A symmetry witness is a subset of the space of selfadjoint trace class operators that allows one to ascertain whether a linear map acting in that space is a symmetry transformation. This notion arises from a certain type of linear preserver problems. Precisely, a symmetry witness is a suitable set which is invariant with respect to an injective linear map in the Banach space of selfadjoint trace class operators where the quantum states live if and only if this map acts as a symmetry transformation. In particular, by a linear version of Wigner's classical theorem, the set of pure states — the rank-one projections — is a symmetry witness. Linearity entails that the usual assumption of preservation of the transition probability between pure states becomes superfluous. This result extends to every set of projections of a fixed (finite) rank, with some suitable constraint on this rank. One then obtains a classification of the sets of projections of a fixed rank that are symmetry witnesses. These symmetry witnesses are projectable. Namely, formulating the mentioned result in terms of quantum states, the sets of 'uniform' density operators of a suitable fixed rank are symmetry witnesses as well.
A symmetry witness is a subset of the space of selfadjoint trace class operators that allows one to ascertain whether a linear map acting in that space is a symmetry transformation. This notion arises from a certain type of linear preserver problems. Precisely, a symmetry witness is a suitable set which is invariant with respect to an injective linear map in the Banach space of selfadjoint trace class operators where the quantum states live if and only if this map acts as a symmetry transformation. In particular, by a linear version of Wigner's classical theorem, the set of pure states — the rank-one projections — is a symmetry witness. Linearity entails that the usual assumption of preservation of the transition probability between pure states becomes superfluous. This result extends to every set of projections of a fixed (finite) rank, with some suitable constraint on this rank. One then obtains a classification of the sets of projections of a fixed rank that are symmetry witnesses. These symmetry witnesses are projectable. Namely, formulating the mentioned result in terms of quantum states, the sets of 'uniform' density operators of a suitable fixed rank are symmetry witnesses as well.
Abstract A symmetry witness is a suitable subset of the space of selfadjoint trace class operators that allows one to determine whether a linear map is a symmetry transformation, in the sense of Wigner. More precisely, such a set is invariant with respect to an injective densely defined linear operator in the Banach space of selfadjoint trace class operators (if and) only if this operator is a symmetry transformation. According to a linear version of Wigner’s theorem, the set of pure states—the rank-one projections—is a symmetry witness. We show that an analogous result holds for the set of projections with a fixed rank (with some mild constraint on this rank, in the finite-dimensional case). It turns out that this result provides a complete classification of the sets of projections with a fixed rank that are symmetry witnesses. These particular symmetry witnesses are projectable ; i.e. reasoning in terms of quantum states, the sets of ‘uniform’ density operators of corresponding fixed rank are symmetry witnesses too.
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).
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.
A cornerstone of quantum mechanics is the characterization of symmetries provided by Wigner's theorem. Wigner's theorem establishes that every symmetry of the quantum state space must be either a unitary transformation or an antiunitary transformation. Here we extend Wigner's theorem from quantum states to quantum evolutions, including both the deterministic evolution associated with the dynamics of closed systems and the stochastic evolutions associated with the outcomes of quantum measurements. We prove that every symmetry of the space of quantum evolutions can be decomposed into two state space symmetries that are either both unitary or both antiunitary. Building on this result, we show that it is impossible to extend the time-reversal symmetry of unitary quantum dynamics to a symmetry of the full set of quantum evolutions. Our no-go theorem implies that any time-symmetric formulation of quantum theory must either restrict the set of the allowed evolutions or modify the operational interpretation of quantum states and processes. Here we propose a time-symmetric formulation of quantum theory where the allowed quantum evolutions are restricted to a suitable set, which includes both unitary evolution and projective measurements but excludes the deterministic preparation of pure states. The standard operational formulation of quantum theory can be retrieved from this time-symmetric version by introducing an operation of conditioning on the outcomes of past experiments.
In recent years, the traditional notion of symmetry in quantum theory was expanded to so-called generalised or categorical symmetries, which, unlike ordinary group symmetries, may be non-invertible. This appears to be at odds with Wigner's theorem, which requires quantum symmetries to be implemented by (anti)unitary -- and hence invertible -- operators in order to preserve probabilities. We resolve this puzzle for (higher) fusion category symmetries C by proposing that, instead of acting by unitary operators on a fixed Hilbert space, symmetry defects in C act as isometries between distinct Hilbert spaces constructed from twisted sectors. As a result, we find that non-invertible symmetries naturally act as trace-preserving quantum channels. Crucially, our construction relies on the symmetry category C being unitary. We illustrate our proposal through several examples that include Tambara-Yamagami, Fibonacci, and Yang-Lee as well as higher categorical symmetries.