Symmetry witnesses

Abstract

A symmetry witness is a suitable subset of the space of selfadjoint trace class operators that allows one to determine whether a linear map is a symmetry transformation, in the sense of Wigner. More precisely, such a set is invariant with respect to an injective densely defined linear operator in the Banach space of selfadjoint trace class operators (if and) only if this operator is a symmetry transformation. According to a linear version of Wigner's theorem, the set of pure states -the rank-one projections -is a symmetry witness. We show that an analogous result holds for the set of projections with a fixed rank (with some mild constraint on this rank, in the finite-dimensional case). It turns out that this result provides a complete classification of the sets of projections with a fixed rank that are symmetry witnesses. These particular symmetry witnesses are projectable; i.e., reasoning in terms of quantum states, the sets of 'uniform' density operators of corresponding fixed rank are symmetry witnesses too.

A class of stochastic products on the convex set of quantum states

“…In our setting of linear maps in B 1 (H), a symmetry transformation will be simply a map in B 1 (H) of the form specified in Corollary 1. This is of course coherent with Wigner's classical theorem [15][16][17][18][19][20], and with a linear version of this theorem [19,20], where the assumption of preservation of the transition probability between pure states becomes immaterial. Remark 5.…”

“…We stress that here the factor s = ±1 cannot depend on the choice of ψ, because the unitary (linear) operators U , V * -or the antiunitary (antilinear) operators W , Z * -generate the same symmetry transformation [16,[18][19][20], so that s must be a constant factor; see, in particular, Theorem 2 of [16]. Therefore, an isometric, adjointpreserving, surjective linear map in B 1 (H) is -possibly up to a factor s = −1 -the bonded linear extension of a symmetry transformation P(H) ∋ P → U P U * (P(H) ∋ P → W P W * ) where, by Wigner's theorem [16,[18][19][20], the unitary operator U (the antiunitary operator W ) is uniquely defined up to a phase factor. Remark 2.…”

“…The structure of pureness-preserving linear maps is determined by the following refinement of Theorem 1 of [19] (in the proofs of these results essentially the same arguments are used):…”

Wigner's type theorem in terms of linear operators which send projections of a fixed rank to projections of other fixed rank

“…This operator is induced by a linear or conjugate linear isometry if dim H = 2k; in the case when dim H = 2k, there is an additional class of operators satisfying the above conditions. This statement is a small generalization of the result by Aniello and Chruściński [1]. In this paper, it will be presented as a simple consequence of [11,Theorem 1] and some arguments from [4].…”

“…Linear operators preserving projections of fixed finite rank were investigated in [1,12,14]. Let L be a linear operator on F s (H) such that…”