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Journal of Physics A Mathematical and Theoretical

Characterization of sum automorphisms and Jordan triple automorphisms of quantum probabilistic maps

Zhaofang Bai, Shuanping Du

Published Apr 1, 2010• 3 citations
Publisher

Abstract

In quantum mechanics, it is often important for the representation of a quantum system to study the structure-preserving bijective maps of the system. Such maps are also called isomorphisms or automorphisms. In this paper, using the Uhlhorn-type of Wigner's theorem, we show that both sum automorphisms and Jordan triple automorphisms of the unit ball of density operators are implemented by either unitary or anti-unitary operators of the underlying Hilbert space.

DOI
10.1088/1751-8113/43/16/165210

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In quantum mechanics, it is often important for the representation of a quantum system to study the structure-preserving bijective maps of the system. Such maps are also called isomorphisms or automorphisms. In this paper, using the Uhlhorn-type of Wigner's theorem, we show that both sum automorphisms and Jordan triple automorphisms of the unit ball of density operators are implemented by either unitary or anti-unitary operators of the underlying Hilbert space.
Abstract Let HHH be a Hilbert space and P(H)P(H)P(H) be the projective space of all quantum pure states. Wigner’s theorem states that every bijection ϕ ⁣:P(H)→P(H)\phi \colon P(H)\to P(H)ϕ:P(H)→P(H) that preserves the quantum angle between pure states is automatically induced by either a unitary or an antiunitary operator U ⁣:H→HU\colon H\to HU:H→H. Uhlhorn’s theorem generalizes this result for bijective maps $\phi $ that are only assumed to preserve the quantum angle π2\frac{\pi }{2}2π​ (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<α≤π40 < \alpha \leq \frac{\pi }{4}0<α≤4π​ holds. In this paper we solve the remaining structural problem for quantum angles $\alpha $ that satisfy π4<α<π2\frac{\pi }{4} < \alpha < \frac{\pi }{2}4π​<α<2π​, 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.
In the usual formulation of quantum mechanics, groups of automorphisms of quantum states have ray representations by unitary and antiunitary operators on complex Hilbert space, in accordance with Wigner's theorem. In the phase-space formulation, they have real, true unitary representations in the space of square-integrable functions on phase space. Each such phase-space representation is a Weyl–Wigner product of the corresponding Hilbert space representation with its contragredient, and these can be recovered by 'factorizing' the Weyl–Wigner product. However, not every real, unitary representation on phase space corresponds to a group of automorphisms, so not every such representation is in the form of a Weyl–Wigner product and can be factorized. The conditions under which this is possible are examined. Examples are presented.
Five conceptually distinct notions of symmetry in quantum theory are studied in the algebraic setting where a quantum system is characterized by a von Neumann algebra of observables and the set of normal states on the algebra. It is shown that all five symmetry notions are closely related and that the glue binding them together is the concept of a Jordan ∗-automorphism. For factor algebras a Jordan ∗-automorphism reduces either to an ∗-automorphism or a ∗-anti-automorphism. If the algebra is put in standard form then a ∗-automorphism is always unitarily implementable, whereas a ∗-anti-automorphism is always anti-unitarily implementable. However, there is no guarantee that a general von Neumann algebra admits ∗-anti-automorphisms or, if it does, that it admits order two (or involutory) ∗-anti-automorphisms). For non-factor algebras there can be genuine Jordan ∗-automorphisms that are neither ∗-automorphisms nor ∗-anti-automorphisms, and implementation is possible only through partial isometries. These developments enable generalized versions of Wigner's theorem on the implementation of transition probability preserving symmetries for von Neumann algebras. This review is largely an exercise in connecting the dots in existing mathematics and physics literature. But in the service of the philosophy of physics it is an exercise worth doing since the practitioners in this field seem largely unaware of or unappreciative of this literature and how it fits together to yield a multifaceted but unified picture of quantum symmetries. Along the way various interpretations issues worthy of further discussion are flagged.
The empirical content of quantum theory is entirely encoded in the Born probability pairing between states and effects. In finite dimension, we study the following structural problem: which reparametrizations of states and effects preserve all Born probabilities? We consider bijective affine maps on the state space and effect space that leave the Born pairing invariant for all state–effect pairs, and provide a complete classification of all such transformations. We prove that every probability-preserving reparametrization is necessarily induced by a Jordan *-automorphism of the matrix algebra B(H). In finite dimension, this restricts the transformation to the form A→UAU† or A→UA⊤U†, for some unitary operator U. In particular, equivalence classes of states under probability-preserving reparametrizations coincide with unitary orbits and are completely characterized by the spectrum of the density operator. The result shows that unitary and antiunitary conjugations exhaust all probability-preserving affine reparametrizations of the state–effect framework.