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Linear and Multilinear Algebra

A quadratic operator equation and a non-commutative uncertainty relation

Fateme Abdollahzade Gonabadi, Hamed Najafi

Published Jan 24, 2025• 0 citations
Publisher

Abstract

In this paper, we explore equivalent conditions for the positivity of a quadratic equation P(x)=Ax2+Bx+C with operator coefficients. Specifically, we show that Ax2+2Bx+C≥0 for all x∈R if and only if A,C≥0, B=B∗, and there exists a self-adjoint operator H such that [A+C+HB+i(A−C)B+i(A−C)A+C−H]≥0.Also, we show that if A and B are two unital C∗-algebras, Φ:A→B is a unital tracial positive linear map, and ϕ:A→A is a linear map satisfying ϕ(X)∗ϕ(X)≤ϕ(X∗X) for all X∈A, then VΨ(A)♯VΨ(B)≥±12Ψ([A,B]),where A and B are self-adjoint elements in A, Ψ=Φ∘ϕ, and VΨ(X)=Ψ(X∗X)−Ψ(X∗)Ψ(X) denotes the generalized variance of an observable X.

DOI
10.1080/03081087.2024.2446783

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In this paper, we explore equivalent conditions for the positivity of a quadratic equation P(x)=Ax2+Bx+C with operator coefficients. Specifically, we show that Ax2+2Bx+C≥0 for all x∈R if and only if A,C≥0, B=B∗, and there exists a self-adjoint operator H such that [A+C+HB+i(A−C)B+i(A−C)A+C−H]≥0.Also, we show that if A and B are two unital C∗-algebras, Φ:A→B is a unital tracial positive linear map, and ϕ:A→A is a linear map satisfying ϕ(X)∗ϕ(X)≤ϕ(X∗X) for all X∈A, then VΨ(A)♯VΨ(B)≥±12Ψ([A,B]),where A and B are self-adjoint elements in A, Ψ=Φ∘ϕ, and VΨ(X)=Ψ(X∗X)−Ψ(X∗)Ψ(X) denotes the generalized variance of an observable X.
Abstract Real Hilbert space is a Euclidean space of infinite dimensionality. Complex Hilbert space is less amenable to elementary geometrical intuitions, but these do serve to give structure to the present argument. The familiar process of finding, relative to a vector x, the expectation Ā of a self-adjoint linear operator A, and then using the deviation A — Ā to obtain a dispersion ΔA, may be interpreted as a projection leading to a unit vector e orthogonal to x. If we have n self-adjoint operators Ar, we thus form n unit vectors er orthogonal to x. From a basic property of Hilbert space, the vector c1e1 + c2e2 + ... + cnen has a non-negative norm for arbitrary complex coefficients. This demands the positive-indefinite character of a real quadratic form in 2n variables; equivalently, all its eigenvalues must be non-negative. This leads to inequalities on the dispersions ΔAr, the expectations of the commutators i[Ar, As] and other expectations. For n = 2, this process leads to Schrödinger’s improvement of the Heisenberg inequality. The case n = 3 is here worked out explicitly.
Observables A and B satisfy the Heisenberg inequality if the product of their variances has a positive lower bound independent of the state of the system. In the Hilbert space formulation of quantum mechanics it is a consequence of the Schwarz inequality that the Heisenberg-type inequality Var(A,φ)⋅Var(B,φ)≥ (1)/(4) ‖〈Aφ‖Bφ〉−〈Bφ‖Aφ〉‖2 holds for any pair of observables A and B (represented as self-adjoint operators) and for any (vector) state (represented as a unit vector). If inf{‖〈Aφ‖Bφ〉−〈Bφ‖Aφ〉‖ ‖φ∈dom(A) ∩dom(B)}≠0 then A and B satisfy the Heisenberg inequality. In the present paper the derivability of the Heisenberg-type inequality is analyzed within the general theoretical frame of a sum logic. It is shown that any real-valued non-negative function (A,α)→f(A,α) of observables A and of states α, which has a symmetry property f(A+B,α)+f(A−B,α)=2f(A,α)+2f(B,α) with respect to observables, satisfies the Heisenberg-type inequality f(A,α)⋅f(B,α)≥ (1)/(4) ‖ f(A+B,α)−f(A,α)−f(B,α)‖2 for all observables A and B and for all states α. The natural probabilistic realizations f1(A,α)=Exp(A2,α) and f2(A,α)=Var(A,α) of such functions are then analyzed. It turns out that only with the complex extension of the theory can the Heisenberg inequality be attained in the Hilbert space realization of the theory. This is used as an argument in favor of the complex field as the scalar field of quantum mechanics.
Let X \mathcal{X} X be a p-adic Hilbert space. Let A:D(A)⊆X→X A: \mathcal{D}(A)\subseteq \mathcal{X}\to \mathcal{X} A:D(A)⊆X→X and B:D(B)⊆X→X B: \mathcal{D}(B)\subseteq \mathcal{X}\to \mathcal{X} B:D(B)⊆X→X be possibly unbounded self-adjoint linear operators. For x∈D(A) x \in \mathcal{D}(A) x∈D(A) with ⟨x,x⟩=1 \langle x, x \rangle =1 ⟨x,x⟩=1, define Δx(A):=∥Ax−⟨Ax,x⟩x∥. \Delta _x(A):= \|Ax- \langle Ax, x \rangle x \|. Δx​(A):=∥Ax−⟨Ax,x⟩x∥. Then for all x∈D(AB)∩D(BA) x \in \mathcal{D}(AB)\cap \mathcal{D}(BA) x∈D(AB)∩D(BA) with ⟨x,x⟩=1 \langle x, x \rangle =1 ⟨x,x⟩=1, we show that (1)max{Δx(A),Δx(B)}≥|⟨[A,B]x,x⟩2+(⟨{A,B}x,x⟩−2⟨Ax,x⟩⟨Bx,x⟩)2||2| and (2)max{Δx(A),Δx(B)}≥|⟨(A+B)x,y⟩|,∀y∈X satisfying ‖y‖≤1,⟨x,y⟩=0. We call Inequality (1) as p-adic Heisenberg-Robertson-Schrodinger uncertainty principle and Inequality (2) as p-adic Maccone-Pati uncertainty principle.
In this paper we study quasi quantum quadratic operators (QQO) acting on the algebra of [Formula: see text] matrices [Formula: see text]. We consider two kinds of quasi QQO the corresponding quadratic operator maps from the unit circle into the sphere and from the sphere into the unit circle, respectively. In our early paper we have defined a q-purity of quasi QQO. This notion is equivalent to the invariance of the unit sphere in [Formula: see text]. But to check this condition, in general, is tricky. Therefore, it would be better to find weaker conditions to check the q-purity. One of the main results of this paper is to provide a criterion of q-purity of quasi QQO in terms of the unit circles. Moreover, we are able to classify all possible kinds of quadratic operators which can produce q-pure quasi QQO. We think that such result will allow one to check whether a given mapping is a pure channel or not. This finding suggests us to study such a class of nonpositive mappings. Correspondingly, the complement of this class will be of potential interest for physicist since this set contains all completely positive mappings.
We show that Robertson uncertainty relation can be refined for q-commutator which is defined by [A,B]q=AB−qBA[A,B]_q = AB-qBA[A,B]q​=AB−qBA, where A,BA, BA,B are self-adjoint operators and real number q∈Rq\in \mathbb{R}q∈R. The coefficient is represented by the eigenvalues of state ρρρ.