Maximally entangled mixed states for a fixed spectrum do not always exist

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 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 non-entangling 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.

arXiv comment5 pages

References in corpus (29)

• Quantum entanglement
• Entanglement of Formation of an Arbitrary State of Two Qubits
• Mixed State Entanglement and Quantum Error Correction
• A computable measure of entanglement
• Quantifying Entanglement
• Quantum Resource Theories
• Conditions for a class of entanglement transformations
• Entanglement monotones
• Information-theoretic aspects of quantum inseparability of mixed states
• Optimal Creation of Entanglement Using a Two--Qubit Gate
• Entanglement of pure states for a single copy
• Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask)
• An improved bound on distillable entanglement
• Maximally entangled mixed states of two qubits
• Maximally entangled mixed states in two qubits
• A rigorous treatment of distillable entanglement
• Minimal conditions for local pure-state entanglement manipulation
• Quantifying entanglement resources
• Entanglement Theory and the Second Law of Thermodynamics
• Entanglement capabilities of non-local Hamiltonians
• How robust is a quantum gate in the presence of noise?
• The maximally entangled set of multipartite quantum states
• Approximate transformations and robust manipulation of bipartite pure state entanglement
• A reversible theory of entanglement and its relation to the second law
• A resource theory of entanglement with a unique multipartite maximally entangled state
• No second law of entanglement manipulation after all
• On a gap in the proof of the generalised quantum Stein's lemma and its consequences for the reversibility of quantum resources
• Optimal local preparation of an arbitrary mixed state of two qubits. Closed expression for the single copy case
• Measurement outcomes that do not occur and their role in entanglement transformations

Cited by in corpus (5)

• Catalytic and asymptotic equivalence for quantum entanglement
• General Concurrence Percolation on Quantum Networks
• Squeezing generation crossing a mean-field critical point: Work statistics, irreversibility and critical fingerprints
• State convertibility under genuinely incoherent operations
• Nonexistence of maximally entangled mixed states for a fixed spectrum