Spectral Properties of Symmetric Quantum States and Symmetric Entanglement Witnesses

Abstract

We introduce and explore two questions concerning spectra of operators that are of interest in the theory of entanglement in symmetric (i.e., bosonic) quantum systems. First, we investigate the inverse eigenvalue problem for symmetric entanglement witnesses-that is, we investigate what their possible spectra are. Second, we investigate the problem of characterizing which separable symmetric quantum states remain separable after conjugation by an arbitrary unitary acting on symmetric space-that is, which states are separable in every orthonormal symmetric basis. Both of these questions have been investigated thoroughly in the non-symmetric setting, and we contrast the answers that we find with their non-symmetric counterparts.

Maximally entangled mixed symmetric states of two qubits

“…1 . The problem of finding the minimal eigenvalue of ρ T A S when the state |ψ is restricted in the symmetric sector |ψ ∈ H ∨2 1 was studied recently in [21], where they also reported the same characterization of SAS states as given in our Corollary 1.…”

“…Our results allow us to characterize the set of separable symmetric two-qubit states that remain separable after arbitrary SU (3) transformations in the symmetric sector, which we call symmetric absolutely separable (SAS) states. The same characterization of the SAS two-qubit states is obtained in [21] using a different technique. The questions mentioned above can be studied for the three-qubit system by applying the sufficient and necessary PPT criterion of the qubit-qutrit system.…”

“…Here, we introduce the notion of symmetric absolutely separable (SAS) states (called absolutely symmetric separable states in Ref. [21]), the set of which will be denoted by A sym . A symmetric state ρ S ∈ B(H ∨N 1 ) will be called SAS if its SU (N + 1)-orbit…”

“…Proof of Theorem 1. The minimal eigenvalue Λ min of the partial transpose of ρ S with respect to one qubit, ρ T A S , is equivalent to [13,21] Λ min = min…”

Absolutely $k$-Incoherent Quantum States and Spectral Inequalities for Factor Width of a Matrix

“…This "absolute" question has been asked for some other resource theories and in some other contexts as well. For example, it has been explored in the resource theory of symmetric separability [CJMP21,SEM21], and it has been explored in the context of the reduction map from quantum information theory [JLNR15].…”