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arXiv:2411.11035 · Open Systems & Information Dynamics

On One-Parameter Families of Hermiticity-Preserving Superoperators which Are Not Positive

Grzegorz Pastuszak, Alicja Jaworska-Pastuszak, Takeo Kamizawa

Published Mar 1, 2025• 0 citations
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Abstract

A one-parameter family of hermiticity-preserving superoperators is a time-dependent family [Formula: see text] of hermiticity-preserving superoperators determined, in a certain sense, by real and complex polynomial functions in the variable [Formula: see text]. The paper studies sufficient computable criteria for nonpositivity of superoperators in one-parameter families. More precisely, we give sufficient conditions for the following assertions to hold: [Formula: see text] every [Formula: see text] is not positive, [Formula: see text] [Formula: see text] is not positive for [Formula: see text] in some [Formula: see text] and [Formula: see text] there is some [Formula: see text] which is not positive. We show that [Formula: see text] implies [Formula: see text]. Our approach to the problem is based on the Descartes rule of signs and the Sturm-Tarski theorem. In order to apply these facts, we introduce the sign variation formulas. These formulas are first order logical formulas in one variable [Formula: see text], generalising sign sequences of polynomials used in Descartes rule of signs.

DOI
10.1142/s1230161225500040

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A one-parameter family of hermiticity-preserving superoperators is a time-dependent family [Formula: see text] of hermiticity-preserving superoperators determined, in a certain sense, by real and complex polynomial functions in the variable [Formula: see text]. The paper studies sufficient computable criteria for nonpositivity of superoperators in one-parameter families. More precisely, we give sufficient conditions for the following assertions to hold: [Formula: see text] every [Formula: see text] is not positive, [Formula: see text] [Formula: see text] is not positive for [Formula: see text] in some [Formula: see text] and [Formula: see text] there is some [Formula: see text] which is not positive. We show that [Formula: see text] implies [Formula: see text]. Our approach to the problem is based on the Descartes rule of signs and the Sturm-Tarski theorem. In order to apply these facts, we introduce the sign variation formulas. These formulas are first order logical formulas in one variable [Formula: see text], generalising sign sequences of polynomials used in Descartes rule of signs.
Assume that [Formula: see text] is a superoperator which preserves hermiticity. We give an algorithm determining whether [Formula: see text] preserves semipositivity (we call [Formula: see text] positive in this case). Our approach to the problem has a model-theoretic nature, namely, we apply techniques of quantifier elimination theory for real numbers. An approach based on these techniques seems to be the only one that allows to decide whether an arbitrary hermiticity-preserving [Formula: see text] is positive. Before we go to detailed analysis of the problem, we argue that quantifier elimination for real numbers (and also for complex numbers) can play a significant role in quantum information theory and other areas as well.
We study [Formula: see text]-positive linear maps on matrix algebras and address two problems, (i) characterizations of [Formula: see text]-positivity and (ii) generation of nondecomposable [Formula: see text]-positive maps. On the characterization side, we derive optimization-based conditions equivalent to [Formula: see text]-positivity that (a) reduce to a simple check when [Formula: see text], (b) reveal a direct link to the spectral norm of certain order-3 tensors (aligning with known NP-hardness barriers for [Formula: see text]), and (c) recast [Formula: see text]-positivity as a novel optimization problem over separable states, thereby connecting it explicitly to separability testing. On the generation side, we introduce a Lie-semigroup-based method that, starting from a single [Formula: see text]-positive map, produces one-parameter families that remain [Formula: see text]-positive and nondecomposable for small enough times. We illustrate this by generating such families for [Formula: see text] and [Formula: see text]. We also formulate a semidefinite program (SDP) to test an equivalent form of the positive partial transpose (PPT) square conjecture (and do not find any violation of the latter). Our results provide practical computational tools for certifying [Formula: see text]-positivity and a systematic way to sample [Formula: see text]-positive nondecomposable maps.
Given any separable complex Hilbert space, any trace-class operator [Formula: see text] which does not have purely imaginary trace, and any generator [Formula: see text] of a norm-continuous one-parameter semigroup of completely positive maps we prove that there exists a unique bounded operator [Formula: see text] and a unique completely positive map [Formula: see text] such that (i) [Formula: see text], (ii) the superoperator [Formula: see text] is trace class and has vanishing trace, and (iii) [Formula: see text] is a real number. Central to our proof is a modified version of the Choi formalism which relates completely positive maps to positive semi-definite operators. We characterize when this correspondence is injective and surjective, respectively, which in turn explains why the proof idea of our main result cannot extend to non-separable Hilbert spaces. In particular, we find examples of positive semi-definite operators which have empty pre-image under the Choi formalism as soon as the underlying Hilbert space is infinite-dimensional.
We present a multi-parameter family of positive maps between spaces of arbitrary but finite dimensions. This framework facilitates the construction of Entanglement Witnesses (EWs) specifically designed for systems living in [Formula: see text] dimensions. A sufficient condition for positivity is presented. Interestingly, it is shown that all EWs constructed this way are equivalent to a single realignment-like criterion which for [Formula: see text] and [Formula: see text] are different is in general stronger than the original realignment criterion. We illustrate effectiveness of this criterion considering examples of Positive Partial Transpose entangled states in [Formula: see text] and [Formula: see text] dimensions.
In this note, we mainly study the necessary and sufficient conditions for the complete positivity of generalizations of depolarizing and transpose-depolarizing channels. Specifically, we define [Formula: see text] and [Formula: see text], where [Formula: see text] (the set of all bounded linear operators on the finite-dimensional Hilbert space [Formula: see text] is given and [Formula: see text] is the transpose of [Formula: see text] in a fixed orthonormal basis of [Formula: see text] First, we show that [Formula: see text] is completely positive if and only if [Formula: see text] is a positive map, which is equivalent to [Formula: see text] Moreover, [Formula: see text] is a completely positive map if and only if [Formula: see text] and [Formula: see text] At last, we also get that [Formula: see text] is a completely positive map if and only if [Formula: see text] with [Formula: see text] for all [Formula: see text] where [Formula: see text] are eigenvalues of [Formula: see text].
This paper discusses the properties of the canonical quantum decomposition of the classical Pearson random variables. We show that this leads to the problem of representing the creation–annihilation–preservation (CAP) operators canonically associated to a real-valued random variable [Formula: see text] with all moments as (normally ordered) differential operators with polynomial coefficients — a problem already studied in the literature (see references in the introduction). We deduce a formula, for the polynomial coefficients in the representation of the CAP operators of [Formula: see text] as pseudo-differential operators, more explicit than the one existing in the literature. We give a new characterization of the Pearson distributions in terms of the Hermitianity of the associated Sturm–Liouville operators. In the second part of the paper, we introduce the notion of finite type random variable [Formula: see text] and characterize type-[Formula: see text] and type-[Formula: see text] real-valued random variables. We prove that a necessary condition for [Formula: see text] to be of finite type is the polynomial growth of the corresponding principal Jacobi sequence. This allows to single out three classes of random variables of infinite type and to prove that the Beta and the uniform distributions are of infinite type.