Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Optimal Quantum Measurements with respect to the Fidelity

Datong Chen, Huangjun Zhu

Abstract

Fidelity is the standard measure for quantifying the similarity between two quantum states. It is equal to the square of the minimum Bhattacharyya coefficient between the probability distributions induced by quantum measurements on the two states. Though established for over thirty years, the structure of fidelity-optimal quantum measurements remains unclear when the two density operators are singular (not invertible). Here we address this gap, with a focus on minimal optimal measurements, which admit no nontrivial coarse graining that is still optimal. We show that there exists either a unique minimal optimal measurement or infinitely many inequivalent choices. Moreover, the first case holds if and only if the two density operators satisfy a weak commutativity condition. In addition, we provide a complete characterization of all minimal optimal measurements when one state is pure, leveraging geometric insights from the Bloch-sphere representation. The connections with quantum incompatibility, operator pencils, and geometric means are highlighted.

Optimal Quantum Measurements with respect to the Fidelity

Abstract

Fidelity is the standard measure for quantifying the similarity between two quantum states. It is equal to the square of the minimum Bhattacharyya coefficient between the probability distributions induced by quantum measurements on the two states. Though established for over thirty years, the structure of fidelity-optimal quantum measurements remains unclear when the two density operators are singular (not invertible). Here we address this gap, with a focus on minimal optimal measurements, which admit no nontrivial coarse graining that is still optimal. We show that there exists either a unique minimal optimal measurement or infinitely many inequivalent choices. Moreover, the first case holds if and only if the two density operators satisfy a weak commutativity condition. In addition, we provide a complete characterization of all minimal optimal measurements when one state is pure, leveraging geometric insights from the Bloch-sphere representation. The connections with quantum incompatibility, operator pencils, and geometric means are highlighted.

Paper Structure

This paper contains 21 sections, 32 theorems, 69 equations, 3 figures.

Table of Contents

  1. An example to support Theorem \ref{['thm:FoptimalEqui']}
  2. F-optimal measurements for distinguishing a pure state and a mixed state
  3. F-optimal measurements when $\rho+\sigma$ is singular
  4. Optimal measurements with respect to the trace distance
  5. Basic properties
  6. Proofs of Propositions \ref{['prop:DCoarse']}-\ref{['pro:ToptimalMinimal']}
  7. Auxiliary lemmas
  8. Properties of operator geometric means
  9. Basic properties
  10. Proofs of Lemmas \ref{['lem:GMsuppNull']}-\ref{['lem:GMcommu2']}
  11. Properties of operator pencils
  12. Basic properties
  13. Proofs of Lemmas \ref{['lem:PencilEigSpace']}-\ref{['lem:PencilEigPiC']}
  14. Proofs of general results on F-optimal measurements
  15. Proof of Proposition \ref{['pro:CoarseIneq']}
  16. ...and 6 more sections

Key Result

Proposition 1

Suppose $\rho, \sigma$ are two quantum states on $\mathcal{H}$ and $\mathscr{A},\mathscr{B}$ are two POVMs on $\mathcal{H}$, where $\mathscr{A}$ is a coarse graining of $\mathscr{B}$. Then

Figures (3)

  • Figure 1: Schematic diagram of the fidelity $F(\rho, \sigma)$ and the classical fidelity $F_\mathscr{E}(\rho,\sigma)$ induced by the POVM $\mathscr{E}$. The POVM $\mathscr{E}$ is F-optimal if $F_\mathscr{E}(\rho,\sigma)=F(\rho, \sigma)$.
  • Figure 2: Weak-commutativity condition and dichotomy between a unique minimal F-optimal POVM and infinitely many inequivalent choices as represented by different colors.
  • Figure 3: F-optimal POVMs for distinguishing two pure states $\rho$ and $\sigma$ illustrated on the Bloch sphere. Here $A$ and $B$ denote the two pure states $\rho$ and $\sigma$, while $M$ and $N$ denote their respective antipodal points. Each normalized POVM element of any F-optimal POVM is represented by a point on the major arc $\wideparen{NABM}$.

Theorems & Definitions (58)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Proposition 5
  • Proposition 6
  • ...and 48 more