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Wigner's Theorem and Spectrum Shrinking Maps

Allen, Meaghan

Published May 1, 2025• 0 citations

Abstract

Wigner's Theorem, an influential result in quantum mechanics established by Eugene Wigner in 1931, states that every symmetry transformation on a ray space is induced by either a unitary or anti-unitary transformation. In this work, we generalize Wigner's Theorem in two directions: first, to maps between different Hilbert spaces, and second, to maps between factors of type II. In both cases, we focus on weakening the assumptions on the map, specifically by utilizing spectrum shrinking conditions. For the first generalization, let $\mathcal{H}_1$ and $\mathcal{H}_2$ be Hilbert spaces, and let $k$ be a positive integer such that $2\leq k < \frac{1}{2} \dim(\mathcal{H}_1)$. We show that if an injective map $\psi: \mathcal{P}_k(\mathcal{H}_1) \to \mathcal{P}_k(\mathcal{H}_2)$ satisfies two spectrum shrinking conditions, then it is induced by a linear or conjugate-linear isometry. For the second generalization, let $\mathcal{M}$ be a type II$_1$ or type II$_\infty$ factor, let $\tau_\mathcal{M}$ be a faithful normal semi-finite tracial weight of $\mathcal M$, and let $0

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•Jan 1, 2009•5 citations

Wigner's Theorem, an influential result in quantum mechanics established by Eugene Wigner in 1931, states that every symmetry transformation on a ray space is induced by either a unitary or anti-unitary transformation. In this work, we generalize Wigner's Theorem in two directions: first, to maps between different Hilbert spaces, and second, to maps between factors of type II. In both cases, we focus on weakening the assumptions on the map, specifically by utilizing spectrum shrinking conditions. For the first generalization, let

\mathcal{H}_1

and

\mathcal{H}_2

be Hilbert spaces, and let

k

be a positive integer such that

2\leq k < \frac{1}{2} \dim(\mathcal{H}_1)

. We show that if an injective map

\psi: \mathcal{P}_k(\mathcal{H}_1) \to \mathcal{P}_k(\mathcal{H}_2)

satisfies two spectrum shrinking conditions, then it is induced by a linear or conjugate-linear isometry. For the second generalization, let

\mathcal{M}

be a type II

_1

or type II

_\infty

factor, let

\tau_\mathcal{M}

be a faithful normal semi-finite tracial weight of

\mathcal M

, and let $0

Let H be a complex Hilbert space and let P ( H ) be the associated projective space (the set of rank-one projections). Suppose dim H ⩾ 3 . We prove the following Wigner-type theorem: if H is finite dimensional, then every orthogonality preserving transformation of P ( H ) is induced by a unitary or anti-unitary operator. This statement will be obtained as a consequence of the following result: every orthogonality preserving lineation of P ( H ) to itself is induced by a linear or conjugate-linear isometry ( H is not assumed to be finite-dimensional). As an application, we describe (not necessarily injective) transformations of Grassmannians preserving some types of principal angles.

Abstract Let

H

be a Hilbert space and

P(H)

be the projective space of all quantum pure states. Wigner’s theorem states that every bijection

\phi \colon P(H)\to P(H)

that preserves the quantum angle between pure states is automatically induced by either a unitary or an antiunitary operator

U\colon H\to H

. Uhlhorn’s theorem generalizes this result for bijective maps $\phi $ that are only assumed to preserve the quantum angle

\frac{\pi }{2}

(orthogonality) in both directions. Recently, two papers, written by Li–Plevnik–Šemrl and Gehér, solved the corresponding structural problem for bijections that preserve only one fixed quantum angle $\alpha $ in both directions, provided that

0 < \alpha \leq \frac{\pi }{4}

holds. In this paper we solve the remaining structural problem for quantum angles $\alpha $ that satisfy

\frac{\pi }{4} < \alpha < \frac{\pi }{2}

, hence complete a programme started by Uhlhorn. In particular, it turns out that these maps are always induced by unitary or antiunitary operators, however, our assumption is much weaker than Wigner’s.

We determine the isometry group of the

n

-qubit state space with respect to the quantum Wasserstein distance induced by the so-called symmetric transport cost for all

n \in \mathbb{N}.

It turns out that the isometries are precisely the Wigner symmetries, that is, the unitary or anti-unitary conjugations.