arXiv:math/0212078 · Journal of Mathematical Physics

Preserving the measure of compatibility between quantum states

Lajos Molnár, Werner Timmermann

Published Feb 28, 2003• 3 citations

Abstract

In this article 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.

DOI: 10.1063/1.1545164

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arXiv:math/0212078 · Journal of Mathematical Physics

Preserving the measure of compatibility between quantum states

Lajos Molnár, Werner Timmermann

Published Feb 28, 2003• 3 citations

arXiv HTML PDF

Abstract

DOI: 10.1063/1.1545164

Related papers

Preserving the measure of compatibility between quantum states

Werner Timmermannn

Jan 1, 20023 citations

Maps on quantum states preserving the Jensen–Shannon divergence

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We show that any bijective map on the space of all quantum states which preserves the Jensen–Shannon divergence is induced by a unitary or antiunitary operator.

Journal of Physics A Mathematical and TheoreticalNov 26, 20086 citations

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Mark Pankov

Cambridge University Press eBooksJan 6, 20200 citations

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.

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Orthogonality preserving transformations on complex projective spaces

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It is well known that transformations of $ \mathbb{C}^n $ preserving the standard inner product are unitary transformations. In this paper, all bijective transformations of isotropic sets of $ \mathbb{C}P^{n} $ preserving $ H $-orthogonality in both directions, called $ H $-orthogonal transformations, have been determined. This is a generalization of Uhlhorn's version of Wigner's unitary-antiunitary theorem. The group of $ H $-orthogonal transformations on some other sets of $ \mathbb{C}P^{n} $ were also determined.It is well known that transformations of $ \mathbb{C}^n $ preserving the standard inner product are unitary transformations. In this paper, all bijective transformations of isotropic sets of $ \mathbb{C}P^{n} $ preserving $ H $-orthogonality in both directions, called $ H $-orthogonal transformations, have been determined. This is a generalization of Uhlhorn's version of Wigner's unitary-antiunitary theorem. The group of $ H $-orthogonal transformations on some other sets of $ \mathbb{C}P^{n} $ were also determined.

AIMS MathematicsJan 1, 20250 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.