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.
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.
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 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→H. Uhlhorn’s theorem generalizes this result for bijective maps $\phi $ that are only assumed to preserve the quantum angle 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≤4π holds. In this paper we solve the remaining structural problem for quantum angles $\alpha $ that satisfy 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.
Let Φ : 𝔖(M1) → 𝔖(M2) be a bijection (not assumed affine nor continuous) between the sets of normal states of two quantum systems, modelled on the self-adjoint parts of von Neumann algebras M1 and M2, respectively. This paper concerns with the situation when Φ preserves (or partially preserves) one of the following three notions of “transition probability” on the normal state spaces: the transition probability PU introduced by Uhlmann [Rep. Math. Phys. 9, 273-279 (1976)], the transition probability PR introduced by Raggio [Lett. Math. Phys. 6, 233-236 (1982)], and an “asymmetric transition probability” P0 (as introduced in this article). It is shown that the two systems are isomorphic, i.e., M1 and M2 are Jordan ∗-isomorphic, if Φ preserves all pairs with zero Uhlmann (respectively, Raggio or asymmetric) transition probability, in the sense that for any normal states μ and ν, we have PΦ(μ),Φ(ν)=0 if and only if P(μ, ν) = 0, where P stands for PU (respectively, PR or P0). Furthermore, as an extension of Wigner’s theorem, it is shown that there is a Jordan ∗-isomorphism Θ : M2 → M1 satisfying Φ = Θ∗|𝔖(M1) if and only if Φ preserves the “asymmetric transition probability.” This is also equivalent to Φ preserving the Raggio transition probability. Consequently, if Φ preserves the Raggio transition probability, it will preserve the Uhlmann transition probability as well. As another application, the sets of normal states equipped with either the usual metric, the Bures metric or “the metric induced by the self-dual cone,” are complete Jordan ∗-invariants for the underlying von Neumann algebras.
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.
Symmetry in quantum mechanicsRoughly speaking, a symmetry of some mathematical object is an invertible transformation that leaves all relevant structure as it is.Thus a symmetry of a set is just a bijection (as sets have no further structure, whence invertibility is the only demand on a symmetry), a symmetry of a topological space is a homeomorphism, a symmetry of a Banach space is a linear isometric isomorphism, and, crucially important for this chapter, a symmetry of a Hilbert space H is a unitary operator, i.e., a linear map u : H → H satisfying one and hence all of the following equivalent conditions:• u is invertible and preserves the inner product, i.e., uϕ, uψ = ϕ, ψ (ϕ, ψ ∈ H).The discussion of symmetries in quantum physics is based on the above idea, but the mathematically obvious choices need not be the physically relevant ones.Even in elementary quantum mechanics, where A = B(H), i.e., the C*-algebra of all bounded operators on some Hilbert space H, the concept of a symmetry is already diverse.The main structures whose symmetries we shall study in this chapter are:1.The normal pure state space P 1 (H), i.e., the set of one-dimensional projections on H, with transition probability τ : P 1 (H) × P 1 (H) → [0, 1] defined by (2.44).2. The normal state space D(H), i.e. the convex set of density operators ρ on H. The self-adjoint operators B(H)sa on H, seen as a Jordan algebra (see below).4. The effects E (H) = [0, 1] B(H) , seen as a convex partially ordered set (poset). 5.The projections P(H) on H, seen as an orthocomplemented lattice.6.The unital commutative C*-subalgebras C (B(H)) of B(H), seen as a poset.Each of these structures comes with its own notion of a symmetry, but the main point of this chapter will be to show these notions are equivalent, corresponding in all cases to either unitary or-surprisingly-anti-unitary operators, both merely defined up to a phase.The latter subtlety will open the world of projective unitary group representation to quantum mechanics (without which the existence of spin-1 2 particles such as electrons, and therewith also of ourselves, would be impossible).