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A universal scheme to self-test any quantum state or measurement

Shubhayan Sarkar, Alexandre C. Orthey,, Remigiusz Augusiak

Abstract

The emergence of quantum devices has raised a significant issue: how to certify the quantum properties of a device without placing trust in it. To characterise quantum states and measurements in a device-independent way, up to some degree of freedom, we can make use of a technique known as self-testing. While schemes have been proposed to self-test all pure multipartite entangled states (up to complex conjugation) and real local projective measurements, little has been done to certify mixed entangled states, composite or non-projective measurements. By employing the framework of quantum networks, we propose a scheme for self-testing (up to complex conjugation) arbitrary extremal measurements, including the projective ones, but also, in an indirect way, any quantum state, including the mixed ones and any quantum measurement, including non-extremal ones. The quantum network considered in this work is the simple star network, which is implementable using current technologies. For our purposes, we also construct a scheme that can be used to self-test the two-dimensional tomographically complete set of measurements with an arbitrary number of parties.

A universal scheme to self-test any quantum state or measurement

Abstract

The emergence of quantum devices has raised a significant issue: how to certify the quantum properties of a device without placing trust in it. To characterise quantum states and measurements in a device-independent way, up to some degree of freedom, we can make use of a technique known as self-testing. While schemes have been proposed to self-test all pure multipartite entangled states (up to complex conjugation) and real local projective measurements, little has been done to certify mixed entangled states, composite or non-projective measurements. By employing the framework of quantum networks, we propose a scheme for self-testing (up to complex conjugation) arbitrary extremal measurements, including the projective ones, but also, in an indirect way, any quantum state, including the mixed ones and any quantum measurement, including non-extremal ones. The quantum network considered in this work is the simple star network, which is implementable using current technologies. For our purposes, we also construct a scheme that can be used to self-test the two-dimensional tomographically complete set of measurements with an arbitrary number of parties.
Paper Structure (7 sections, 9 theorems, 211 equations, 2 figures)

This paper contains 7 sections, 9 theorems, 211 equations, 2 figures.

Table of Contents

  1. Appendix A: Self-testing of N-GHZ bases, measurements of external parties, and the states prepared by the preparation devices
  2. Bell inequalities
  3. Self-testing
  4. Appendix B: Self-testing any extremal POVM
  5. Alternative proof for projective measurements
  6. Appendix C: Self-testing any quantum state
  7. Appendix D: Robustness analysis of self-testing any extremal measurement

Key Result

Theorem 1

Consider the scenario depicted in Fig. fig1. If the Bell inequalities $\mathcal{I}_l$BE1Nm for any $l$ are maximally violated when Eve chooses the input $e=0$ with each outcome occurring with probability $\overline{P}(l|0)=1/2^N$, then the measurements of the external parties are equivalent, accordi

Figures (2)

  • Figure 1: Depiction of the quantum network scenario. It consists of $N+1$ parties, namely, $A_i$$(i=1,\ldots,N)$, and $E$, and $N$ independent sources distributing bipartite quantum states $\rho_{A_iE_i}$ among the parties as shown in the figure. The central party $E$ shares quantum states with each one of the other external parties $A_i$. While each $A_i$ has three inputs and two outcomes, $E$ has two inputs, The first Eve's measurement has $2^N-1$ outcomes, whereas the second one has $K$ outcomes.
  • Figure 2: Self-testing any quantum measurement. First, we have self-tested all the measurements performed by each external party $A_i$ as \ref{['GHZObsm']}, as well as, the sources as the maximally entangled two-qubit state $|\phi_{A_iE_i}^+\rangle$. Thus, Eve's first measurement is certified to the one in the GHZ basis \ref{['GHZvecsm']}. These are the ingredients sufficient to certify Eve's arbitrary second measurement.

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • proof
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • ...and 7 more