2009/07/31 by Jon Tyson · 40 citations
Computer Science · Mathematics · Physics and Astronomy · #Bipartite graph #Iterated function #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum algorithm #Quantum discord #Quantum entanglement #Quantum error correction #Quantum operation #Quantum state #Simple (philosophy) #math-ph #math.MP #quant-ph
paper · pdf · doi:10.1063/1.3463451
published in Journal of Mathematical Physics 51(9) (American Institute of Physics) · Extensively revised & new content added. Improved min-entropy bounds. Notation made more accessible. Minimax theorem used to clarify relationship between "worst case" bounds and "single instance" bounds. Improved motivation of the choice of "small angle" guess. Eliminated spurious factor appearing when overlap bounds are applied to state distinction. Work connected to that of Beny and Oreshkov
arxiv created 2010/06/02 · openalex publication_date 2010/09/01 · arxiv updated 2010/09/29 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
In a unified framework, we estimate the following quantities of interest in quantum information theory: (1) the minimum-error distinguishability of arbitrary ensembles of mixed quantum states; (2) the approximate reversibility of quantum dynamics in terms of entanglement fidelity (This is referred to as “channel-adapted quantum error recovery” when applied to the composition of an encoding operation and a noise channel.); (3) the maximum overlap between a bipartite pure quantum state and a bipartite mixed-state that may be achieved by applying a local quantum operation to one part of the mixed-state; and (4) the conditional min-entropy of bipartite quantum states. A refined version of the author’s techniques [J. Tyson, J. Math. Phys. 50, 032016 (2009)] for bounding the first quantity is employed to give two-sided estimates of the remaining three quantities. We obtain a closed-form approximate reversal channel. Using a state-dependent Kraus decomposition, our reversal may be interpreted as a quadratically weighted version of that of Barnum and Knill [J. Math. Phys. 43, 2097 (2002)]. The relationship between our reversal and Barnum and Knill’s is therefore similar to the relationship between Holevo’s asymptotically optimal measurement [A. S. Kholevo, Theor. Probab. Appl. 23, 411 (1978)] and the “pretty good” measurement of Belavkin [Stochastics 1, 315 (1975)] and Hausladen and Wootters [J. Mod. Opt. 41, 2385 (1994)]. In particular, we obtain relatively simple reversibility estimates without negative matrix-powers at no cost in tightness of our bounds. Our recovery operation is found to significantly outperform the so-called “transpose channel” in the simple case of depolarizing noise acting on half of a maximally entangled state. Furthermore, our overlap results allow the entangled input state and the output target state to differ, thus obtaining estimates in a somewhat more general setting. Using a result of König et al. [IEEE Trans. Inf. Theory 55, 4337 (2009)], our maximum overlap estimate is used to bound the conditional min-entropy of arbitrary bipartite states. Our primary tool is “small angle” initialization of an abstract generalization of the iterative schemes of Ježek et al. [Phys. Rev. A 65, 060301 (2002)], Ježek et al. [Phys. Rev. A 68, 012305 (2003)], and Reimpell and Werner [Phys. Rev. Lett. 94, 080501 (2005)]. The monotonicity result of Reimpell [Ph.D. thesis, Technishe Universität, 2007] follows in greater generality.