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arXiv:quant-ph/0409009 · Physical Review A

Comparison of the relative entropy of entanglement and negativity

Adam Miranowicz, Satoshi Ishizaka, Bohdan Horst, Andrzej Grudka

Published Nov 5, 2008• 30 citations
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Abstract

It is well known that for two qubits the upper bounds of the relative entropy of entanglement (REE) for a given concurrence as well as the negativity for a given concurrence are reached by pure states. We show that, by contrast, there are two-qubit mixed states for which the REE for some range of a fixed negativity is higher than that for pure states. Moreover, we demonstrate that a mixture of a pure entangled state and pure separable state orthogonal to it is likely to give the maximal REE. By noting that the negativity is a measure of entanglement cost under operations preserving positivity of partial transpose, our results provide an explicit example of operations such that, even though the entanglement cost for an exact preparation is the same, the entanglement of distillation of a mixed state can exceed that of pure states. This means that the entanglement manipulation via a pure state can result in a larger entanglement loss than that via a mixed state.

DOI
10.1103/physreva.78.052308

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It is well known that for two qubits the upper bounds of the relative entropy of entanglement (REE) for a given concurrence as well as the negativity for a given concurrence are reached by pure states. We show that, by contrast, there are two-qubit mixed states for which the REE for some range of a fixed negativity is higher than that for pure states. Moreover, we demonstrate that a mixture of a pure entangled state and pure separable state orthogonal to it is likely to give the maximal REE. By noting that the negativity is a measure of entanglement cost under operations preserving positivity of partial transpose, our results provide an explicit example of operations such that, even though the entanglement cost for an exact preparation is the same, the entanglement of distillation of a mixed state can exceed that of pure states. This means that the entanglement manipulation via a pure state can result in a larger entanglement loss than that via a mixed state.
Amplitude damping changes entangled pure states into usually less-entangled mixed states. We show, however, that even local amplitude damping of one or two qubits can result in mixed states more entangled than pure states if one compares the relative entropy of entanglement (REE) for a given degree of the Bell--Clauser-Horne-Shimony-Holt inequality violation (referred to as nonlocality). By applying Monte Carlo simulations, we find the maximally entangled mixed states and show that they are likely to be optimal by checking the Karush-Kuhn-Tucker conditions, which generalize the method of Lagrange multipliers for this nonlinear optimization problem. We show that the REE for mixed states can exceed that of pure states if the nonlocality is in the range (0,0.82) and the maximal difference between these REEs is 0.4. A former comparison [Phys. Rev. A 78, 052308 (2008)] of the REE for a given negativity showed analogous property but the corresponding maximal difference in the REEs is one order smaller (i.e., 0.039) and the negativity range is (0,0.53) only. For appropriate comparison, we normalized the nonlocality measure to be equal to the standard entanglement measures, including the negativity, for arbitrary two-qubit pure states. We also analyze the influence of the phase-damping channel on the entanglement of the initially pure states. We show that the minimum of the REE for a given nonlocality can be achieved by this channel, contrary to the amplitude-damping channel.
In this paper we investigate two different entanglement measures in the case of mixed states of two qubits. We prove that the negativity of a state can never exceed its concurrence and is always larger than √[(1 − C ) 2 + C 2 ] − (1 − C ), where C is the concurrence of the state. Furthermore, we derive an explicit expression for the states for which the upper or lower bound is satisfied. Finally we show that similar results hold if the relative entropy of entanglement and the entanglement of formation are compared.
We study the ordering of two-qubit states with respect to the degree of bipartite entanglement using the Wootters concurrence---a measure of the entanglement of formation---and the negativity---a measure of the entanglement cost under the positive-partial-transpose-preserving operations. For two-qubit pure states, the negativity is the same as the concurrence. However, we demonstrate analytically on simple examples of various mixtures of Bell and separable states that the entanglement measures can impose different orderings on the states. We show which states, in general, give the maximally different predictions (i) when one of the states has the concurrence greater but the negativity smaller than those for the other state and (ii) when the states are entangled to the same degree according to one of the measures, but differently according to the other.
We prove that the binegativity is always positive for any two-qubit state. As a result, and as suggested by previous work, the asymptotic relative entropy of entanglement in two qubits does not exceed the Rains bound, and the positive partial transposed-entanglement cost for any two-qubit state is determined to be the logarithmic negativity of the state. Further, the proof reveals some geometrical characteristics of the entangled states, and shows that partial transposition can give another separable approximation of the entangled state in two qubits.
The long-standing problem of finding a closed formula for the relative entropy of entanglement (REE) for two qubits is addressed. A compact-form solution to the inverse problem, which characterizes an entangled state for a given closest separable state, is obtained. Analysis of the formula for a large class of entangled states strongly suggests that a compact analytical solution of the original problem, which corresponds to finding the closest separable state for a given entangled state, can be given only in some special cases. A few applications of the compact-form formula are given to show additivity of the REE, to relate the REE with the Rains upper bound for distillable entanglement, and to show that a Bell state does not have a unique closest separable state.
Numerous work had been done to quantify the entanglement of a two-qubit quantum state, but it can be seen that previous works were based on joint measurements on two copies or more than two copies of the quantum state under consideration. In this work, we show that a single copy and two measurements are enough to estimate the entanglement quantifier like entanglement negativity and concurrence. To achieve our goal, we establish a relationship between the entanglement negativity and the minimum eigenvalue of the structural physical approximation of the partial transpose of an arbitrary two-qubit state. The derived relation makes it possible to estimate entanglement negativity experimentally by Hong-Ou-Mandel interferometry with only two detectors. Also, we derive the upper bound of the concurrence of an arbitrary two-qubit state and have shown that the upper bound can be realized in experiment. We will further show that the concurrence of i) an arbitrary pure two-qubit state and ii) a particular class of mixed states, namely, rank-2 quasi-distillable mixed states, can be exactly estimated with two measurements.