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1.   Introduction and statement of the main results

Let $H, K$ be complex or real separable Hilbert spaces and $n$ a positive integer. As usual, the symbols $P_{n}\left (H\right)$ and $I_{H}$ stand for the set of all rank-$n$ self-adjoint projections on $H$, and the identity operator on $H$, respectively. For $S, T\in P_{n}\left (H\right)$, we say $S$ is orthogonal to $T$ iff $ST = 0$ and the quantity $\text{Tr} \left (ST \right)$ is the transition probability between $S, T$. Plainly, $S\bot T$ is equivalent to $\text{Tr} \left (ST \right) = 0$. If $u\in H$ is a unit vector, then the rank-$1$ projection onto $\text{span} \left \{ u\right \}$ will be denoted by $u\otimes u$. The transition probability associated with a pair of rank-$1$ projections (pure states) is the commonly used concept in quantum theory. We call a family $\left \{ S_{i} \right \} \subseteq P_{n}\left (H\right)$ a complete orthogonal system of rank-$n$ projections (briefly, $\text{COSP} _n$) iff

● $S_{i}\bot S_{j}$ whenever $i\ne j$.

● There is no rank-$1$ projection $T$ orthogonal to each $S_{i}$.

The celebrated Wigner's theorem [1, pp.251–254] states that if $\phi: P_{1}\left (H\right) \to P_{1}\left (H\right)$ is a bijection satisfying

equivalently, if $\phi$ preserves the transition probability between $S$ and $T$, then there exists a unitary or an anti-unitary $U:H\to H$ such that $\phi\left(A\right) = UAU^{*}$. Recently, there has been considerable interest in improving and reproving this vital result in many ways (referred to in ^{[2,3,4,5,6,7]}).

Wigner's theorem also serves as a frequently used tool for investigating the symmetries in some mathematical structures of quantum mechanics. Suppose that $\phi$ is a bijection on the set of all observables/the state space/the effect algebra, and such a map preserves a certain property/relation/operation relevant in quantum mechanics. The given problem is to characterize the form of such maps (symmetries), and a classical approach to this problem is to first show that $\phi$ preserves the rank-$1$ projections and the corresponding transition probability. This is the crucial step of the proof. Applying Wigner's theorem, one may immediately see that the restriction of $\phi$ to $P_{1}\left (H\right)$ has a nice behavior. Then the final step to prove that $\phi$ takes the desired form on the entire quantum structure is usually considered as an easier part of the proof. The interested readers are referred to [8, Chapter 2] and references therein for more examples of this approach and some background for the so-called preservers problems.

When using the above method, sometimes we may not ensure that $\phi$ maps $P_{1}\left (H\right)$ into itself, and quite often we merely know that it preserves the zero-transition probability. This motivates us to search for a stronger version of the classical Wigner's theorem. The main aim of this paper is to provide the generalizations of Wigner's theorem in which instead of assuming that $\phi$ maps $P_{1}\left (H\right)$ into itself, we assume that $\phi$ maps $P_{1}\left (H\right)$ into $P_{n}\left (K\right)$.

Theorem 1.1. If $\phi :P_{1}\left (H\right) \to P_{n}\left (K\right)$ is a map satisfying

then there exists a collection $\left \{ V_1, \dots, V_n \right \}$ of linear or conjugate linear isometries from $H$ into $K$ with mutually orthogonal ranges, such that

Notice that the property $\left (1.2\right)$ is equivalent to the following condition:

where $\left \| \cdot \right \|_{HS}$ represents the Hilbert–Schmidt norm. Namely, our result describes the general form of maps from $P_{1} \left(H\right)$ into $P_{n}\left (K\right)$ multiplying $\sqrt{n}$ times the distance induced by this special norm. We point out that several papers ^[9,10] studied the isometries of $P_{n} \left (H\right)$ with respect to the operator norm.

For the case of $\dim H \ge 3$, Uhlhorn ^[11] significantly generalized Wigner's theorem by replacing the assumption (1.1) with a weaker one: $\text{Tr} \left (ST \right) = 0 \Leftrightarrow \text{Tr} \left(\phi \left (S\right) \phi\left (T\right)\right) = 0$. Uhlhorn's result has been further improved in ^[12,13]: It is proved that the bijectivity assumption can be relaxed when $\dim H < \infty$. Unfortunately, when $\dim H = \infty$, it is shown in ^[14] that there exist injective maps preserving orthogonality in both directions, which behave quite wildly. Thus, an additional hypothesis will be needed in the infinite-dimensional case.

Theorem 1.2. Let $\dim H \ge 3$. If $\phi :P_{1}\left (H\right) \rightarrow P_{n}\left (K\right)$ is a map that sends each complete orthogonal system of rank-$1$ projections to some complete orthogonal system of rank-$n$ projections, then there exists a collection $\left \{ V_1, \dots, V_n \right \}$ of linear or conjugate linear isometries from $H$ into $K$, which have mutually orthogonal ranges and satisfy ${ \sum_{i = 1}^{n}}V_iV_i^{*} = I$, such that

If $3\le \dim H < \infty$ and $\dim K = n\dim H$, then a map $\phi :P_{1}\left (H\right) \to P_{n}\left (K\right)$ that preserves orthogonality only in one direction automatically sends each $\text{COSP} _1$ to some $\text{COSP} _n$. Therefore, a generalization (without bijectivity either) of Uhlhorn's theorem in matrix algebra is a direct consequence of Theorem 1.2.

Corollary 1.3. Let $3\le \dim H < \infty$ and $\dim K = n\dim H$. If $\phi :P_{1}\left (H\right) \to P_{n}\left (K\right)$ is a map that preserves orthogonality in one direction, then $\phi$ has the form $\left (1.3\right)$.

2.   Proofs

In what follows, we denote by $C\left (H\right)$, $F\left (H\right)$, and $F_s\left (H\right)$ the set of compact operators, finite-rank operators, and finite-rank self-adjoint operators on $H$. The following lemma will be used to prove Theorem 1.1.

Lemma 2.1. If $\phi :F_s\left (H\right) \to F_s\left (K\right)$ is a linear map that sends rank-$1$ projections to rank-$n$ projections and satisfies

then there exists a collection $\left \{ V_1, \dots, V_n \right \}$ of linear or conjugate linear isometries from $H$ into $K$ with mutually orthogonal ranges, such that

To prove Lemma 2.1, we need the following lemmas. For $S, T\in F_s\left (H\right)$, we write $S\le T$ if $T-S$ is positive.

Lemma 2.2. Let $\phi :F_s\left (H\right) \to F_s\left (K\right)$ be a linear map that preserves projections. If $S, T\in F_s\left (H\right)$ are projections with $S\ge T$, then $\phi \left (S\right) \ge \phi \left (T\right)$.

Proof. Since $S, T$ are projections with $S\ge T$, there exists some projection $R$ orthogonal to $T$, such that $S = T+R$. Thus, $\phi \left (S\right) = \phi \left (T\right) +\phi \left (R\right) \ge \phi \left (T\right)$. □

Lemma 2.3. (see [15, Theorem 1.9.1]) Let $\mathcal{M}$ be a dense subspace of a normed space $\mathcal{V}$, and $\mathcal{W}$ a Banach space. If $\phi:\mathcal{M} \to \mathcal{W}$ is a continuous linear map, then $\phi$ has a unique continuous linear extension $\phi^{\prime} :\mathcal{V} \to \mathcal{W}$.

Proof of Lemma 2.1. By Eq $\left (2.1\right)$, we see that $\phi$ sends orthogonal rank-$1$ projections to orthogonal rank-$n$ projections. Clearly, any finite-rank projection is the sum of mutually orthogonal rank-$1$ projections. Consequently, $\phi$ preserves the projections.

Assume that the underlying space $H$ is complex. Extend $\phi$ to a complex linear map from $F\left (H\right)$ into $F\left (K\right)$ by setting

Let $A = { \sum_{i}}\alpha _{i}P_{i}$, $\alpha _{i}\in \mathbb{R}$, $P_{i}\in P_{1}\left (H\right)$, denote the spectral decomposition of any operator $A\in F_s\left (H\right)$. Then $\tilde{ \phi}\left (P_{i}\right) \tilde{ \phi}\left (P_{j}\right) = 0$ for each $i\ne j$, and hence $\tilde{ \phi}\left (A^{2}\right) = \tilde{ \phi}\left (A\right) ^{2}$. Replacing $A$ by $A+B$, with $A, B\in F_s\left (H\right)$, we obtain that $\tilde{\phi} \left (AB+BA \right) = \tilde{\phi} \left (A \right) \tilde{\phi} \left (B \right) +\tilde{\phi} \left (B \right)\tilde{\phi} \left (A \right)$. Then it follows that

This implies that $\tilde{ \phi}$ is a Jordan homomorphism. Since $\tilde{ \phi}$ preserves the self-adjoint operators, we infer that $\tilde{ \phi}$ is a (continuous) Jordan $*$ - homomorphism. It is known that $F\left (H \right)$ is dense in the $C^{*}$ - algebra $C\left (H\right)$. By Lemma 2.3, $\tilde{ \phi}$ can be uniquely extended to a Jordan $*$ - homomorphism from $C\left (H\right)$ into $C\left (K\right)$. According to [8, Theorem A.6], each Jordan $*$ - homomorphism of the $C^{*}$ - algebra is a direct sum of a $*$-antihomomorphism and a $*$ - homomorphism. Every $*$ - homomorphism of $C\left (H\right)$ is in fact a direct sum of inner homomorphisms (see [16, Theorem 10.4.7]). Then $\tilde{ \phi}$ has the asserted form.

The case when $H$ is real demands an other approach (this idea is borrowed from [17, Theorem 2.2] below). Assume that $\left\{ u _{i} \right\}_{i\in \Omega}$ is an orthonormal basis for $H$ and denote $\text{rng} \phi \left (u _{i}\otimes u _{i}\right) = K_{i}$, $i \in \Omega$. For any $i, j\in \Omega$ with $i\neq j$, since $\left[\left (u _{i}+u_j\right)\otimes \left (u _{i}+ u _{j}\right) \right]/2$ is a projection with range lying within that of $u _{i}\otimes u _{i}+ u _{j}\otimes u _{j}$, it follows by Lemma 2.2 that

Therefore, we may write

for some linear operator $P_{ij}^{\prime }:K_{j}\to K_{i}$. For any nonzero $\alpha\in \mathbb{R}$, consider

By directly computing, $Q_{1}Q_{2} = 0$. It follows that

Because this equation holds true for any nonzero $\alpha\in \mathbb{R}$, we see that $P_{11}^{\prime }$ and $P_{22}^{\prime }$ are zeroes. Hence, we obtain

It follows that $P_{12}^{\prime } = {P_{21}^{\prime }}^{-1} = {P_{21}^{\prime }}^{\ast }$ and we have similar conclusions for all $P_{ij}$.

Since all $K_i$ are isomorphic to $\mathbb{R} ^{n}$, there exists an isomorphism from $H\otimes \mathbb{R} ^{n}$ to $\bigoplus _{\Omega }K_{i}$, given by ${ \sum_{i}^{}} u_{i}\otimes \eta _{i} \to \left (\cdots \eta _{i} \cdots \right)$. Write $K = \left (H\otimes \mathbb{R} ^{n} \right) \oplus K_{s}$ and replace $\phi$ by the mapping

such that

We are going to prove that

To see this, let $Z = \left [\left (u _{1}+ u _{i}+ u _{j}\right) \otimes \left (u _{1}+ u _{i}+ u _{j}\right) \right ] /3$. Then $Z$ is a rank-$1$ projection such that, up to unitary similarity, $\phi \left (Z\right)$ is equal to a direct sum of $0$ and

As $Y^{2} = Y$, it follows that $I_{\mathbb{R} ^{n}}+2P_{ij}^{\prime } = 3P_{ij}^{\prime }$. Thus $P_{ij}^{\prime } = {P_{ij}^{\prime }}^{-1} = I_{\mathbb{R} ^{n}}$.

Since $\text{span} _{\mathbb{R}}\left\{ u_{i}\otimes u_{j}+u_{j}\otimes u_{i}:i, j\in \Omega \right\}$ is dense in $F_s\left (H\right)$ when $H$ is a real Hilbert space, we can prove that

where $U:K\to K$ is a unitary. We arrive at the conclusion. □

Proof of Theorem 1.1. Since the whole $F_s\left (H\right)$ is real linearly generated by $P_{1}\left (H\right)$, we may extend $\phi$ to a real-linear map $\tilde{ \phi} :F_s\left (H\right) \to F_s\left (K\right)$ by setting

where $\left \{ \lambda _{i} \right \}\subseteq \mathbb{R}$ and $\left \{ S_{i} \right \} \subseteq P_{1} \left (H\right)$ are finite subsets. We claim that $\tilde{ \phi}$ is well-defined. Assume that ${ \sum_{{i}}^{} \lambda _{i}}S_{i} = { \sum_{{j}}^{} \mu _{j}}T_{j}$, $\left \{ \mu _{j} \right \}\subseteq \mathbb{R}$, $\left \{ T_{j} \right \} \subseteq P_{1} \left (H\right)$. Then for each $A\in P_{1}\left (H\right)$, it follows by Eq $\left (1.2\right)$ that

This implies that

Based on the linearity of the function $\text{Tr}$, we can replace $\phi \left (A\right)$ by its linear combination. Then we obtain

Since the square of Hermitian operator $\left ({ \sum_{i}}\lambda _{i}\phi \left (S_{i}\right) -{ \sum_{j}}\mu _{j}\phi \left (T_{j}\right) \right) ^{2}$ is positive with zero trace, we deduce that

It means that $\tilde{ \phi}$ is well-defined. Then the form of this linear map $\tilde{ \phi}$ is given by Lemma 2.1. □

Proof of Theorem 1.2. First, let us recall Gleason's theorem ^[18]. A positive and trace-class operator $\sigma :H \to H$ with $\text{Tr} \left(\sigma \right) = 1$ is called a density operator. We restate Gleason's theorem as follows: Suppose that $\dim H \ge 3$ and $f:P_{1}\left (H\right) \to \left [ 0, 1 \right ]$ is a function such that for each complete orthogonal system of rank-$1$ projections $\left \{ S_{i} \right\} \subseteq P_{1}\left (H\right)$, one has

Then there is a density operator $\sigma:H\to H$ for which

To prove Theorem 1.2, we need to choose an arbitrary density operator $\varrho :K\to K$ and define the function $f _{\varrho } :P_{1} \left (H\right) \to \left [ 0, 1 \right ]$ by

It follows from our assumption that Gleason's theorem can be used. Therefore, for each density operator $\varrho :K \to K$, there exists a density operator $\sigma :H \to H$ such that

In particular, pick $\varrho = \phi \left (T\right) /n$ for some fixed $T\in P_{1}\left (H\right)$. Then we obtain

where $\sigma _{T}$ is the density operator corresponding to $T$. Taking $S = T$, we infer that

It is easy to verify that if $u$ is a unit vector such that $T = u\otimes u$, then

As $0\le \sigma _{T}\le I$, it follows by the operator theory that $\sigma _{T}u = u$. Therefore, $1$ is an eigenvalue of $\sigma _{T}$ and $u$ belongs to the corresponding eigenspace. Under the decomposition $H = \text{span} \left \{ u\right \}\oplus \left \{ u \right \}^{\bot }$, the operator $\sigma _{T}$ has the following matrix representation:

where $X$ is the positive operator acting on $\left \{ u \right \}^{\bot }$ with zero trace. Thus, $X = 0$, which means $\sigma _{T} = T$.

Hence, for each $S\in P_{1}\left (H\right)$, we have $\text{Tr} \left(\phi \left (S\right) \phi \left (T\right)\right) = n\text{Tr} \left (ST \right)$. Since $T$ was chosen arbitrarily, we deduce that $\phi$ multiplies $n$ times the transition probability. Then Theorem 1.1 tells us the form of the map $\phi$. □

3.   Discussion

The conclusion in Theorem 1.2 does not hold when $\dim H = 2$, as demonstrated in the following example: In fact, we can identify $H$ with $\mathbb{C} ^{2}$ and hence $F\left (H\right ) = M_{2} \left (\mathbb{C} \right )$, the set of $2\times 2$ complex matrices. All the rank-$1$ projections in $M_{2} \left (\mathbb{C} \right )$ are in $1$-to-$1$ correspondence with the unit vectors in the Bloch sphere in $\mathbb{R} ^{3}$, $i.e.$: $

It is straightforward to compute the orthogonal complement of

Consider the bijective transformation $\phi: P_{1} \left (\mathbb{C}^{2}\right) \to P_{1} \left (\mathbb{C}^{2}\right)$, which fix all rank-$1$ projections, but change the role of $\begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix}$ and $\begin{bmatrix} 0 & 0\\ 0 & 1 \end{bmatrix}$. Obviously, the only $\text{COSP} _1$ in $M_{2} \left (\mathbb{C} \right)$ that contains $\begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix}$ is $\left \{ \begin{bmatrix} 1 & 0\\ 0 & 0 \end{bmatrix}, \begin{bmatrix} 0 & 0\\ 0 & 1 \end{bmatrix} \right \}$ and hence $\phi$ preserves orthogonality. However, this discontinuous transformation $\phi$ can not be extended to any linear transformation (in fact, any Jordan $*$ - homomorphism also) on the whole matrix space $M_{2} \left (\mathbb{C} \right)$.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This study was funded by Fundamental Research Funds for the Central Universities of China (Grant No. 2572022DJ07).

Conflict of interest

The authors declare there is no conflicts of interest.