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arXiv:1310.7038

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

Samuel R. Hedemann

Published Oct 25, 2013• 14 citations
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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.

DOI
10.48550/arxiv.1310.7038

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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 NNN-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 2×32\times 32×3, including the proposition of the 2×32\times 32×3 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.
Strong evidence is presented suggesting that all two-qubit mixed and pure 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 the entanglement of a general state 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 the 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 2x3, including the proposition of the 2x3 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 utilizing the EPU-equivalence between general states and TGX-states.
A multipartite entanglement measure called the ent is presented and shown to be an entanglement monotone, with the special property of automatic normalization. Necessary and sufficient conditions are developed for constructing maximally entangled states in every multipartite system such that they are true-generalized X states (TGX) states, a generalization of the Bell states, and are extended to general nonTGX states as well. These results are then used to prove the existence of maximally entangled basis (MEB) sets in all systems. A parameterization of general pure states of all ent values is given, and proposed as a multipartite Schmidt decomposition. Finally, we develop an ent vector and ent array to handle more general definitions of multipartite entanglement, and the ent is extended to general mixed states, providing a general multipartite entanglement measure.
The monogamy of entanglement is generally discussed using a bipartite entanglement measure as an upper bound. Here we discuss a new kind of monogamous relation where the upper bound is given by a multipartite measure of entanglement, the generalized concurrence. We show a monogamous equality involving the multipartite concurrence, all the bipartite concurrences, and the genuine tripartite entanglement for pure three-qubit systems. The result extends to mixed states in an inequality involving the generalized concurrence and all the bipartite concurrences. We provide a counterexample showing that the result cannot be extended for systems with more than three qubits.
We consider mixed states of two qubits and show under which global unitary operations their entanglement is maximized. This leads to a class of states that is a generalization of the Bell states. Three measures of entanglement are considered: entanglement of formation, negativity, and relative entropy of entanglement. Surprisingly all states that maximize one measure also maximize the others. We give a complete characterization of these generalized Bell states and prove that these states for fixed eigenvalues are all equivalent under local unitary transformations. Furthermore we characterize all nearly entangled states closest to the maximally mixed state and derive a lower bound on the volume of separable mixed states.
Local unitary equivalence is an important ingredient for quantifying and classifying entanglement. Verifying whether or not two quantum states are local unitary equivalent is a crucial problem, where only the case of multipartite pure states is solved. For mixed states, however, the verification of local unitary equivalence is still a challenging problem. In this paper, based on the coefficient matrices of generalized Bloch representations of quantum states, we find a variety of local unitary invariants for arbitrary-dimensional bipartite quantum states. These invariants are operational and can be used as necessary conditions for verifying the local unitary equivalence of two quantum states. Furthermore, we extend the construction to the arbitrary-dimensional multipartite case. We finally apply these invariants to estimate concurrence, a vital entanglement measure, showing the practicability of local unitary invariants in characterizing entanglement.