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Quantum Relativity of Subsystems

Shadi Ali Ahmad, Thomas D. Galley, Philipp A. Hoehn, Maximilian P. E. Lock, Alexander R. H. Smith

TL;DR

The paper develops a gauge-invariant, perspective-neutral framework for subsystems and entanglement in quantum systems with constraints, using quantum reference frames (QRFs) and relational Dirac observables. It shows that subsystem locality and observable factorization are generally frame-dependent, and that QRF transformations can map commuting subalgebras into non-commuting ones, altering correlations. A precise necessary-and-sufficient condition is derived for when kinematical subsystems inherit physical factorizability, expressed via a Minkowski-sum condition on constraint spectra. The authors illustrate the concepts with explicit examples, including two qubits becoming a physical qutrit under a clock-like constraint, and discuss implications for gauge theories and gravity. Overall, the work reframes subsystems and entanglement as inherently reference-frame-dependent, perspective-neutral constructs in quantum theory.

Abstract

One of the most basic notions in physics is the partitioning of a system into subsystems, and the study of correlations among its parts. In this work, we explore these notions in the context of quantum reference frame (QRF) covariance, in which this partitioning is subject to a symmetry constraint. We demonstrate that different reference frame perspectives induce different sets of subsystem observable algebras, which leads to a gauge-invariant, frame-dependent notion of subsystems and entanglement. We further demonstrate that subalgebras which commute before imposing the symmetry constraint can translate into non-commuting algebras in a given QRF perspective after symmetry imposition. Such a QRF perspective does not inherit the distinction between subsystems in terms of the corresponding tensor factorizability of the kinematical Hilbert space and observable algebra. Since the condition for this to occur is contingent on the choice of QRF, the notion of subsystem locality is frame-dependent.

Quantum Relativity of Subsystems

TL;DR

The paper develops a gauge-invariant, perspective-neutral framework for subsystems and entanglement in quantum systems with constraints, using quantum reference frames (QRFs) and relational Dirac observables. It shows that subsystem locality and observable factorization are generally frame-dependent, and that QRF transformations can map commuting subalgebras into non-commuting ones, altering correlations. A precise necessary-and-sufficient condition is derived for when kinematical subsystems inherit physical factorizability, expressed via a Minkowski-sum condition on constraint spectra. The authors illustrate the concepts with explicit examples, including two qubits becoming a physical qutrit under a clock-like constraint, and discuss implications for gauge theories and gravity. Overall, the work reframes subsystems and entanglement as inherently reference-frame-dependent, perspective-neutral constructs in quantum theory.

Abstract

One of the most basic notions in physics is the partitioning of a system into subsystems, and the study of correlations among its parts. In this work, we explore these notions in the context of quantum reference frame (QRF) covariance, in which this partitioning is subject to a symmetry constraint. We demonstrate that different reference frame perspectives induce different sets of subsystem observable algebras, which leads to a gauge-invariant, frame-dependent notion of subsystems and entanglement. We further demonstrate that subalgebras which commute before imposing the symmetry constraint can translate into non-commuting algebras in a given QRF perspective after symmetry imposition. Such a QRF perspective does not inherit the distinction between subsystems in terms of the corresponding tensor factorizability of the kinematical Hilbert space and observable algebra. Since the condition for this to occur is contingent on the choice of QRF, the notion of subsystem locality is frame-dependent.

Paper Structure

This paper contains 9 sections, 4 theorems, 51 equations, 1 figure.

Table of Contents

  1. Supplemental Material
  2. From physical states to Quantum Reference Frames (QRFs), including degeneracy
  3. Commuting subalgebras of relational observables
  4. Example: QRF dependent subsystem locality and correlations
  5. A necessary and sufficient condition for physical factorizability according to the kinematical subsystems
  6. (Non-)factorizability of the observable algebra in frame perspectives
  7. Kinematically commuting subsystem pairs can reduce to physically non-commuting subsystem pairs
  8. Two kinematical qubits can be one physical qutrit
  9. Frame-dependent factorizability

Key Result

Theorem 1

The algebra $\mathcal{A}^{C|A}_{\rm phys}$ of relational observables of $C$ relative to $A$, is distinct from the algebra $\mathcal{A}^{C|B}_{\rm phys}$ of relational observables of $C$ relative to $B$, $\mathcal{A}_{\rm phys}^{C\vert A}\neq\mathcal{A}_{\rm phys}^{C\vert B}$.

Figures (1)

  • Figure 1: The change from perspective $A$ to perspective $B$ takes the compositional form of a 'quantum coordinate transformation', $\Lambda_{A\to B}\colonequals\mathcal{R}_{B}^{(H)}\circ (\mathcal{R}^{(H)}_{A})^{-1}$vanrietvelde2020changeVanrietvelde:2018dithohn2020switchHoehn:2018whnhoehn2019trinityHLSrelativisticperiodtrinity. This induces a transformation on the algebra observables from the perspective of $A$ to the perspective of $B$, namely $\Lambda_{A\to B} \mathcal{A}_{BC|A} \Lambda_{A\to B}^{-1} \subseteq \mathcal{A}_{AC|B}$, where $\mathcal{A}_{ij|k} \simeq \mathcal{B}(\mathcal{H}_{ij|k})$.

Theorems & Definitions (7)

  • Theorem 1
  • Theorem 2
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof