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arXiv:0903.1340 · Physical Review A

Concurrence and entanglement entropy of stochastic one-qubit maps

Meik Hellmund, Armin Uhlmann

Published May 15, 2009• 7 citations
arXivHTMLPDF

Abstract

Explicit expressions for the concurrence of all positive and trace-preserving (``stochastic'') one-qubit maps are presented. Our method allows to construct the relevant convex roof patterns. We conclude that two-component optimal decompositions always exist. Our results can be transferred to $2\ifmmode\times\else\texttimes\fi{}n$-quantum systems providing the concurrence for all rank-two density operators, as well as lower and upper bounds for their entanglement of formation. We apply these results to a study of the entanglement entropy of one-qubit stochastic maps which preserve axial symmetry. Using analytic and numeric results we analyze the bifurcation patterns appearing in the convex roof of optimal decompositions and give results for the one-shot (Holevo-Schumacher-Westmoreland) capacity of those maps.

DOI
10.1103/physreva.79.052319

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Explicit expressions for the concurrence of all positive and trace-preserving (``stochastic'') one-qubit maps are presented. Our method allows to construct the relevant convex roof patterns. We conclude that two-component optimal decompositions always exist. Our results can be transferred to 2\ifmmode×\else\texttimes\fin2\ifmmode\times\else\texttimes\fi{}n2\ifmmode×\else\texttimes\fin-quantum systems providing the concurrence for all rank-two density operators, as well as lower and upper bounds for their entanglement of formation. We apply these results to a study of the entanglement entropy of one-qubit stochastic maps which preserve axial symmetry. Using analytic and numeric results we analyze the bifurcation patterns appearing in the convex roof of optimal decompositions and give results for the one-shot (Holevo-Schumacher-Westmoreland) capacity of those maps.
Explicit expressions for the concurrence of all positive and trace-preserving ("stochastic") 1-qubit maps are presented. By a new method we find the relevant convex roof pattern. We conclude that two component optimal decompositions always exist. Our results can be transferred to 2×n2 \times n2×n-quantum systems providing the concurrence for all rank two density operators as well as a lower bound for their entanglement of formation.
We review some properties of the convex roof extension, a construction used, e.g., in the definition of the entanglement of formation. Especially we consider the use of symmetries of channels and states for the construction of the convex roof. As an application we study the entanglement entropy of the diagonal map for permutation symmetric real N=3 states ω(z)ω(z)ω(z) and solve the case z<0z<0z<0 where zzz is the non-diagonal entry in the density matrix. We also report a surprising result about the behaviour of the output entropy of the diagonal map for arbitrary dimensions NNN; showing a bifurcation at N=6.
We review some properties of the convex roof extension, a construction used, e.g., in the definition of the entanglement of formation. Especially we consider the use of symmetries of channels and states for the construction of the convex roof. As an application we study the entanglement entropy of the diagonal map for permutation symmetric real N=3N=3N=3 states ω(z)\omega(z)ω(z) and solve the case z<0z<0z<0 where zzz is the non-diagonal entry in the density matrix. We also report a surprising result about the behavior of the output entropy of the diagonal map for arbitrary dimensions NNN; showing a bifurcation at N=6N=6N=6.
We consider the minimal entropy of qubit states transmitted through two uses of a noisy quantum channel, which is modeled by the action of a completely positive trace-preserving (or stochastic) map. We provide strong support for the conjecture that this minimal entropy is additive, namely, that the minimum entropy can be achieved when product states are transmitted. Explicitly, we prove that for a tensor product of two unital stochastic maps on qubit states, using an entanglement that involves only states which emerge with minimal entropy cannot decrease the entropy below the minimum achievable using product states. We give a separate argument, based on the geometry of the image of the set of density matrices under stochastic maps, which suggests that the minimal entropy conjecture holds for nonunital as well as for unital maps. We also show that the maximal norm of the output states is multiplicative for most product maps on n-qubit states, including all those for which at least one map is unital. For the class of unital channels on C/sup 2/, we show that additivity of minimal entropy implies that the Holevo (see IEEE Trans. Inform. Theory, vol.44, p.269-73, 1998 and Russian Math. Surv., p.1295-1331, 1999) capacity of the channel is additive over two inputs, achievable with orthogonal states, and equal to the Shannon capacity. This implies that superadditivity of the capacity is possible only for nonunital channels.
Abstract We study the closed expressions of the convex roof coherence measures for one-qubit states in this paper. We present the analytical expressions for the convex roof coherence measures, C f ( ρ ) , of one-qubit states with C f ( φ ) := f ( c 0 2 , c 1 2 ) (where | φ ⟩ = c 0 | 0 ⟩ + c 1 | 1 ⟩ ) being convex with respect to the l 1 norm of coherence of ϕ (i.e. C l 1 ( φ ) ), such coherence measures including the coherence of formation, the geometric measure of coherence, the coherence concurrence, and the coherence rank. We further present the operational interpretations of these measures. Finally, we present the usefulness of the convex roof coherence measures C f ( φ ) being non-convex with respect to C l 1 ( φ ) by giving the necessary and sufficient conditions for the transformations p φ 1 ⊕ ( 1 − p ) φ 2 → q ϕ 1 ⊕ ( 1 − q ) ϕ 2 via incoherent operations, where ϕ i , φ j ( i , j = 1 , 2 ) are one-qubit pure states and 0 ⩽ p , q ⩽ 1 .
We establish a general operational one-to-one mapping between coherence measures and entanglement measures: Any entanglement measure of bipartite pure states is the minimum of a suitable coherence measure over product bases. Any coherence measure of pure states, with extension to mixed states by convex roof, is the maximum entanglement generated by incoherent operations acting on the system and an incoherent ancilla. Remarkably, the generalized cnot gate is the universal optimal incoherent operation. In this way, all convex-roof coherence measures, including the coherence of formation, are endowed with (additional) operational interpretations. By virtue of this connection, many results on entanglement can be translated to the coherence setting, and vice versa. As applications, we provide tight observable lower bounds for generalized entanglement concurrence and coherence concurrence, which enable experimentalists to quantify entanglement and coherence of the maximal dimension in real experiments.