Preserving the measure of compatibility between quantum states

Werner Timmermannn

Published Jan 1, 2002• 3 citations

Abstract

In this paper after defining the abstract concept of compatibility-like functions on quantum states, we prove that every bijective transformation on the set of all states which preserves such a function is implemented by an either unitary or antiunitary operator.

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Preserving the measure of compatibility between quantum states

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Preserving the measure of compatibility between quantum states

Werner Timmermannn

Published Jan 1, 2002• 3 citations

OpenAlex

Abstract

Related papers

Preserving the measure of compatibility between quantum states

Lajos Molnár, Werner Timmermann

arXiv:math/0212078Journal of Mathematical PhysicsFeb 28, 20033 citations

Maps on quantum states preserving the Jensen–Shannon divergence

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Journal of Physics A Mathematical and TheoreticalNov 26, 20086 citations

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Mark Pankov

Cambridge University Press eBooksJan 6, 20200 citations

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Lajos Molnár

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Transformations on the set of all $n$ -dimensional subspaces of a Hilbert space preserving orthogonalityTransformations on the set of all $n$-dimensional subspaces of a Hilbert space preserving orthogonality

Máté Győry

Publicationes Mathematicae DebrecenJul 1, 200424 citations

The structure of maps on the space of all quantum pure states that preserve a fixed quantum angle

György Pál Gehér, Michiya Mori

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.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.

arXiv:2102.05780International Mathematics Research NoticesFeb 10, 20216 citations

Transformations on the set of all

n

-dimensional subspaces of a Hilbert space preserving orthogonality

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.