Reference frames, superselection rules, and quantum information

Abstract

Recently, there has been much interest in a new kind of ``unspeakable'' quantum information that stands to regular quantum information in the same way that a direction in space or a moment in time stands to a classical bit string: the former can only be encoded using particular degrees of freedom while the latter are indifferent to the physical nature of the information carriers. The problem of correlating distant reference frames, of which aligning Cartesian axes and synchronizing clocks are important instances, is an example of a task that requires the exchange of unspeakable information and for which it is interesting to determine the fundamental quantum limit of efficiency. There have also been many investigations into the information theory that is appropriate for parties that lack reference frames or that lack correlation between their reference frames, restrictions that result in global and local superselection rules. In the presence of these, quantum unspeakable information becomes a new kind of resource that can be manipulated, depleted, quantified, etcetera. Methods have also been developed to contend with these restrictions using relational encodings, particularly in the context of computation, cryptography, communication, and the manipulation of entanglement. This article reviews the role of reference frames and superselection rules in the theory of quantum information processing.Comment: 55 pages, published versio

Extending Noether’s theorem by quantifying the asymmetry of quantum states

“…Note that in this case the regular representation of the group is not finite dimensional. Nonetheless, as has been noted previously 4,27 , there still exists a sequence of finite d-dimensional spaces, {H d : dAN}, and for each H d an over-complete basis {|gS: gAG} such that the unitary representation g-U(g) of the symmetry group G acts as 8g; h 2 G : UðgÞ jhi ¼jghi: ð16Þ Furthermore, for a given pair of distinct group elements, g 1 ag 2 , one can make the inner product /g 2 |g 1 S arbitrarily close to zero in the limit of large d 4,27 . In this limit, the state |eS/e| has the maximal asymmetry in the sense that for any given state r on an arbitrary Hilbert space H, there exists a symmetric channel E r such that…”

“…The figure of merit for a metrology task is therefore a measure of asymmetry and dictates the optimal r. Suppose, for instance, that one seeks an unbiased estimatorf of f and the figure of merit is the variance inf, denoted VarðfÞ. It has previously been shown 19 that for a state r, we have VarðfÞ 1=4S N; 1 2 ðrÞ (a quantum generalization of the Cramer-Rao bound) where S N; 1 2 ðrÞ is the Wigner-Araki-Dyson skew information of order s ¼ 1/2, defined in Equation (4). Hence, it is the latter measure of asymmetry that is relevant in this case.…”

“…A symmetric evolution is one that commutes with the action of the symmetry group 4 . For instance, a rotationally invariant dynamics is such that a rotation of the state before the dynamics has the same effect as doing so after the dynamics.…”

Quantum time

“…The first refers to superselection [39][40][41]: the observables of a theory must commute with the theory's constraints. Whenever one of the constraints is the total energy, such as in canonical general relativity, then all observables must be stationary as they commute with the Hamiltonian.…”

Spectral properties of reduced fermionic density operators and parity superselection rule

“…Given a system with a varying number of particles, it is therefore possible to have a superposition of half-integer and integer total spins. However, it was recognized later that the comparison of phases between states with halfinteger and integer angular momenta cannot be reconciled with the requirement of relativistic invariance [12], which resulted in the formulation of parity (spinor, univalence) superselection rule (see, e.g., the reviews [13][14][15]). Any superselection rule relies on a group of physical transformations, for example the parity superselection rule relies on the rotational invariance [16] whereas the mass superselection rule relies on the Galilean invariance of nonrelativistic quantum mechanics [17].…”