Optimal Entanglement Criterion for Mixed Quantum States

Abstract

We develop a strong and computationally simple entanglement criterion. The criterion is based on an elementary positive map Φ which operates on state spaces with even dimension N ≥ 4. It is shown that Φ detects many entangled states with positive partial transposition (PPT) and that it leads to a class of optimal entanglement witnesses. This implies that there are no other witnesses which can detect more entangled PPT states. The map Φ yields a systematic method for the explicit construction of high-dimensional manifolds of bound entangled states.PACS numbers: 03.67. Mn,03.65.Ud,03.65.Yz Entanglement and quantum inseparability are key features of quantum mechanics which are connected to the tensor product structure of the state spaces of composite systems. A mixed state ρ of a bipartite system, for instance, is defined to be separable or classically correlated if it can be written as a convex linear combination of uncorrelated product states, i. e., if it can be represented in the form ρ = i p i ρ i 1 ⊗ρ i 2 , where {p i } is a probability distribution and the ρ i 1 , ρ i 2 are density matrices describing states of the first and the second subsystem, respectively [1]. States which cannot be written in this way are called inseparable or entangled. Much effort in quantum information theory has been devoted to the problems of the characterization, the classification and the quantification of mixed state entanglement [2,3]. Although considerable progress has been made in recent years (see, e. g., Refs.[4]), we are still far away from a true understanding of many aspects of these problems.A problem of central importance in entanglement theory is the development of computationally efficient criteria which allow us to decide whether or not a given state is entangled. Peres [5] has developed a very strong entanglement criterion which is known as criterion of positive partial transposition (PPT). It states that a necessary condition for a given state ρ to be separable is that its partial transpose is a positive operator. Usually, one writes this condition as (I ⊗ T )ρ ≥ 0, where T denotes the transposition of operators in a chosen basis and I is the identity map, indicating that the transposition is carried out only on the second part of the composite system. The PPT condition represents a necessary and sufficient separability criterion for certain low-dimensional systems [6], but it is only necessary in higher dimensions. Hence, there are generally entangled PPT states which belong to the class of bound entangled states [7].The transposition T is a distinguished example of a positive but not completely positive map. This means that T maps all positive operators on the subsystems to positive operators, while there exist states ρ of the combined system for which (I ⊗T )ρ has negative eigenvalues. There are many other maps with this property. The significance of positive maps in entanglement theory is provided by a fundamental theorem established in Ref. [6]. This theorem states that a necessary and sufficient conditio...

Many-body physics from a quantum information perspective

“…However, unlike the reduction map, this one is indecomposable as examples of PPT entangled states detected by Λ U ext can be found [37,38]. Let us summarize our considerations with the following two theorems.…”

Entanglement of formation and concurrence for mixed states

“…Another lower bound of EOF for bipartite states on even dimensional Hilbert spaces N has been presented [29] from a new separability criterion [30]. On even dimensional spaces there exist antisymmetric unitary operations V T 5 2V.…”

“…In Ref. [29], another lower bound on EOF for bipartite states has been presented from a new separability criterion [30]. A lower bound on concurrence based on a local uncertainty relations (LURs) criterion is obtained in Ref.…”

Necessary and sufficient conditions for bipartite entanglement

“…This approach gives no insight into bound entangled states remaining positive after PT. To overcome this limitation, to the matrices of moments other kinds of PCNP maps have been applied [10], including Kossakowsky, Choi and Breuer maps [11,12,13]. The identification of bound entanglement in this way, however, turned out to be a cumbersome problem.…”