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arXiv:2402.05673 · Physical Review Letters

Maximally Entangled Mixed States for a Fixed Spectrum Do Not Always Exist

Julio I. de Vicente

Published Jul 30, 2024• 11 citations
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Abstract

Entanglement is a resource under local operations assisted by classical communication (LOCC). Given a set of states S, if there is one state in S that can be transformed by LOCC into all other states in S, then this state is maximally entangled in S. It is a well-known result that the d-dimensional Bell state is the maximally entangled state in the set of all bipartite states of local dimension d. Since in practical applications noise renders every state mixed, it is interesting to study whether sets of mixed states of relevance enable the notion of a maximally entangled state. A natural choice is the set of all states with the same spectrum. In fact, for any given spectrum distribution on two-qubit states, previous work has shown that several entanglement measures are all maximized by one particular state in this set. This has led us to consider the possibility that this family of states could be the maximally entangled states in the set of all states with the same spectrum, which should then maximize all entanglement measures. In this work, I answer this question in the negative: There are no maximally entangled states for a fixed spectrum in general, i.e., for every possible choice of the spectrum. In order to do so, I consider the case of rank-2 states and show that for particular values of the eigenvalues there exists no state that can be transformed to all other isospectral states not only under LOCC but also under the larger class of nonentangling operations. This in particular implies that in these cases the state that maximizes a given entanglement measure among all states with the same spectrum depends on the choice of entanglement measure; i.e., it cannot be that the aforementioned family of states maximizes all entanglement measures.

DOI
10.1103/physrevlett.133.050202

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Entanglement is a resource under local operations assisted by classical communication (LOCC). Given a set of states S, if there is one state in S that can be transformed by LOCC into all other states in S, then this state is maximally entangled in S. It is a well-known result that the d-dimensional Bell state is the maximally entangled state in the set of all bipartite states of local dimension d. Since in practical applications noise renders every state mixed, it is interesting to study whether sets of mixed states of relevance enable the notion of a maximally entangled state. A natural choice is the set of all states with the same spectrum. In fact, for any given spectrum distribution on two-qubit states, previous work has shown that several entanglement measures are all maximized by one particular state in this set. This has led us to consider the possibility that this family of states could be the maximally entangled states in the set of all states with the same spectrum, which should then maximize all entanglement measures. In this work, I answer this question in the negative: There are no maximally entangled states for a fixed spectrum in general, i.e., for every possible choice of the spectrum. In order to do so, I consider the case of rank-2 states and show that for particular values of the eigenvalues there exists no state that can be transformed to all other isospectral states not only under LOCC but also under the larger class of nonentangling operations. This in particular implies that in these cases the state that maximizes a given entanglement measure among all states with the same spectrum depends on the choice of entanglement measure; i.e., it cannot be that the aforementioned family of states maximizes all entanglement measures.
The existence of a maximally entangled pure state is a cornerstone result of entanglement theory that has paramount consequences in quantum information theory. A natural generalization of this property is to consider whether a notion of maximal entanglement is possible among all states with the same spectrum (where the aforementioned case of pure states corresponds to the particular choice in which the spectrum is a delta distribution, i.e., rank-1 states). Despite positive evidence in the past that such a notion might exist at least in the case of two-qubit states, it was recently shown in [Phys. Rev. Lett. 133, 050202 (2024)] that the answer to the above question is negative. This reference proved this for particular choices of the spectrum in the case of rank-2 two-qubit density matrices. While this settles the problem in general, it still leaves open whether there are other choices of the spectrum outside the case of pure states where a maximally entangled state for a fixed spectrum might exist. In this work we extend this impossibility result to all rank-2 and rank-3 two-qubit states as well as for a large class of eigenvalue distributions in the case where the rank equals four.
Entanglement is a resource in quantum information theory when state manipulation is restricted to local operations assisted by classical communication (LOCC). It is therefore of paramount importance to decide which LOCC transformations are possible and, particularly, which states are maximally useful under this restriction. While the bipartite maximally entangled state is well known (it is the only state that cannot be obtained from any other and, at the same time, it can be transformed to any other by LOCC), no such state exists in the multipartite case. In order to cope with this fact, we introduce here the notion of the maximally entangled set (MES) of n-partite states. This is the set of states which are maximally useful under LOCC manipulation; i.e., any state outside of this set can be obtained via LOCC from one of the states within the set and no state in the set can be obtained from any other state via LOCC. We determine the MES for states of three and four qubits and provide a simple characterization for them. In both cases, infinitely many states are required. However, while the MES is of measure zero for 3-qubit states, almost all 4-qubit states are in the MES. This is because, in contrast to the 3-qubit case, deterministic LOCC transformations are almost never possible among fully entangled four-partite states. We determine the measure-zero subset of the MES of LOCC convertible states. This is the only relevant class of states for entanglement manipulation.
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.
We show that there exist sets of three mutually orthogonal ddd-dimensional maximally entangled states which cannot be perfectly distinguished using one-way local operations and classical communication (LOCC) for arbitrarily large values of ddd. This contrasts with several well-known families of maximally entangled states, for which any three states can be perfectly distinguished. We then show that two-way LOCC is sufficient to distinguish these examples. We also show that any three mutually orthogonal ddd-dimensional maximally entangled states can be perfectly distinguished using measurements with a positive partial transpose (PPT) and can be distinguished with one-way LOCC with high probability. These results circle around the question of whether there exist three maximally entangled states which cannot be distinguished using the full power of LOCC; we discuss possible approaches to answer this question.
Entanglement is a resource to overcome the natural restriction of operations used for state manipulation to Local Operations assisted by Classical Communication (LOCC). Hence, a bipartite maximally entangled state is a state which can be transformed deterministically into any other state via LOCC. In the multipartite setting no such state exists. There, rather a whole set, the Maximally Entangled Set of states (MES), which we recently introduced, is required. This set has on the one hand the property that any state outside of this set can be obtained via LOCC from one of the states within the set and on the other hand, no state in the set can be obtained from any other state via LOCC. Recently, we studied LOCC transformations among pure multipartite states and derived the MES for three and generic four qubit states. Here, we consider the non-generic four qubit states and analyze their properties regarding local transformations. As already the most coarse grained classification, due to Stochastic LOCC (SLOCC), of four qubit states is much richer than in case of three qubits, the investigation of possible LOCC transformations is correspondingly more difficult. We prove that most SLOCC classes show a similar behavior as the generic states, however we also identify here three classes with very distinct properties. The first consists of the GHZ and W class, where any state can be transformed into some other state non-trivially. In particular, there exists no isolation. On the other hand, there also exist classes where all states are isolated. Last but not least we identify an additional class of states, whose transformation properties differ drastically from all the other classes. Although the possibility of transforming states into local-unitary inequivalent states by LOCC turns out to be very rare, we identify those states (with exception of the latter class) which are in the MES and those, which can be obtained (transformed) non-trivially from (into) other states respectively. These investigations do not only identify the most relevant classes of states for LOCC entanglement manipulation, but also reveal new insight into the similarities and differences between separable and LOCC transformations and enable the investigation of LOCC transformations among arbitrary four qubit states.
We consider the problems of maximizing the entanglement negativity of X-form qubit-qutrit density matrices with (i) a fixed spectrum and (ii) a fixed purity. In the first case, the problem is solved in full generality whereas, in the latter, partial solutions are obtained by imposing extra spectral constraints such as rank deficiency and degeneracy, which enable a semidefinite programming treatment for the optimization problem at hand. Despite the technically motivated assumptions, we provide strong numerical evidence that threefold degenerate X states of purity PPP reach the highest entanglement negativity accessible to arbitrary qubit-qutrit density matrices of the same purity, hence characterizing a sparse family of likely qubit-qutrit maximally entangled mixed states.