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arXiv:quant-ph/0410083 · Physical Review A

Quantifying the resource of sharing a reference frame

S. J. van Enk

Published Mar 24, 2005• 42 citations
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Abstract

We define a quantity called a refbit, which allows one to quantify the resource of sharing a reference frame in quantum communication protocols. By considering both asymptotic and nonasymptotic protocols we find relations between refbits and other communication resources. We also consider the same resources in encoded, reference-frame-independent, form. This allows one to rephrase and unify previous work on phase references, reference frames, and superselection rules.

DOI
10.1103/physreva.71.032339

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We define a quantity called a refbit, which allows one to quantify the resource of sharing a reference frame in quantum communication protocols. By considering both asymptotic and nonasymptotic protocols we find relations between refbits and other communication resources. We also consider the same resources in encoded, reference-frame-independent, form. This allows one to rephrase and unify previous work on phase references, reference frames, and superselection rules.
We consider quantum communication in the case that the communicating parties not only do not share a reference frame but use imperfect quantum communication channels, in that each channel applies some fixed but unknown unitary rotation to each qubit. We discuss similarities and differences between reference frames within that quantum communication model and gauge fields in gauge theory. We generalize the concept of refbits and analyze various quantum communication protocols within the communication model.
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, etc. 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 paper reviews the role of reference frames and superselection rules in the theory of quantum-information processing.
We propose that observables in quantum theory are properly understood as representatives of symmetry-invariant quantities relating one system to another, the latter to be called a reference system. We provide a rigorous mathematical language to introduce and study quantum reference systems, showing that the orthodox “absolute” quantities are good representatives of observable relative quantities if the reference state is suitably localised. We use this relational formalism to critique the literature on the relationship between reference frames and superselection rules, settling a long-standing debate on the subject.
We propose that observables in quantum theory are properly understood as representatives of symmetry-invariant quantities relating one system to another, the latter to be called a reference system. We provide a rigorous mathematical language to introduce and study quantum reference systems, showing that the orthodox "absolute" quantities are good representatives of observable relative quantities if the reference state is suitably localised. We use this relational formalism to critique the literature on the relationship between reference frames and superselection rules, settling a long-standing debate on the subject.
This note shows that several apparently different problems in quantum physics share one underlying structure: the role of physical reference frames and the information that becomes inaccessible when the required reference frame is unavailable. A companion program studied exact frame-dependent information deficits in solvable models, including charge-locked information in Hawking-type radiation, the loss of timing information in constrained quantum systems, and the quantum cost of interacting clocks. This work identifies these results as instances of the resource theory of asymmetry, where inaccessible information is quantified by the relative entropy of asymmetry. The note proves that the previously computed deficits are evaluations of the same asymmetry resource, including the information deficit in charge sectors, the timing deficit in constrained systems, and related entropy quantities. It also shows that the matching conditions for recovering hidden information are unified: a physical reference frame must contain the appropriate symmetry content of the locked coherence. A further result is a general resolution-disturbance tradeoff for physical reference frames. For pure reference frames, the product of measurement resolution and induced decoherence is fixed by the coupling strength and the energy or generator gap. Mixed reference frames perform strictly worse, showing that imperfect reference frames pay an additional quantum cost. The work establishes a unified framework connecting superselection rules, quantum reference frames, relational time, information accessibility, and decoherence. The results are derived and verified in finite exactly solvable models, with declared limits: no claim of a complete theory of dynamical gravity is made. Verification and reproducibility materials are included.
Every restriction on quantum operations defines a resource theory, determining how quantum states that cannot be prepared under the restriction may be manipulated and used to circumvent the restriction. A superselection rule (SSR) is a restriction that arises through the lack of a classical reference frame and the states that circumvent it (the resource) are quantum reference frames. We consider the resource theories that arise from three types of SSRs, associated respectively with lacking: (i) a phase reference, (ii) a frame for chirality, and (iii) a frame for spatial orientation. Focusing on pure unipartite quantum states (and in some cases restricting our attention even further to subsets of these), we explore single-copy and asymptotic manipulations. In particular, we identify the necessary and sufficient conditions for a deterministic transformation between two resource states to be possible and, when these conditions are not met, the maximum probability with which the transformation can be achieved. We also determine when a particular transformation can be achieved reversibly in the limit of arbitrarily many copies and find the maximum rate of conversion. A comparison of the three resource theories demonstrates that the extent to which resources can be interconverted decreases as the strength of the restriction increases. Along the way, we introduce several measures of frameness and prove that these are monotonically non-increasing under various classes of operations that are permitted by the SSR.