AIMS Mathematics

Orthogonality preserving transformations on complex projective spaces

Kai Zhou, Zhenhua Gu, Hongfeng Wu

Published Jan 1, 2025• 0 citations

Abstract

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.

DOI: 10.3934/math.2025519

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Classical Wigner’s theorem characterizes unitary and anti-unitary operators as symmetries of pure states of quantum mechanical systems, i.e. rank one projections. We consider a non-injective version of Wigner’s theorem as well as Uhlhorn’s version concerning orthogonality preserving transformations and describe various extensions of these results onto other Grassmannians.

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AIMS Mathematics

Orthogonality preserving transformations on complex projective spaces

Kai Zhou, Zhenhua Gu, Hongfeng Wu

Published Jan 1, 2025• 0 citations

Publisher

Abstract

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.

DOI: 10.3934/math.2025519

Related papers

Symmetries of Projective Spaces and Spheres

György Pál Gehér

Abstract Let H be either a complex inner product space of dimension at least two or a real inner product space of dimension at least three, and let us fix an $\alpha \in (0,\tfrac{\pi }{2} )$ . The purpose of this paper is to characterise all bijective transformations on the projective space P(H) which preserve the quantum angle $\alpha$ (or Fubini–Study distance $\alpha$ ) between lines in both directions. (Let us emphasise that we do not assume anything about the preservation of other quantum angles). For real inner product spaces and when $H=\mathbb{C}^2$ we do this for every $\alpha$ , and when H is a complex inner product space of dimension at least three we describe the structure of such transformations for $\alpha \leq \tfrac{\pi }{4}$ . Our result immediately gives an Uhlhorn-type generalisation of Wigner’s theorem on quantum mechanical symmetry transformations, that is considered to be a cornerstone of the mathematical foundations of quantum mechanics. Namely, under the above assumptions, every bijective map on the set of pure states of a quantum mechanical system that preserves the transition probability $\cos ^2\alpha$ in both directions is a Wigner symmetry (thus automatically preserves all transition probabilities), except for the case when $H=\mathbb{C}^2$ and $\alpha = \tfrac{\pi }{4}$ where an additional possibility occurs. (Note that the classical theorem of Uhlhorn is the solution for the $\alpha = \tfrac{\pi }{2}$ case). Usually in the literature, results which are connected to Wigner’s theorem are discussed under the assumption of completeness of H; however, here we shall remove this unnecessary hypothesis in our investigation. Our main tool is a characterisation of bijective maps on unit spheres of real inner product spaces which preserve one spherical angle in both directions.Abstract Let H be either a complex inner product space of dimension at least two or a real inner product space of dimension at least three, and let us fix an $\alpha \in (0,\tfrac{\pi }{2} )$. The purpose of this paper is to characterise all bijective transformations on the projective space P(H) which preserve the quantum angle $\alpha$ (or Fubini–Study distance $\alpha$) between lines in both directions. (Let us emphasise that we do not assume anything about the preservation of other quantum angles). For real inner product spaces and when $H=\mathbb{C}^2$ we do this for every $\alpha$, and when H is a complex inner product space of dimension at least three we describe the structure of such transformations for $\alpha \leq \tfrac{\pi }{4}$. Our result immediately gives an Uhlhorn-type generalisation of Wigner’s theorem on quantum mechanical symmetry transformations, that is considered to be a cornerstone of the mathematical foundations of quantum mechanics. Namely, under the above assumptions, every bijective map on the set of pure states of a quantum mechanical system that preserves the transition probability $\cos ^2\alpha$ in both directions is a Wigner symmetry (thus automatically preserves all transition probabilities), except for the case when $H=\mathbb{C}^2$ and $\alpha = \tfrac{\pi }{4}$ where an additional possibility occurs. (Note that the classical theorem of Uhlhorn is the solution for the $\alpha = \tfrac{\pi }{2}$ case). Usually in the literature, results which are connected to Wigner’s theorem are discussed under the assumption of completeness of H; however, here we shall remove this unnecessary hypothesis in our investigation. Our main tool is a characterisation of bijective maps on unit spheres of real inner product spaces which preserve one spherical angle in both directions.

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A geometric approach to Wigner‐type theorems

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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.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.

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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

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Mark Pankov

Cambridge University Press eBooksJan 6, 20200 citations

Wigner symmetries and Gleason’s theorem *

Peter Šemrl

Abstract We prove an optimal version of Uhlhorn’s generalization of Wigner’s unitary–antiunitary theorem. The main tool in our proof is Gleason’s theorem.

arXiv:2106.06182Journal of Physics A Mathematical and TheoreticalJun 21, 20217 citations

Maps on quantum states preserving the Jensen–Shannon divergence

Lajos Molnár, Werner Timmermann

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

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.

Abstract Let H be either a complex inner product space of dimension at least two or a real inner product space of dimension at least three, and let us fix an

\alpha \in (0,\tfrac{\pi }{2} )

. The purpose of this paper is to characterise all bijective transformations on the projective space P(H) which preserve the quantum angle

\alpha

(or Fubini–Study distance

\alpha

) between lines in both directions. (Let us emphasise that we do not assume anything about the preservation of other quantum angles). For real inner product spaces and when

H=\mathbb{C}^2

we do this for every

\alpha

, and when H is a complex inner product space of dimension at least three we describe the structure of such transformations for

\alpha \leq \tfrac{\pi }{4}

. Our result immediately gives an Uhlhorn-type generalisation of Wigner’s theorem on quantum mechanical symmetry transformations, that is considered to be a cornerstone of the mathematical foundations of quantum mechanics. Namely, under the above assumptions, every bijective map on the set of pure states of a quantum mechanical system that preserves the transition probability

\cos ^2\alpha

in both directions is a Wigner symmetry (thus automatically preserves all transition probabilities), except for the case when

H=\mathbb{C}^2

and

\alpha = \tfrac{\pi }{4}

where an additional possibility occurs. (Note that the classical theorem of Uhlhorn is the solution for the

\alpha = \tfrac{\pi }{2}

case). Usually in the literature, results which are connected to Wigner’s theorem are discussed under the assumption of completeness of H; however, here we shall remove this unnecessary hypothesis in our investigation. Our main tool is a characterisation of bijective maps on unit spheres of real inner product spaces which preserve one spherical angle in both directions.

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.