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Tomographically-nonlocal entanglement

Roberto D. Baldijão, Marco Erba, David Schmid, John H. Selby, Ana Belén Sainz

Abstract

Entanglement is a central and subtle feature of quantum theory, whose structure and operational behavior can change dramatically when additional physical constraints, such as symmetries or superselection rules, are imposed. Such constraints can give rise to striking and counter-intuitive phenomena, including local broadcasting of entangled states and failures of entanglement monogamy. These effects naturally arise in tomographically nonlocal theories (like real quantum theory, twirled worlds, or fermionic quantum theory), where composite systems possess holistic degrees of freedom that are inaccessible to local measurements. In this work, we study entanglement in such theories within the framework of generalized probabilistic theories. We show that the failure of tomographic locality leads to two qualitatively distinct forms of entanglement, which we term $\textit{tomographically-local}$ entanglement and $\textit{tomographically-nonlocal}$ entanglement. We analyze the operational consequences of this distinction, proving that tomographically-nonlocal entanglement is useless for Bell nonlocality, steering, and teleportation, but sufficient for dense coding and perfectly secure data hiding. This framework clarifies the origin of several previously puzzling features of entanglement that arise when tomographic locality fails, as can happen even in quantum theory when one considers fermions or fundamental superselection rules.

Tomographically-nonlocal entanglement

Abstract

Entanglement is a central and subtle feature of quantum theory, whose structure and operational behavior can change dramatically when additional physical constraints, such as symmetries or superselection rules, are imposed. Such constraints can give rise to striking and counter-intuitive phenomena, including local broadcasting of entangled states and failures of entanglement monogamy. These effects naturally arise in tomographically nonlocal theories (like real quantum theory, twirled worlds, or fermionic quantum theory), where composite systems possess holistic degrees of freedom that are inaccessible to local measurements. In this work, we study entanglement in such theories within the framework of generalized probabilistic theories. We show that the failure of tomographic locality leads to two qualitatively distinct forms of entanglement, which we term entanglement and entanglement. We analyze the operational consequences of this distinction, proving that tomographically-nonlocal entanglement is useless for Bell nonlocality, steering, and teleportation, but sufficient for dense coding and perfectly secure data hiding. This framework clarifies the origin of several previously puzzling features of entanglement that arise when tomographic locality fails, as can happen even in quantum theory when one considers fermions or fundamental superselection rules.
Paper Structure (26 sections, 20 theorems, 113 equations)

This paper contains 26 sections, 20 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Composite GPT systems
  4. Tomographic locality and its failure
  5. Entanglement
  6. Tools for exploring the absence of tomographic locality
  7. Decomposition of vector spaces and holistic subspaces
  8. Projections onto the tomographically local and holistic subspaces
  9. Tomographically-nonlocal entanglement
  10. Example: Real Quantum Theory
  11. Systems in Real Quantum Theory
  12. Composite systems in real quantum theory
  13. Entanglement in real quantum theory
  14. TNL entanglement is useless for nonlocality, steering, and teleportation
  15. Tasks for which TNL entanglement is useful
  16. ...and 11 more sections

Key Result

Lemma 1

Given any composite system $\mathcal{AB}$ of two GPT systems $\mathcal{A}$ and $\mathcal{B}$, the span of product states (effects) is isomorphic to the algebraic tensor product $A\otimes B$ (resp., $A^*\otimes B^*$). i.e., the following isomorhpisms hold: and where $\otimes$ denotes the algebraic tensor product of real vector spaces. Moreover, the real vector spaces associated with the composite

Theorems & Definitions (50)

  • Definition 1: Composite GPT systems
  • Definition 2: Tomographic Locality (TL)
  • Definition 3: Entangled states
  • Definition 4: Entangled effects
  • Lemma 1: $AB$ contains $A\otimes B$
  • proof
  • Definition 5: Holistic subspaces
  • Lemma 2
  • Definition 6: Projection onto the tomographically local subspace
  • Proposition 1: Dual of $\Pi_{\rm TL}$
  • ...and 40 more