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arXiv:1302.3356

Characterization of affine automorphisms and ortho-order automorphisms of quantum probabilistic maps

Zhaofang Bai, Shuanping Du

Published Feb 14, 2013• 0 citations
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Abstract

In quantum mechanics, often it is important for the representation of quantum system to study the structure-preserving bijective maps of the quantum system. Such maps are also called isomorphisms or automorphisms. In this note, using the Uhlhorn-type of Wigner's theorem, we characterize all affine automorphisms and ortho-order automorphisms of quantum probabilistic maps.

DOI
10.48550/arxiv.1302.3356

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In quantum mechanics, often it is important for the representation of quantum system to study the structure-preserving bijective maps of the quantum system. Such maps are also called isomorphisms or automorphisms. In this note, using the Uhlhorn-type of Wigner's theorem, we characterize all affine automorphisms and ortho-order automorphisms of quantum probabilistic maps.
Let H be a Hilbert space and E ( H ) the effect algebra on H . A bijective map is called an ortho-order automorphism of E ( H ) if for every we have and . The classical theorem of Ludwig states that every such ϕ is of the form , , for some unitary or antiunitary operator U . It is also known that each bijective map on E ( H ) preserving order and coexistency in both directions is of the same form. Can we improve these two theorems by relaxing the bijectivity assumption and/or replacing the above preserving properties by the weaker assumptions of preserving above relations in one direction only and still get the same conclusion? For both characterizations of automorphisms of effect algebras we will prove the optimal versions and give counterexamples showing the optimality of the obtained results.
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.
This work is a study on the symmetry groups of quantum mechanics accompanied by its applications on the coexistence of qubit effects. We surveyed among some of the mappings known as preservers on the set of quantum effects for their role in conserving specific structures that are important in quantum mechanics. Since the preserves in this work mainly target the quantum effects, we provided principals of its corresponding algebra known as effect algebra. Then we advanced, to study the relations between some of these preservers (here we mean automorphisms) by channels operate on qubit effects. The channels are in convex form and we investigate the role of their convex parameters in conserving or breaking the coexistence of two qubit effects. The preserver that we are interested in is a bijective map which preserves the order and coexistence of effects in both directions. Of course, the coexistence of qubit effects can also be studied in the language of coexistence and compatibility of observables. Then our task is to study the coexistence of observables when there are only two effects in their outcome sets. This task has been studied for observables with arbitrary number of effects in their outcome sets [48]. The bounds they have found for the channels convex parameters are independent of that number. However, when the case narrows for limited number outcomes no analytic solution is available, and the study should be carried numerically. Here, we considered observables each having only two nontrivial effects in their out come sets, and asked for the demanding conditions that break their compatibility partially or completely.