Evidence that All States Are Unitarily Equivalent to X States of the Same Entanglement

Abstract

Strong numerical evidence is presented suggesting that all two-qubit mixed states are equivalent to X states by a single entanglement-preserving unitary (EPU) transformation, so that the concurrence of such an X state equals that of the original general state. An X-state parameterization of a general two-qubit state is given, allowing all states to have their concurrence parametrically specified. A new kind of entanglement measure is proposed, relating a general state's entanglement to that of a pure state in the same system. New states called "H States" are presented, having fully parametric concurrence and purity, with the intention of using them to construct entanglement-preserving depolarization channels, which may aid development of the new entanglement measure. A theory of "true-generalized" X states (TGX states) is proposed for the general case of $N$-partite systems. While such states do not generally have the literal "X" shape, evidence is shown that they are the true generalizations of X states in larger systems, since they appear to always be EPU-equivalent to general states of all ranks, whereas literal X states generally are not. An example of this is given for $2\times 3$, including the proposition of the $2\times 3$ maximally entangled mixed states (MEMS). If the claim that TGX states are universal is valid, then any entanglement measure may be computable in a simpler form by using the EPU-equivalence between general states and TGX states.

21 pages 9 figures. v2: Term "local-unitary" replaced by "entanglement-preserving unitary" (EPU) since product-form not nec. to preserve X-state entanglement. v3: Formally defined EPU, added proof for rank-2 family and Appendix proving EPUs can be nonloc. for X states, and EPU-rotated X states have enough DOF for gen. mixed states. v4: Success on 10,000 consecutive states. Basic results unchanged

References in corpus (1)

• X-matrices provide a lower bound of concurrence

Cited by in corpus (9)

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• Separability and entanglement of the qudit X-state with j = 3/2
• New Minkowski type inequalities and entropic inequalities for quantum states of qudits
• Maximally genuine multipartite entangled mixed X-states of N-qubits
• Heuristic for estimation of multiqubit genuine multipartite entanglement
• Separability and entanglement in the Hilbert space reference frames related through the generic unitary transform for four level system
• Inequalities for purity parameters for multipartite and single qudit states