We determine the quantum states and measurements that optimize the accessible information in a reference frame alignment protocol associated with the groups U (1), corresponding to a phase reference, and , the cyclic group of M elements. Our result provides an operational interpretation of the G -asymmetry which is information-theoretic and which was thus far lacking. In particular, we show that in the limit of many copies of the bounded-size quantum reference frame, the accessible information approaches the Holevo bound. This implies that the rate of alignment of reference frames, measured by the (linearized) accessible information per system, is equal to the regularized, linearized G -asymmetry. The latter quantity is equal to the number variance in the case where G = U (1). Quite surprisingly, for the case where and M ⩾ 4, it is equal to a quantity that is not additive in general, but instead can be super-additive under tensor product of two distinct bounded-size reference frames. This remarkable phenomenon is purely quantum and has no classical analogue.
We determine the quantum states and measurements that optimize the accessible information in a reference frame alignment protocol associated with the groups U (1), corresponding to a phase reference, and , the cyclic group of M elements. Our result provides an operational interpretation of the G -asymmetry which is information-theoretic and which was thus far lacking. In particular, we show that in the limit of many copies of the bounded-size quantum reference frame, the accessible information approaches the Holevo bound. This implies that the rate of alignment of reference frames, measured by the (linearized) accessible information per system, is equal to the regularized, linearized G -asymmetry. The latter quantity is equal to the number variance in the case where G = U (1). Quite surprisingly, for the case where and M ⩾ 4, it is equal to a quantity that is not additive in general, but instead can be super-additive under tensor product of two distinct bounded-size reference frames. This remarkable phenomenon is purely quantum and has no classical analogue.
In the absence of a reference frame for transformations associated with group G, any quantum state that is noninvariant under the action of G may serve as a token of the missing reference frame. We here present a measure of the quality of such a token: the relative entropy of frameness. This is defined as the relative entropy distance between the state of interest and the nearest G-invariant state. Unlike the relative entropy of entanglement, this quantity is straightforward to calculate, and we find it to be precisely equal to the G-asymmetry, a measure of frameness introduced by Vaccaro et al. It is shown to provide an upper bound on the mutual information between the group element encoded into the token and the group element that may be extracted from it by measurement. In this sense, it quantifies the extent to which the token successfully simulates a full reference frame. We also show that despite a suggestive analogy from entanglement theory, the regularized relative entropy of frameness is zero and therefore does not quantify the rate of interconversion between the token and some standard form of quantum reference frame. Finally, we show how these investigations yield an approach to bounding the relative entropy of entanglement.
This note clarifies a fundamental but often-misunderstood limitation of quantum computation: regardless of the exponentially large Hilbert space of an N-qubit register, the amount of classical information that can be extracted from a single run is bounded by the quantum-to-classical measurement interface. Using the accessible information framework and the Holevo bound, we show that for a system of dimension d (in particular d=2N), the accessible classical information per run satisfies Iacc≤logd, i.e. at most N bits for N qubits (log base 2). We emphasize that quantum advantage is therefore not about “reading out” exponentially many classical bits, but about how interference structures the distribution of O(N) measured bits so that they encode useful answers. Finally, we discuss why temporal/phase coordination becomes an architectural requirement in *distributed* quantum computing and quantum networks: without a shared phase/time reference, operations are symmetry-restricted and measurement events across nodes may fail to compose reliably. This motivates cycle-anchored phase-window coordination and reference-frame/asymmetry viewpoints as practical foundations for scalable distributed systems. Keywordsquantum measurement, Holevo bound, accessible information, readout bottleneck, distributed quantum computing, quantum networks, phase coordination, U(1) asymmetry.
Abstract Symmetry principles are fundamental in physics, and while they are well understood within Lagrangian mechanics, their impact on quantum channels has a range of open questions. The theory of asymmetry grew out of information-theoretic work on entanglement and quantum reference frames, and allows us to quantify the degree to which a quantum system encodes coordinates of a symmetry group. Recently, a complete set of entropic conditions was found for asymmetry in terms of correlations relative to infinitely many quantum reference frames. However, these conditions are difficult to use in practice and their physical implications unclear. In the present theoretical work, we show that this set of conditions has extensive redundancy, and one can restrict to reference frames forming any closed surface in the state space that has the maximally mixed state in its interior. This in turn implies that asymmetry can be reduced to just a single entropic condition evaluated at the maximally mixed state. Contrary to intuition, this shows that we do not need macroscopic, classical reference frames to determine the asymmetry properties of a quantum system, but instead infinitesimally small frames suffice. Building on this analysis, we provide simple, closed conditions to estimate the minimal depolarization needed to make a given quantum state accessible under channels covariant with any given symmetry group.
Typical quantum communication schemes are such that to achieve perfect decoding the receiver must share a reference frame (RF) with the sender. Indeed, if the receiver only possesses a bounded-size quantum token of the sender's RF, then the decoding is imperfect, and we can describe this effect as a noisy quantum channel. We seek here to characterize the performance of such schemes, or equivalently, to determine the effective decoherence induced by having a bounded-size RF. We assume that the token is prepared in a special state that has particularly nice group-theoretic properties and that is near-optimal for transmitting information about the sender's frame. We present a decoding operation, which can be proven to be near-optimal in this case, and we demonstrate that there are two distinct ways of implementing it (corresponding to two distinct Kraus decompositions). In one, the receiver measures the orientation of the RF token and reorients the system appropriately. In the other, the receiver extracts the encoded information from the virtual subsystems that describe the relational degrees of freedom of the system and token. Finally, we provide explicit characterizations of these decoding schemes when the system is a single qubit and for three standard kinds of RF: a phase reference, a Cartesian frame (representing an orthogonal triad of spatial directions), and a reference direction (representing a single spatial direction).
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.
We formulate quantum reference frames within the Dirac quantization of gauge theories, where physical states are obtained from a kinematical Hilbert space Hkin by group averaging. We assume the gauge group G is unimodular, and we define a quantum reference frame as a system of coherent states associated with a group GR of frame orientations acting on a tensor factor of Hkin. In direct analogy with the Faddeev-Popov procedure in path integrals, we show that the physical inner product can be written in terms of a choice of frame orientation with an appropriate operator measure, but that the physical Hilbert space is independent of this choice of frame. This yields a generalization of the Page-Wootters reduction map used to define conditional states and conditional probabilities of measurement outcomes given a frame orientation. These probabilities correspond to the expectation values of relational Dirac observables obtained from generalizations of previously considered relativization maps. Our main result is that the map that implements a change of frame is unitary, even when G does not act as a tensor-product representation across a factorization of Hkin. The formalism therefore accommodates interactions between reference frames and the systems described relative to them. We apply this formalism to temporal frames, showing that a unitary Schrödinger equation emerges for an interacting clock and system, and that a previously reported time-nonlocal Schrödinger equation is recovered in an appropriate limit. With this, problems of non-unitarity are avoided by the use of the gauge-fixing measure in the physical inner product. We also discuss the case of interacting spatial frames, where the known correspondence between coherence in one frame and entanglement in another is shown to depend on the coupling.