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arXiv:quant-ph/0412210 · Open Systems & Information Dynamics

Simplifying Monotonicity Conditions for Entanglement Measures

Michał Horodecki

Published Aug 24, 2005• 43 citations
arXivHTMLPDF

Abstract

We show that for a convex function the following, rather modest conditions, are equivalent to monotonicity under local operations and classical communication. The conditions are: (i) invariance under local unitaries, (ii) invariance under adding local ancilla in arbitrary state (iii) on mixtures of states possessing local orthogonal flags the function is equal to its average. The result holds for multipartite systems. It is intriguing that the obtained conditions are equalities. The only inequality is hidden in the condition of convexity.

DOI
10.1007/s11080-005-0920-5

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We show that for a convex function the following, rather modest conditions, are equivalent to monotonicity under local operations and classical communication. The conditions are: (i) invariance under local unitaries, (ii) invariance under adding local ancilla in arbitrary state (iii) on mixtures of states possessing local orthogonal flags the function is equal to its average. The result holds for multipartite systems. It is intriguing that the obtained conditions are equalities. The only inequality is hidden in the condition of convexity.
Much of the theory of entanglement concerns the transformations that are possible to a state under local operations with classical communication (LOCC); however, this set of operations is complicated and difficult to describe mathematically. An idea which has proven very useful is that of the entanglement monotone: a function of the state which is invariant under local unitary transformations and always decreases (or increases) on average after any local operation. In this paper we look on LOCC as the set of operations generated by infinitesimal local operations, operations which can be performed locally and which leave the state little changed. We show that a necessary and sufficient condition for a function of the state to be an entanglement monotone under local operations that do not involve information loss is that the function be a monotone under infinitesimal local operations. We then derive necessary and sufficient differential conditions for a function of the state to be an entanglement monotone. We first derive two conditions for local operations without information loss, and then show that they can be extended to more general operations by adding the requirement of convexity. We then demonstrate that a number of known entanglement monotones satisfy these differential criteria. Finally, as an application, we use the differential conditions to construct a new polynomial entanglement monotone for three-qubit pure states. It is our hope that this approach will avoid some of the difficulties in the theory of multipartite and mixed-state entanglement.
Entanglement monotone is defined as a convex measure of entanglement that does not increase on average under local operations and classical communication (LOCC). Here we call an entanglement monotone a strict entanglement monotone (SEM) if it decreases strictly on average under LOCC. We show that, for any convex roof extended entanglement monotone that on pure states is given by a function of the reduced states, if the function is strictly concave, then it is a SEM. Moreover, we prove that the negativity and the relative entropy of entanglement, which are not defined by the convex roof structure, are also SEMs. In addition, if the squashed entanglement could be obtained by some optimal extension, then it is a SEM as well. Our results imply that entanglement is strictly decreasing on average under LOCC.
The conditions for transforming pure entangled states under local operations and classical communication (LOCC) are well understood. A natural question then arises: Can we determine the transformation conditions for mixed entangled states under LOCC based on the properties of their pure-state ensembles? While much effort has been devoted to this issue, in this paper, we rule out this possibility. Our findings address several open questions, including: (i) The conditions Efcr(ρ)≥Efcr(σ) E_f^{cr}(ρ) \geq E_f^{cr}(σ) Efcr​(ρ)≥Efcr​(σ) for all convex roof entanglement measures EfcrE_f^{cr}Efcr​ is insufficient to guarantee the existence of an LOCC transformation ΛL(⋅)Λ^L(\cdot)ΛL(⋅) from ρρρ to σσσ; and (ii) The inequalities ∑jpjE(φj)≥∑lqlE(ψl)\sum_j p_j E(\varphi_j) \geq \sum_l q_l E(ψ_l)∑j​pj​E(φj​)≥∑l​ql​E(ψl​) for all entanglement monotones EEE are not sufficient to ensure the existence of an LOCC transformation from {pj,∣φj⟩}\{p_j, \ket{\varphi_j}\}{pj​,∣φj​⟩} to {ql,∣ψl⟩}\{q_l, \ket{ψ_l}\}{ql​,∣ψl​⟩}.
We propose general conditions for a measure of total correlations as an entanglement monotone using its pure and mixed convex-roof extension. In doing so, we derive crucial theorems and propose a concrete candidate for a total correlation measure which is a multi-additive multipartite entanglement monotone. We perform numerical simulations that show, comparatively, the dependence of the proposed coherent total correlation measure on the size of the register for several different quantum states and for global and pairwise correlation measures. We also show how to simplify some of the conditions required for monotonicity, in particular that a function must be non-increasing under local operations and classical communications.
Local operations assisted by classical communication (LOCC) play a fundamental role in the resource-theoretic formulation of entanglement theory inducing an operationally meaningful ordering in the set of entangled states. The particular class within LOCC of local unitary (LU) transformations is obviously closed under inversion. Thus, LU-equivalent states are interconvertible by LOCC and have therefore the same entanglement. For this reason LU-equivalence has been thoroughly studied in the context of entanglement theory. However, as I note here with a simple example, the converse is not true: there exist entangled states that are interconvertible by LOCC but which are not LU-equivalent. This motivates studying under which conditions LOCC interconvertibility is characterized by LU-equivalence. I provide a sufficient condition for this to hold. When one considers multipartite states with full entanglement dimensionality (as quantified by the Schmidt number vector) LOCC interconvertibility can only hold under LU-equivalence.
If we want to transform the quantum state of a system to another using local measurement processes, what is the probability of success? This probability is bounded by quantifying entanglement in both the states. In this paper, we construct a family of local unitary invariants of multipartite states that are monotonic under local operations and classical communication on average. These monotones are constructed from local unitary invariant polynomials of the state and its conjugate, and hence are easy to compute for pure states. Using these measures we bound the success probability of transforming a given state into another state using local quantum operations and classical communication.