Entanglement universality of two-qubit X-states

Abstract

We demonstrate that for every two-qubit state there is a X-counterpart, i.e., a corresponding two-qubit X-state of same spectrum and entanglement, as measured by concurrence, negativity or relative entropy of entanglement. By parametrizing the set of two-qubit X-states and a family of unitary transformations that preserve the sparse structure of a two-qubit X-state density matrix, we obtain the parametric form of a unitary transformation that converts arbitrary two-qubit states into their X-counterparts. Moreover, we provide a semi-analytic prescription on how to set the parameters of this unitary transformation in order to preserve concurrence or negativity. We also explicitly construct a set of X-state density matrices, parametrized by their purity and concurrence, whose elements are in one-to-one correspondence with the points of the concurrence versus purity (CP) diagram for generic two-qubit states.Comment: 24 pages, 6 figures. v2 includes new references and minor changes (accepted version

Maximally entangled mixed states for qubit-qutrit systems

“…In the next section, we explore the results of Theorems 1 and 2 to show that, unlike it occurs for two-qubit X states [21,23], it is not generally possible to map an arbitrary 2 × 3 state into a 2 × 3 X state of same negativity and purity by means of a unitary transformation, this time supporting the conjecture in Ref. [21].…”

“…This simple observation reveals the set of 2 × 3 X states to be a "less universal" set than that of 2 × 2 X states, since any 2 × 2 state can be mapped into an X state via an entanglementpreserving unitary transformation; a property that has been recently entitled "entanglement universality of twoqubit X states" [21,23].…”

“…[21], Hedemann proposed that it may be possible to unitarily transform any two-qubit state into an X state of the same entanglement; a conjecture that was later confirmed in Ref. [23] (with entanglement measured by concurrence, negativity, or relative entropy of entanglement). Furthermore, Ref.…”

“…That is for two reasons: First, from a technical viewpoint, the sparsity of X states allows the relevant maximizations of negativity to be carried out analytically in many circumstances. Second, from the aforementioned results on two-qubit systems [16,17,19,20] we learned that 2 × 2 MEMS can 2 always be written in X form (in fact, any 2 × 2 state can be unitarily transformed into an X state of the same entanglement [21,23]), and so the hypothesis that 2 × 3 MEMS will also fall within this subset feels like a reasonable place to start. Notwithstanding, since this is ultimately an unconfirmed hypothesis, we name X-MEMS (as opposed to simply MEMS) the optimal density matrices resulting from optimization problems constrained to output X states.…”

Formulas for Generalized Two-Qubit Separability Probabilities

“…, 1465 analysis, a slope estimate of −0.864025, again converging in the direction of log(27/64). (Let us remark, regarding the generalized two-qubit version of the [simpler, lower-dimensional] X-states model [20,30,31], that it has been shown that the slope of a [now, log-log] plot of log (prob(| PT | > 0)) versus log tends to −1/2, as → ∞. )…”