Quantum Hadamard channels (II) Vlad Gheorghiu Department of Physics Carnegie Mellon University Pittsburgh, PA 15213, USA August 5, 2010 Vlad Gheorghiu (CMU) Quantum Hadamard channels (II) August 5, 2010 1 / 14
Outline 1 Brief review of part I 2 Trade-off capacity regions 3 Single letterization of the CQ and CE trade-off curves for Hadamard channels 4 Examples 5 References Reference: Brádler et al, Phys Rev A 81, 062312 (2010) A summary of this talk (both parts) is available online at http://quantumphyscmuedu/qip Vlad Gheorghiu (CMU) Quantum Hadamard channels (II) August 5, 2010 2 / 14
Brief review of part I Summary of part I Noisy quantum channel, Kraus representation N (σ) = N i σn i, N i N i = I i i (1) Isometric extension N A B (σ) = Tr E {U N σu N }, U N U N = I (2) The Kraus operators provide a straightforward method for constructing an isometric extension U A BE N = i N A B i i E (3) Vlad Gheorghiu (CMU) Quantum Hadamard channels (II) August 5, 2010 3 / 14
Brief review of part I Complementary channel, unique up to isometries on the system E N c (σ) := Tr B {U N σu N } (4) Useful fact: a valid complementary channel for the channel introduced before N c (σ) = i,j Tr{N i σn j } i j E (5) Degradable channel: if there exists a degrading channel D B E that simulates the action of the complementary channel (N c ) A E, ie D B E st σ, D B E N A B (σ) = (N c ) A E (σ) (6) Vlad Gheorghiu (CMU) Quantum Hadamard channels (II) August 5, 2010 4 / 14
Brief review of part I Entanglement-braking channels: N A B is entanglement-breaking if and only if N A B ( Φ AA ) = p X (x)ρ A x ρ B x, (7) x where Φ AA is MES The action of an entanglement-breaking channel can always be written as N EB (ρ) = Tr{Λ x ρ}σ x, (8) x where {Λ x } is a POVM and the states σ x depend on the channel A Hadamard channel is a quantum channel whose complementary channel is entanglement breaking Vlad Gheorghiu (CMU) Quantum Hadamard channels (II) August 5, 2010 5 / 14
Trade-off capacity regions Information-theoretic quantities von-neumann entropy H(ρ) := Tr{ρ log 2 ρ} Notation H(A) ρ := H(ρ A ) Conditional entropy H(A B) := H(AB) H(B) Mutual information I (A; B) := H(A) H(A B) Coherent information I (A B) := H(A B) Vlad Gheorghiu (CMU) Quantum Hadamard channels (II) August 5, 2010 6 / 14
Trade-off capacity regions Classically enhanced quantum capacity region (CQ) Consider a protocol that exploits a noisy quantum channel N to transmit both classical and quantum information The protocol transmits one classical message from a set of M messages and an arbitrary quantum state of dimension K using a large number n uses of the quantum channel The classical rate is C := log 2 (M) n quantum rate is Q := log 2 (K) n bits per channel use, and the qubits per channel use If there is a scheme that transmits classical data at rate C with vanishing error probability and quantum data at rate Q with fidelity approaching unity in the limit of large n, we say that the rates C and Q form an achievable rate pair (C, Q) Vlad Gheorghiu (CMU) Quantum Hadamard channels (II) August 5, 2010 7 / 14
Trade-off capacity regions Entanglement-assisted classical capacity region (CE) The protocol transmits one classical message from a set of M messages using a large number n uses of the quantum channel and a MES of Schmidt rank D shared between the sender and the receiver The classical rate is C := log 2 (M) n entanglement consumption is E := log 2 (D) n bits per channel use, and the rate of ebits per channel use If there is a scheme that transmits classical data at rate C with vanishing error probability and consumes entanglement at rate E in the limit of large n we say that the rates C and E form an achievable rate pair (C, E) Vlad Gheorghiu (CMU) Quantum Hadamard channels (II) August 5, 2010 8 / 14
Trade-off capacity regions Capacity region for entanglement-assisted transmission of classical and quantum information (CQE) The natural generalization of the previous two scenarios The goal is to transmit as much classical information with vanishing error probability and quantum information with fidelity approaching unity while consuming as little entanglement as possible in the limit of large number of uses of the channel N The classical rate is C := log 2 (M) n rate is Q := log 2 (K) n bits per channel use, the quantum qubits per channel use, and the rate of entanglement consumption is E := log 2 (D) n The achievable rates are denoted by (C, Q, E) ebits per channel use Vlad Gheorghiu (CMU) Quantum Hadamard channels (II) August 5, 2010 9 / 14
Single letter formulas CQ trade-off curve In general, the trade-off curves do not admit a simple single-letter formula Things are simpler for Hadamard channels CQ parametrization: Theorem For any fixed λ 1, the function f λ (N ) := max I (X ; B) ρ + λi (A BX ) ρ (9) ρ leads to a point (I (X ; B) ρ, I (A BX ) ρ ) on the CQ trade-off curve Here the maximization is taken over classical-quantum state of the form ρ XABE := x p X (x) x x X U A BE N (ϕ AA x ), (10) where the states ϕ AA x are pure Vlad Gheorghiu (CMU) Quantum Hadamard channels (II) August 5, 2010 10 / 14
Single letter formulas CE trade-off curve Theorem (CE parametrization) For 0 λ 1, the function g λ (N ) := max I (AX ; B) ρ λh(a X ) ρ (11) ρ leads to a point (I (AX ; B) ρ, H(A X ) ρ ) on the CE trade-off curve Here the maximization is taken over classical-quantum state of the form ρ XABE := x p X (x) x x X U A BE N (ϕ AA x ), (12) where the states ϕ AA x are pure Theorem: Suppose CQ and CE trade-off curves of a quantum channel single-letterize Then the full CQE capacity region of N single-letterizes! Vlad Gheorghiu (CMU) Quantum Hadamard channels (II) August 5, 2010 11 / 14
Examples Qubit dephasing channel The qubit p-dephasing channel: N (σ) = (1 p)σ + p (σ), (σ) = 1 (σ + ZσZ) (13) 2 For µ [0, 1/2] define γ(µ, p) := 1 2 + 1 2 1 16 p 2 (1 p )µ(1 µ) (14) 2 CQ trade-off curve: (1 H 2 (µ), H 2 (µ) H 2 [γ(µ, p)]) (15) CE trade-off curve: (1 + H 2 (µ) H 2 [γ(µ, p)], H 2 (µ)) (16) Vlad Gheorghiu (CMU) Quantum Hadamard channels (II) August 5, 2010 12 / 14
Examples Figure: Trade-off capacities: from Brádler et al, Phys Rev A 81, 062312 (2010) Vlad Gheorghiu (CMU) Quantum Hadamard channels (II) August 5, 2010 13 / 14
References References CQ capacity region: Devetak and Shor, Commun Math Phys 256, 287 (2005) CE capacity region: Shor, in Quantum Information, Statistics, Probability, edited by O Hirota (Rinton Press, Princeton, NJ, 2004), pp 144-152, e-print arxiv: quant-ph/0402129 CQE capacity region: Hsieh and Wilde, IEEE Trans Inf Theory, vol 56, no 9, September 2010 (to be published), e-print arxiv:08114227 [quant-ph] Vlad Gheorghiu (CMU) Quantum Hadamard channels (II) August 5, 2010 14 / 14