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arXiv:quant-ph/0505137 · Physical Review A

Distillation protocols that involve local distinguishing: Composing upper and lower bounds on locally accessible information

Aditi Sen, Ujjwal Sen, Maciej Lewenstein

Published Nov 22, 2006• 13 citations
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Abstract

We find a universal lower bound on locally accessible information for arbitrary bipartite quantum ensembles, when one of the parties is two dimensional. In higher dimensions and in a higher number of parties, the bound is on accessible information by separable operations. We show that for any given density matrix (of an arbitrary number of parties and dimensions), there exists an ensemble, which averages to the given density matrix and whose locally accessible information saturates the lower bound. Moreover, we give a general method to obtain bounds on the yield of singlets in distillation protocols that involves local distinguishing, by using lower and upper bounds on locally accessible information. We then illustrate it by using our lower bound, along with a previously obtained upper bound, on locally accessible information.

DOI
10.1103/physreva.74.052332

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We find a universal lower bound on locally accessible information for arbitrary bipartite quantum ensembles, when one of the parties is two dimensional. In higher dimensions and in a higher number of parties, the bound is on accessible information by separable operations. We show that for any given density matrix (of an arbitrary number of parties and dimensions), there exists an ensemble, which averages to the given density matrix and whose locally accessible information saturates the lower bound. Moreover, we give a general method to obtain bounds on the yield of singlets in distillation protocols that involves local distinguishing, by using lower and upper bounds on locally accessible information. We then illustrate it by using our lower bound, along with a previously obtained upper bound, on locally accessible information.
We find a universal lower bound on locally accessible information for arbitrary bipartite ensembles, when one of the parties is two-dimensional. In higher dimensions and in higher number of parties, the lower bound is on accessible information by separable operations. We evaluate this bound for several ensembles, for which the exact locally accessible information is not known. We show that for any given density matrix (of arbitrary number of parties and dimensions), there exists an ensemble, the so-called ``Scrooge ensemble'', which averages to the given density matrix and whose locally accessible information saturates the lower bound.
We analyze the problem of distillation of entanglement of mixed states in higher-dimensional compound systems. Employing the positive maps method [M. Horodecki et al., Phys. Lett. A 223, 1 (1996)] we introduce and analyze a criterion of separability that relates the structures of the total density matrix and its reductions. We show that any state violating the criterion can be distilled by suitable generalization of the two-qubit protocol that distills any inseparable two-qubit state. In particular, this means that any state $\ensuremath{\varrho}$ of two N-level systems with $〈{\ensuremath{\psi}}_{+}|\ensuremath{\varrho}|{\ensuremath{\psi}}_{+}〉>1/N$ can be distilled $({\ensuremath{\psi}}_{+}$ is the singlet state generalized to higher dimension). The criterion also singles out all the states that can be distilled by a class of protocols. The proof involves construction of the family of states that are invariant under transformation $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\varrho}}U\ensuremath{\bigotimes}{U}^{*}\ensuremath{\varrho}{U}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\bigotimes}{U}^{*\ifmmode\dagger\else\textdagger\fi{}},$ where U is a unitary transformation and the asterisk denotes complex conjugation. The states are related to the depolarizing channel generalized to the nonbinary case.
In the topic of perfect local distinguishability of orthogonal multipartite quantum states, most results obtained so far pertain to bipartite systems whose subsystems are of specific dimensions. In contrast very few results for bipartite systems whose subsystems are of arbitrary dimensions, are known. This is because a rich variety of (algebraic or geometric) structure is exhibited by different sets of orthogonal states owing to which it is difficult to associate some common property underlying them all, i.e., a common property that would play a crucial role in the local distinguishability of these states. In this Rapid Communication, I propose a framework for the distinguishability by one-way local operations and classical communication (1-LOCC) of sets of orthogonal bipartite states in a ${d}_{A}\ensuremath{\bigotimes}{d}_{B}$ bipartite system, where dA,dB{d}_{A},{d}_{B}dA​,dB​ are the dimensions of both subsytems, labeled as AAA and BBB. I show that if the ithi\mathrm{th}ith party (where i=A,Bi=A,Bi=A,B) can initiate a 1-LOCC protocol to perfectly distinguish among a set of orthogonal bipartite states, then the information of the existence of such a 1-LOCC protocol lies in a subspace of ${d}_{i}\ifmmode\times\else\texttimes\fi{}{d}_{i}$ Hermitian matrices, denoted by ${\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$, and that the method to extract this information (of the existence of this 1-LOCC protocol) from ${\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$ depends on the value of $\mathrm{dim}{\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$. In this way one can give sweeping results for the 1-LOCC (in)distinguishability of all sets of orthogonal bipartite states corresponding to certain values of $\mathrm{dim}{\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$. Thus I propose that the value of $\mathrm{dim}{\mathcal{T}}_{\ensuremath{\perp}}^{(i)}$ gives the common underlying property based on which sweeping results for the 1-LOCC (in)distinguishability of orthogonal bipartite quantum states can be made.
We present a lower bound of concurrence for arbitrary-dimensional bipartite quantum states. This lower bound may be used to improve all the known lower bounds of concurrence. Moreover, the lower bound gives rise to an operational sufficient criterion of distillability of quantum entanglement. The significance of our result is illustrated by quantitative evaluation of entanglement for entangled states that fail to be identified by the usual concurrence estimation method and by showing the distillability of mixed states that cannot be recognized by other distillability criteria.