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Bulletin of the American Physical Society

Operational Interpretation of the G-asymmetry for Abelian Groups

Michalis Skotiniotis, Gilad Gour

Published Feb 14, 2012• 0 citations
OpenAlex

Abstract

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 $\mathbb{Z}_M$, the cyclic group of M elements. Our result provides an operational interpretation for 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 variance in the case where G=U(1). Quite surprisingly, for the case where $G=\mathbb{Z}_{M}$ and $M\geq 4$, it is equal to a quantity that is not additive in general, but instead can be superadditive under tensor product of two distinct bounded-size reference frames. This remarkable phenomenon is purely quantum and has no classical analog.

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Stephen D. Bartlett, Terry Rudolph, Robert W. Spekkens +1 more

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).

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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 ZM\mathbb{Z}_MZM​, the cyclic group of M elements. Our result provides an operational interpretation for 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 variance in the case where G=U(1). Quite surprisingly, for the case where G=ZMG=\mathbb{Z}_{M}G=ZM​ and M≥4M\geq 4M≥4, it is equal to a quantity that is not additive in general, but instead can be superadditive under tensor product of two distinct bounded-size reference frames. This remarkable phenomenon is purely quantum and has no classical analog.
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 GGG, any quantum state that is noninvariant under the action of GGG 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 GGG-invariant state. Unlike the relative entropy of entanglement, this quantity is straightforward to calculate, and we find it to be precisely equal to the GGG-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 NNN-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 ddd (in particular d=2Nd=2^Nd=2N), the accessible classical information per run satisfies Iacc≤log⁡dI_{\mathrm{acc}}\le \log dIacc​≤logd, i.e. at most NNN bits for NNN 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)O(N)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)U(1)U(1) asymmetry.
Employing internal quantum systems as reference frames is a crucial concept in quantum gravity, gauge theories, and quantum foundations whenever external relata are unavailable. In this work, we give a comprehensive and self-contained treatment of such quantum reference frames (QRFs) for the case when the underlying configuration space is a finite Abelian group, significantly extending our previous work [M. Krumm, P. A. Höhn, and M. P. Müller, Quantum 5, 530 (2021)]. The simplicity of this setup admits a fully rigorous quantum information–theoretic analysis, while maintaining sufficient structure for exploring many of the conceptual and structural questions also pertinent to more complicated setups. We exploit this to derive several important structures of constraint quantization with quantum information–theoretic methods and to reveal the relation between different approaches to QRF covariance. In particular, we characterize the “physical Hilbert space”—the arena of the “perspective-neutral” approach—as the maximal subspace that admits frame-independent descriptions of purifications of states. We then demonstrate the kinematical equivalence and, surprising, dynamical inequivalence of the “perspective-neutral” and the “alignability” approach to QRFs. While the former admits unitaries generating transitions between arbitrary subsystem relations, the latter, remarkably, admits no such dynamics when requiring symmetry-preservation. We illustrate these findings by example of interacting discrete particles, including how dynamics can be described “relative to one of the subystems.”
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).