Q(N ) Hadamard channel are degradable. N (ρ) = h i ρ h j i j. Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 3 / 23

Degradable channels N is degradable if M s.t. N E = M N [Devetak-Shor, 05] shows that if N is degradable then Q (1) (N ) = Q(N ) Hadamard channel are degradable. N (ρ) = h i ρ h j i j. Question Other (nontrivial) channels which has Q (1) (N ) = Q(N )? Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 3 / 23

Some general upper bounds on Q [Smith-Smolin-Winter, 08] Quantum capacity with symmetric side channels [Tomamichel-Wilde-Winter, 15] Rains relative entropy [Sutter et al, 15] Approximate degradable channels [Wang-Duan, 15] A semidefinite programming upper bound Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 4 / 23

Some general upper bounds on Q [Smith-Smolin-Winter, 08] Quantum capacity with symmetric side channels [Tomamichel-Wilde-Winter, 15] Rains relative entropy [Sutter et al, 15] Approximate degradable channels [Wang-Duan, 15] A semidefinite programming upper bound New approach An upper bound via analysis on Stinespring dilation. Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 4 / 23

Outline 1 Introduction 2 Main Results Two examples TRO channels and their modifications Comparison Theorem 3 Further Applications Private capacity Superadditivity Strong converse rates Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 5 / 23

Example I Let G be a finite group and { g g G} be an orthonormal basis. Denote by λ(g) the unitary λ(g) h = gh. For a probability distribution f on G, N f (ρ) = g f (g)λ(g)ρλ(g). Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 6 / 23

Example I Let G be a finite group and { g g G} be an orthonormal basis. Denote by λ(g) the unitary λ(g) h = gh. For a probability distribution f on G, N f (ρ) = g f (g)λ(g)ρλ(g). max{log G H(f ), Q(N 1 )} Q(N f ) Q(N 1 ) + log G H(f ). Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 6 / 23

Example I Let G be a finite group and { g g G} be an orthonormal basis. Denote by λ(g) the unitary λ(g) h = gh. For a probability distribution f on G, N f (ρ) = g f (g)λ(g)ρλ(g). max{log G H(f ), Q(N 1 )} Q(N f ) Q(N 1 ) + log G H(f ). Not degradable if G is not commutative. Q(N 1 ) = logarithm of the largest degree of G s irreducible representations. Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 6 / 23

Example II Let 1 α 1. N α ( a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 a 41 a 42 a 43 a 44 ) = a 11 + a 22 αa 13 αa 24 αa 31 a 33 0 αa 42 0 a 44. Q (1) (N α ) = Q(N α ) = 1 h( 1 + α ), 2 where h(p) = p log p (1 p) log(1 p) is the binary entropy function. Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 7 / 23

Example II Let 1 α 1. N α ( a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 a 41 a 42 a 43 a 44 ) = a 11 + a 22 αa 13 αa 24 αa 31 a 33 0 αa 42 0 a 44. Q (1) (N α ) = Q(N α ) = 1 h( 1 + α ), 2 where h(p) = p log p (1 p) log(1 p) is the binary entropy function. Not degradable because the first 2 2 block goes to the environment. Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 7 / 23

Stinespring space N (ρ A ) = tr E (V ρ A V ) with V : H A H E H B. Definition The Stinespring space of N is defined as X N = VH A H B H E. X N = H A as Hilbert space,. N is determined by X N H B H E (up to a unitary equivalence). Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 8 / 23

Stinespring space N (ρ A ) = tr E (V ρ A V ) with V : H A H E H B. Definition The Stinespring space of N is defined as X N = VH A H B H E. X N = H A as Hilbert space,. N is determined by X N H B H E (up to a unitary equivalence). Via H B H E = B(HE, H B ), x H A V x X N x B(H E, H B ) N ( x y ) = xy, N E ( x y ) = y x. Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 8 / 23

Stinespring space (cont d) Let M n,m be n n complex matrices and M n = M n,n. a) N = id : M n M n, X = M n,1. N = tr : M n C, X = M 1,n Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 9 / 23

Stinespring space (cont d) Let M n,m be n n complex matrices and M n = M n,n. a) N = id : M n M n, X = M n,1. N = tr : M n C, X = M 1,n b) N = id n tr m : M n M m M n, X = M n,m Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 9 / 23

Stinespring space (cont d) Let M n,m be n n complex matrices and M n = M n,n. a) N = id : M n M n, X = M n,1. N = tr : M n C, X = M 1,n b) N = id n tr m : M n M m M n, X = M n,m c) Let N = k id nk tr mk be a direct sum of partial traces i.e. ρ 11 ρ 1k N (..... ρ k1 ρ kk then X = M nk,m k = ) = id n1 tr m1 (ρ 11 ) 0 0 0... 0 0 0 id nk tr mk (ρ kk )... Mnk,mk...., Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 9 / 23

Stinespring space (cont d) Let M n,m be n n complex matrices and M n = M n,n. a) N = id : M n M n, X = M n,1. N = tr : M n C, X = M 1,n b) N = id n tr m : M n M m M n, X = M n,m c) Let N = k id nk tr mk be a direct sum of partial traces i.e. ρ 11 ρ 1k N (..... ρ k1 ρ kk then X = M nk,m k = ) = id n1 tr m1 (ρ 11 ) 0 0 0... 0 0 0 id nk tr mk (ρ kk )... Mnk,mk.... This is the so-called conditional expectation. By [Fukuda-Wolf, 07] Q (1) (N ) = Q(N ) = log max n k. k, Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 9 / 23

Ternary ring of operators (TRO) X = k (M nk,m k C1 lk ) is a (finite dimensional) ternary ring of operators. Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 10 / 23

Ternary ring of operators (TRO) X = k (M nk,m k C1 lk ) is a (finite dimensional) ternary ring of operators. Definition [Hestenes, 79] A ternary ring of operators (TRO) X is a closed subspace of B(H, K) such that x, y, z X xy z X. Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 10 / 23

Ternary ring of operators (TRO) X = k (M nk,m k C1 lk ) is a (finite dimensional) ternary ring of operators. Definition [Hestenes, 79] A ternary ring of operators (TRO) X is a closed subspace of B(H, K) such that x, y, z X xy z X. We call N a TRO channel if X N is a TRO. If X N = k M nk,m k, then N = k (id nk tr mk ). Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 10 / 23

Modification of TRO channels Modification by matrix multiplication: H A X N B(H E, H B ) x V x x a Question. When this gives a channel? x x a an isometry? Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 11 / 23

Modification of TRO channels Modification by matrix multiplication: H A X N B(H E, H B ) x V x x a Question. When this gives a channel? x x a an isometry? The normalized trace τ(ρ) = 1 n tr(ρ) on M n. A positive f M n is a normalized density if τ(f ) = 1. Definition Two -subalgebra A, B M n are independent if τ(xy) = τ(x)τ(y), x A, y B. An element f M n is independent of A if the C -algebra generated by f (the closure of all polynomials of f and f ) is independent of A. Answer. When f = aa is a normalized density independent of Ran(N E ). Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 11 / 23

Modification of TRO channels (cont d) Let N be a TRO channel N ( x y ) = xy and f B(H E ) be a normalized density independent of Ran(N E ). Then the map is again a channel (cptp). N f ( x y ) = x f y Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 12 / 23

Modification of TRO channels (cont d) Let N be a TRO channel N ( x y ) = xy and f B(H E ) be a normalized density independent of Ran(N E ). Then the map is again a channel (cptp). N 1 = N. N f ( x y ) = x f y N f is a lifting of N, i.e. E N f = N where E is the conditional expectation from B(H B ) onto Ran(N ). Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 12 / 23

Comparison Theorem (main technical result) Let 1 < p and 1 p + 1 p = 1. Recall f τ,p = τ( f p ) 1 p. Theorem Let N : B(H A ) B(H B ) be a TRO channel and f a normalized density independent of Ran(N E ). For any positive ρ B(H A ) and σ Ran(N ), D p (N (ρ) σ) D p (N f (ρ) σ) D p (N (ρ) σ) + p log f τ,p Sandwiched Renyi-p divergence: {p D p (ρ σ) = log σ 1 2p ρσ 1 2p p if supp(ρ) supp(σ) + else Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 13 / 23

Sandwiched Rényi-p divergence D p ρ p = tr( ρ p ) 1 p. For ρ, σ 0, {p D p (ρ σ) = log σ 1 2p ρσ 1 2p p if supp(ρ) supp(σ) + else lim p 1 D p (ρ σ) = D(ρ σ) := tr(ρ(log ρ log σ)) Data processing inequality: D p (ρ σ) D p (N (ρ) N (σ)). (will be used) Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 14 / 23

Sandwiched Rényi-p divergence D p ρ p = tr( ρ p ) 1 p. For ρ, σ 0, {p D p (ρ σ) = log σ 1 2p ρσ 1 2p p if supp(ρ) supp(σ) + else lim p 1 D p (ρ σ) = D(ρ σ) := tr(ρ(log ρ log σ)) Data processing inequality: D p (ρ σ) D p (N (ρ) N (σ)). (will be used) Coherent information: Q (1) (N ) = sup ρ S(AA ) I c (A B) σ, where σ A B = id A N (ρ AA ) and I c (A B) ρ = H(B) H(AB) = D(ρ AB 1 ρ B ) = inf σ B D(ρ AB 1 ρ B ). Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 14 / 23

Capacity bounds Corollary Q(N ) Q(N f ) Q(N ) + τ(f log f ) In particular, Q(N ) = log max k n k if X N = k (M nk,m k ). Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 15 / 23

Capacity bounds Corollary Q(N ) Q(N f ) Q(N ) + τ(f log f ) In particular, Q(N ) = log max k n k if X N = k (M nk,m k ). Proof. Apply the comparison to (id N f ) = (id N ) f, D p (N (ρ AA ) σ) D p (N f (ρ AA ) σ) D p (N (ρ AA ) σ) + p log f τ,p Take limit p 1, choose σ = 1 N (ρ A ) and then take supremum over all bipartite states ρ AA. Q (1) (N ) Q (1) (N f ) Q (1) (N ) + τ(f log f ) Regularization: (N f ) k = (N k ) f k, τ(f k log f k ) = kτ(f log f ). Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 15 / 23

Proof of Comparison Theorem The lower bound follows from data processing inequality. E N f = N D p (N f (ρ) σ) D p (E N f (ρ) E(σ)) = D p (N (ρ) σ). Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 16 / 23

Proof of Comparison Theorem The lower bound follows from data processing inequality. E N f = N D p (N f (ρ) σ) D p (E N f (ρ) E(σ)) = D p (N (ρ) σ). Upper bound via complex interpolation. Consider the right matrix multiplication: x x a Complex interpolation (Stein s theorem). xa x a, xa 2= x 2 a 2,τ (independence). xa 2p x 2p a 2p,τ. Let ρ = ηη and f = aa. Then f p= a 2 2p and σ 1 2p N f (ρ)σ 1 2p p= σ 1 2p ηa 2 2p σ 1 2p η 2 2p a 2 2p,τ = σ 1 2p N (ρ)σ 1 2p p f τ,p Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 16 / 23

Capacity bounds Corollary Q(N ) Q(N f ) Q(N ) + τ(f log f ) In particular, if X N = k (M nk,m k ), Q(N ) = log max k n k. Remarks. Entropy with respect to τ:. lim p 1 p log f p = τ(f log f ) = log E H( 1 E f ) Local comparison: for any subspace A 0 A, Q(N A0 ) Q(N f A0 ) Q(N A0 ) + τ(f log f ). Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 17 / 23

Outline 1 Introduction 2 Main Results Two examples TRO channels and their modifications Comparison Theorem 3 Further Applications Private capacity Superadditivity Strong converse rates Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 18 / 23

Classical private capacity of a channel The private capacity of N is P(N ) = lim k 1 k P(1) (N k ), P (1) (N ) = sup ρ XA I (X :B) σ I (X :E) σ, where I (A:B) = H(A) + H(B) H(AB), and σ XBE = x p(x) x x V ρa x V. Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 19 / 23

Classical private capacity of a channel The private capacity of N is P(N ) = lim k 1 k P(1) (N k ), P (1) (N ) = sup ρ XA I (X :B) σ I (X :E) σ, where I (A:B) = H(A) + H(B) H(AB), and σ XBE = x p(x) x x V ρa x V. P (1) (N ) < P(N ) possible [Smith-Renes-Smolin, 08] Q (1) (N ) P (1) (N ) and Q(N ) P(N ) (1 ebit 1 kbit). For degradable channels, Q (1) (N ) = Q(N ) = P (1) (N ) = P(N ). [Smolin, 08] Corollary Q(N ) = P(N ) P(N f ) P(N ) + τ(f log f ) = Q(N ) + τ(f log f ) Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 19 / 23

Superadditivity [Winter-Yang, 16] introduced the potential capacities Q (p) (N ) = sup Q (1) (N M) Q (1) (M), M P (p) (N ) = sup P (1) (N M) P (1) (M). M N is strongly additive for Q (1) (resp. P (1) ) iff Q (p) (N ) = Q (1) (N ) (resp. P (p) (N ) = P (1) (N )). Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 20 / 23

Superadditivity [Winter-Yang, 16] introduced the potential capacities Q (p) (N ) = sup Q (1) (N M) Q (1) (M), M P (p) (N ) = sup P (1) (N M) P (1) (M). M N is strongly additive for Q (1) (resp. P (1) ) iff Q (p) (N ) = Q (1) (N ) (resp. P (p) (N ) = P (1) (N )). Corollary Q(N ) = Q (p) (N ) Q (p) (N f ) P (p) (N f ) P (p) (N ) + τ(f log f ) = Q(N ) + τ(f log f ) Observe that N f M = (N M) f 1. Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 20 / 23

Strong converse rates Let P (resp. Q ) denote the smallest strong converse rate for private (resp. quantum) communication. [Wilde-Tomamichel-Berta, 16] shows that for any channel P (N ) E R (N ), where E R the relative entropy of entanglement E R (N ) = sup E R (id N (ρ)), E R (ρ AB ) = inf D(ρ σ). ρ σ S(A:B) Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 21 / 23

Strong converse rates Let P (resp. Q ) denote the smallest strong converse rate for private (resp. quantum) communication. [Wilde-Tomamichel-Berta, 16] shows that for any channel P (N ) E R (N ), where E R the relative entropy of entanglement E R (N ) = sup E R (id N (ρ)), E R (ρ AB ) = inf D(ρ σ). ρ σ S(A:B) Corollary E R (N f ) E R (N ) + τ(f log f ), E R (N ) = Q(N ) and in particular Q(N ) Q (N f ) P (N f ) Q(N ) + τ(f log f ) Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 21 / 23

Final Example ρ 11 ρ 14 Write ρ =..... with ρ jk being 4 4 matrices. ρ 41 ρ 44 where f = X, Y, Z Pauli matrices. N f (ρ) = 1 j,k 4 tr(ρ jk f jk ) j k 1 α 3 Z Z iα 2 Y Y α 1 X X α 3 Z Z 1 α 1 X X iα 2 Y Y iα 2 Y Y α 1 X X 1 α 3 Z Z α 1 X X iα 2 Y Y α 3 Z Z 1. Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 22 / 23

Final Example ρ 11 ρ 14 Write ρ =..... with ρ jk being 4 4 matrices. ρ 41 ρ 44 where f = X, Y, Z Pauli matrices. N f (ρ) = 1 j,k 4 tr(ρ jk f jk ) j k 1 α 3 Z Z iα 2 Y Y α 1 X X α 3 Z Z 1 α 1 X X iα 2 Y Y iα 2 Y Y α 1 X X 1 α 3 Z Z α 1 X X iα 2 Y Y α 3 Z Z 1 Q (1) (N f ) = P (1) (N f ) = Q (N f ) = P (N f ). = Q (p) (N f ) = P (p) (N f ) = 1 h( 1 + α ). 2 where α = α 1 2 + α 2 2 + α 3 2 1 (Bloch sphere). Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 22 / 23

Summary TRO channels are orthogonal sums of partial traces. Using analysis on Stinespring space, we give capacities estimates on modifications of TRO channels. Examples wit nonzero capacity that is strongly additive, has strong converse but not degradable. Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 23 / 23

Summary TRO channels are orthogonal sums of partial traces. Using analysis on Stinespring space, we give capacities estimates on modifications of TRO channels. Examples wit nonzero capacity that is strongly additive, has strong converse but not degradable. Thanks! Marius Junge Nicholas LaRacuente Li Gao (UIUC) TRO Channels QIP 2017, Microsoft Research, Seattle 23 / 23