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Self-testing of semisymmetric informationally complete measurements in a qubit prepare-and-measure scenario

Gábor Drótos, Károly F. Pál, Tamás Vértesi

TL;DR

This work develops an analytic framework for self-testing a one-parameter family of four-outcome qubit POVMs, the semi-SIC POVMs, in a semi-device-independent prepare-and-measure scenario with a dimension bound. The authors first self-test four pure qubit states using a two-parameter witness, then relate these states to the semi-SIC Bloch vectors by matching dot products, deriving explicit relations c1(B) and c2(B) that certify the semi-SIC structure for $B\in(1/16,1/12]$. They extend the witness to include a fourth measurement setting, showing that the extended POVM must also be semi-SIC with opposite Bloch vectors, thereby self-testing the full semi-SIC POVM in the PM framework. In the SIC limit ($B=1/12$) they recover $Q=4\sqrt{3}$, illustrating the method's consistency with known SIC self-testing results and highlighting a path toward self-testing any extremal qubit POVM in a minimal PM configuration. The results provide a concrete, analytic route to certify extremal measurements in a minimal semi-DI setting, with potential extensions to noisy scenarios and broader classes of extremal POVMs.

Abstract

Self-testing is a powerful method for certifying quantum systems. Initially proposed in the device-independent (DI) setting, self-testing has since been relaxed to the semi-device-independent (semi-DI) setting. In this study, we focus on the self-testing of a specific type of non-projective qubit measurements belonging to a one-parameter family, using the semi-DI prepare-and-measure (PM) scenario. Remarkably, we identify the simplest PM scenario discovered so far, involving only four preparations and four measurements, for self-testing the fourth measurement. This particular measurement is a four-outcome non-projective positive operator-valued measure (POVM) and falls in the class of semisymmetric informationally complete (semi-SIC) POVMs introduced by Geng et al. [Phys. Rev. Lett. 126, 100401 (2021)]. To achieve this, we develop analytical techniques for semi-DI self-testing in the PM scenario. Our results shall pave the way towards self-testing any extremal qubit POVM within a potentially minimal PM scenario.

Self-testing of semisymmetric informationally complete measurements in a qubit prepare-and-measure scenario

TL;DR

This work develops an analytic framework for self-testing a one-parameter family of four-outcome qubit POVMs, the semi-SIC POVMs, in a semi-device-independent prepare-and-measure scenario with a dimension bound. The authors first self-test four pure qubit states using a two-parameter witness, then relate these states to the semi-SIC Bloch vectors by matching dot products, deriving explicit relations c1(B) and c2(B) that certify the semi-SIC structure for . They extend the witness to include a fourth measurement setting, showing that the extended POVM must also be semi-SIC with opposite Bloch vectors, thereby self-testing the full semi-SIC POVM in the PM framework. In the SIC limit () they recover , illustrating the method's consistency with known SIC self-testing results and highlighting a path toward self-testing any extremal qubit POVM in a minimal PM configuration. The results provide a concrete, analytic route to certify extremal measurements in a minimal semi-DI setting, with potential extensions to noisy scenarios and broader classes of extremal POVMs.

Abstract

Self-testing is a powerful method for certifying quantum systems. Initially proposed in the device-independent (DI) setting, self-testing has since been relaxed to the semi-device-independent (semi-DI) setting. In this study, we focus on the self-testing of a specific type of non-projective qubit measurements belonging to a one-parameter family, using the semi-DI prepare-and-measure (PM) scenario. Remarkably, we identify the simplest PM scenario discovered so far, involving only four preparations and four measurements, for self-testing the fourth measurement. This particular measurement is a four-outcome non-projective positive operator-valued measure (POVM) and falls in the class of semisymmetric informationally complete (semi-SIC) POVMs introduced by Geng et al. [Phys. Rev. Lett. 126, 100401 (2021)]. To achieve this, we develop analytical techniques for semi-DI self-testing in the PM scenario. Our results shall pave the way towards self-testing any extremal qubit POVM within a potentially minimal PM scenario.
Paper Structure (11 sections, 66 equations, 2 figures)

This paper contains 11 sections, 66 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The qubit semi-SIC POVM
  3. Self-testing in the qubit PM scenario
  4. Self-testing specific qubit states in the PM scenario
  5. Correspondence between the specific states and semi-SIC POVMs
  6. Self-testing semi-SIC POVMs
  7. Discussions
  8. Finding the global maximum via the relationship between arithmetic and quadratic means
  9. Proof about the configuration of the optimal vectors
  10. Finding the global maximum by differentiation
  11. Excluding generic two-outcome POVMs as Bob's measurements

Figures (2)

  • Figure 1: The Bloch vectors \ref{['h1to4']} of a semi-SIC POVM for (a) $B = 1/15$ and (b) $B = 1/12$. Note that panel (b) corresponds to a SIC POVM.
  • Figure 2: The prepare-and-measure scenario considered in this study. Panel (a) shows the setup corresponding to the first part of the self-testing problem, while panel (b) shows its extension.