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Reference frames, superselection rules, and quantum information

Stephen D. Bartlett, Terry Rudolph, Robert W. Spekkens

TL;DR

This paper surveys how reference frames and superselection rules shape quantum information processing, recasting the lack of a shared RF as a decoherence effect and introducing G-twirling as a unifying tool. It shows that many tasks—communication, QKD, and entanglement distribution—remain feasible without a shared RF by exploiting relational encodings and decoherence-free subsystems, and it develops a quantum treatment of RFs that lifts SSRs via quantum reference frames. The authors discuss RF alignment, bounds with finite resources, and the asymptotic efficiency of RF-based protocols, revealing Heisenberg-limited scaling in several alignment tasks. They also explore bounded RFs, data hiding, and ancilla-free cryptography, highlighting both practical opportunities and fundamental limits, and they outline a broad research program for a general theory of unspeakable information across arbitrary groups. The work connects quantum metrology, cryptography, and relational quantum theory, offering a framework with potential impact on quantum technologies and foundational questions about reference frames in quantum mechanics.

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.

Reference frames, superselection rules, and quantum information

TL;DR

This paper surveys how reference frames and superselection rules shape quantum information processing, recasting the lack of a shared RF as a decoherence effect and introducing G-twirling as a unifying tool. It shows that many tasks—communication, QKD, and entanglement distribution—remain feasible without a shared RF by exploiting relational encodings and decoherence-free subsystems, and it develops a quantum treatment of RFs that lifts SSRs via quantum reference frames. The authors discuss RF alignment, bounds with finite resources, and the asymptotic efficiency of RF-based protocols, revealing Heisenberg-limited scaling in several alignment tasks. They also explore bounded RFs, data hiding, and ancilla-free cryptography, highlighting both practical opportunities and fundamental limits, and they outline a broad research program for a general theory of unspeakable information across arbitrary groups. The work connects quantum metrology, cryptography, and relational quantum theory, offering a framework with potential impact on quantum technologies and foundational questions about reference frames in quantum mechanics.

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.

Paper Structure

This paper contains 69 sections, 3 theorems, 224 equations.

Table of Contents

  1. Introduction -- Why consider reference frames in quantum information?
  2. Formalizing reference frames and superselection rules
  3. Reference frames in quantum theory
  4. Lacking a phase reference implies a photon-number superselection rule
  5. A general framework for reference frames and superselection rules
  6. Quantum information without a shared reference frame
  7. Communication without a shared reference frame
  8. Communication using photons without a shared phase reference
  9. Communication without a shared Cartesian frame
  10. One transmitted qubit.
  11. Two transmitted qubits: a classical channel.
  12. Three transmitted qubits: a quantum channel.
  13. Asymptotic behaviour.
  14. Relativistic considerations.
  15. Consequences for quantum information processing
  16. ...and 54 more sections

Key Result

Theorem 1

The action of $\mathcal{G}$ in terms of the decomposition is given by where $\mathcal{P}_q$ is the superoperator associated with projection into the charge sector $q$, that is, $\mathcal{P}_q[\rho]= \Pi_q \rho \Pi_q$ with $\Pi_q$ the projection onto $\mathcal{H}_q = \mathcal{M}_q \otimes \mathcal{N}_q$, $\mathcal{D}_{\mathcal{M}}$ denotes the trace-preserving operatio

Theorems & Definitions (4)

  • Theorem
  • proof : Proof of Theorem 1
  • Lemma : Schur's first
  • Lemma : Schur's second