Universal observable detecting all two-qubit entanglement and determinant-based separability tests

Abstract

We construct a single observable measurement of which mean value on four copies of an unknown two-qubit state is sufficient for unambiguous decision whether the state is separable or entangled. In other words, there exists a universal collective entanglement witness detecting all two-qubit entanglement. The test is directly linked to a function which characterizes to some extent the entanglement quantitatively. This function is an entanglement monotone under so-called local pure operations and classical communication (pLOCC) which preserve local dimensions. Moreover it provides tight upper and lower bounds for negativity and concurrence. Elementary quantum computing device estimating unknown two-qubit entanglement is designed.PACS numbers: 03.65.-wIntroduction .-One of the main challenges of both theoretical and experimental Quantum Information Theory is a determination of entanglement properties of a given state. There is an extensive literature covering the problem of deciding entanglement of a state [1,2,3,4,5,6,7]. As one knows from the seminal paper of Peres and Wootters [8] collective measurement on several copies of a system in a given quantum state may provide better results than measurements performed on each copy separately. This fact was reflected in the method of entanglement detection with collective measurements. The method initiated for pure states [9, 10], then developed for mixed states with help of quantum networks [11,12,13,14,15,16,17] and the concept of collective entanglement witnesses [18], has found its first experimental demonstration in coalescence-anti coalescence coincidence experiment [19]. In particular, somewhat surprisingly, it was shown how to estimate and/or even measure amount of entanglement (concurrence) without prior state reconstruction [11,12,13]. Recently the method got the new twist thanks to application of such collective measurements [20,21,22,23] that are directly related to quantum concurrence (see [24]) including photon polarization-momentum experimental demonstration for pure states in distant laboratories paradigm [20]. Recently collective entanglement witnesses were also shown to lead to easily measurable lower bounds on entanglement [21]. The idea of collective entanglement witnesses was also implemented in continuous variables setup [22].We show that a single observable if measured on four copies of a unknown two-qubit state is sufficient for discrimination between entanglement and separability of it. Moreover it can serve for limited quantitative purposes. To this aim we explore the two-qubit separability test (equivalent to the PPT one [2,25]) stating that a state is separable iff the determinant of its partially transposed density matrix is nonnegative [26,27]. The result, known for a few years, was barely mentioned in the literature in that form (see e.g. [28]) and up to our knowledge this is the first time an operative physical meaning is assigned

Moment-based evidence for simple rational-valued Hilbert–Schmidt generic 2 × 2 separability probabilities

“…(The nonnegativity of |ρ P T |-as a corollary of the celebrated Peres-Horodeccy results [5,6]-constitutes a necessary and sufficient condition for separability/disentanglement, when ρ is a 4 × 4 density matrix [17,18].) At this point of our presentation, we note that three of these seventeen extensions are expressible-incorporating as the last factors on their right-hand sides, the two formulas above ((1), (2))-as …”

“…[32]). Also, Dunkl has raised the issue of whether or not nonnegativity of the determinant of the partial transpose is equivalent to separability, as it is known to be in the two-rebit and two-qubit cases [17].) The lower estimate based on 2,325 moments is 0.080495355 (which is 1.000000000049 times the corresponding estimate based on 2,324 moments).…”

Entanglement detection

“…The quantity det(̺ TB ) is a fourth order polynomial of the matrix elements of ̺ and can hence be measured by a single witness on a fourfold copy of ̺ [491]. Similarly, for rotationally invariant states on a 2 × d-system, the PPT criterion is necessary and sufficient for separability [492], and if d is even, Eq.…”

Quantum properties and dynamics ofXstates

“…In [22] this is shown with complete generality, here we show it for X states by elementary means. Notice that if for instance z ≥ √ ad then from (5) one has √ bc ≥ z and √ ad ≥ w which automatically implies that √ bc ≥ w. The argument is reversed if w ≥ √ bc.…”