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arXiv:2609.37133

Stability of isometries of quantum states

Sam Looi

math.FAmath-phquant-phSubmitted Sep 29, 2026• Announced Sep 30, 2026

v1 · Sep 29, 2026 v2 · Oct 13, 2026

Abstract

We prove dimension-independent Hyers-Ulam stability for surjective approximate isometries of the positive trace-class cone in both the trace and Bures metrics. Here $\varepsilon\geq0$ is a uniform upper bound on the additive error in preserving distances. Such an approximate isometry is uniformly approximated by a Wigner symmetry (a unitary or antiunitary conjugation), with error at most $9\varepsilon/2$ in the trace metric and $2\sqrt2\varepsilon$ in the Bures metric, even if it does not fix zero. For maps fixing zero, the trace bound improves to $3\varepsilon$ even under the hypothesis of $δ$-surjectivity, independently of $δ$. In infinite dimensions, we construct bijections of the full state space, including mixed states, whose distortions tend to zero in both metrics but which stay a fixed uniform distance from every Wigner symmetry. For state spaces, stability holds in each fixed finite dimension with a dimension-dependent modulus tending to zero as $\varepsilon\to0$, but no such modulus can be chosen independently of dimension.

Comments: 41 pages. Comments welcome

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We prove dimension-independent Hyers-Ulam stability for surjective approximate isometries of the positive trace-class cone in both the trace and Bures metrics. Here

\varepsilon\geq0

is a uniform upper bound on the additive error in preserving distances. Such an approximate isometry is uniformly approximated by a Wigner symmetry (a unitary or antiunitary conjugation), with error at most

9\varepsilon/2

in the trace metric and

2\sqrt2\varepsilon

in the Bures metric, even if it does not fix zero. For maps fixing zero, the trace bound improves to

3\varepsilon

even under the hypothesis of

δ

-surjectivity, independently of

δ

. In infinite dimensions, we construct bijections of the full state space, including mixed states, whose distortions tend to zero in both metrics but which stay a fixed uniform distance from every Wigner symmetry. For state spaces, stability holds in each fixed finite dimension with a dimension-dependent modulus tending to zero as

\varepsilon\to0

, but no such modulus can be chosen independently of dimension.

Wigner’s theorem characterizes isometries of the set of all rank one projections on a Hilbert space. In metric geometry, nonexpansive maps and noncontractive maps are well-studied generalizations of isometries. We show that under certain conditions Wigner symmetries can be characterized as nonexpansive or noncontractive maps on the set of all projections of rank one. The assumptions required for such characterizations are injectivity or surjectivity and they differ in the finite and the infinite-dimensional case. Motivated by a recently obtained optimal version of Uhlhorn’s generalization of Wigner’s theorem, we also give a description of nonexpansive maps which satisfy a condition that is much weaker than surjectivity. Such maps do not need to be Wigner symmetries. The optimality of all presented results is shown by counterexamples.

In a unital C*-algebra with a faithful trace functional

τ

, the set

D_τ(A)

of positive

ρ\in A

of trace τ(ρ)=1 is an algebraic analogue of the space of density matrices (the set of all positive matrices of a fixed dimension of unit trace). Motivated by the literature concerning the metric properties of the space of density matrices, the present paper studies the density space

D_τ(A)

in terms of the Bures metric. Linear maps on A that map

D_τ(A)

back into itself are positive and trace preserving, hence, they may be viewed as an algebraic analogue of a quantum channel, which are studied intensely in the literature on quantum computing and quantum information theory. The main results in this paper are: (i) to establish that the Bures metric is indeed a metric, (ii) to prove that channels induce nonexpansive maps of the density space

D_τ(A)

, (iii) to introduce and study channels on A that are locally contractive maps (which we call Bures contractions) on the metric space

D_τ(A)

, and (iv) to analyse Bures contractions from the point of view of the Frobenius theory of cone preserving linear maps. Although the focus is on unital C*-algebras, an important class of examples is furnished by finite von Neumann algebras. Indeed, several of the C*-algebra results are established by first proving them for finite von Neumann algebras and then proving them for C*-algebras by embedding a C*-algebra A into its enveloping von Neumann algebra

A^{**}

.

It d-pends. Wigner’s symmetry theorem implies that transformations that preserve transition probabilities of pure quantum states are linear maps on the level of density operators. We investigate the stability of this implication. On the one hand, we show that any transformation that preserves transition probabilities up to an additive ε in a separable Hilbert space admits a weak linear approximation, i.e., one relative to any fixed observable. This implies the existence of a linear approximation that is 4εd–close in Hilbert-Schmidt norm, with d the Hilbert space dimension. On the other hand, we prove that a linear approximation that is close in norm and independent of d does not exist in general. To this end, we provide a lower bound that depends logarithmically on d.

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.

Completely positive trace-preserving maps

S

, also known as quantum channels, arise in quantum physics as a description of how the density operator

ρ

of a system changes in a given time interval, allowing not only for unitary evolution but arbitrary operations including measurements or other interaction with an environment. It is known that if the Hilbert space

\mathscr{H}

that

ρ

acts on is finite-dimensional, then every

S

must have a fixed point, i.e., a density operator

ρ_0

with

S(ρ_0)=ρ_0

. In infinite dimension,

S

need not have a fixed point in general. However, we prove here the existence of a fixed point under a certain additional assumption which is, roughly speaking, that

S

leaves invariant a certain set of density operators with bounded ``cost'' of preparation. The proof is an application of the Schauder-Tychonoff fixed point theorem. Our motivation for this question comes from a proposal of Deutsch for how to define quantum theory in a space-time with closed timelike curves; our result supports the viability of Deutsch's proposal.