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Quantum complementarity

Davide Rolino, Paolo Perinotti, Alessandro Tosini

Abstract

We propose an operational definition of complementarity, pinning down the concept originally introduced by Bohr. Two properties of a system are considered complementary if they cannot be simultaneously well defined. We further show that, within quantum theory, this notion is equivalent to the incompatibility of operations -- that is, their inability to be performed simultaneously.

Quantum complementarity

Abstract

We propose an operational definition of complementarity, pinning down the concept originally introduced by Bohr. Two properties of a system are considered complementary if they cannot be simultaneously well defined. We further show that, within quantum theory, this notion is equivalent to the incompatibility of operations -- that is, their inability to be performed simultaneously.

Paper Structure

This paper contains 7 sections, 4 theorems, 43 equations.

Table of Contents

  1. Operational Probabilistic Theories
  2. Useful definitions
  3. Causality
  4. Conditional instruments
  5. Compatibility
  6. Complementarity in OPTs
  7. Relation between complementarity and incompatibility

Key Result

Lemma 1

Let $\mathsf{P}$ and $\mathsf{P'} \in \mathsf{Instr\left(\mathrm{A}\right)}$ be two elementary properties, and let $\nu \in \Upsilon\!\left(\mathsf{P}\right)$. Then $\nu$ is a verifier for $\mathsf{P'}$ if and only if where the entropy $H_{\nu}\left( \text{e}_{k} \right)$ is defined in eqt:compl:deg:shannon.

Theorems & Definitions (20)

  • Definition 1: Atomic transformation
  • Definition 2: Atomic instrument
  • Definition 3: Causal OPT
  • Definition 4: Conditional instruments
  • Remark 1
  • Definition 5: Does not exclude
  • Definition 6: Weakly-compatible instrument
  • Definition 7: Compatibility of observation-instruments
  • Definition 8: Repeatable instruments
  • Definition 9: Verifier state
  • ...and 10 more