Self-testing nonprojective quantum measurements in prepare-and-measure experiments

Abstract

Self-testing represents the strongest form of certification of a quantum system. Here we investigate theoretically and experimentally the question of self-testing non-projective quantum measurements. That is, how can one certify, from observed data only, that an uncharacterised measurement device implements a desired non-projective positive-operator-valued-measure (POVM). We consider a prepare-and-measure scenario with a bound on the Hilbert space dimension, which we argue is natural for this problem since any measurement can be made projective by artificially increasing the Hilbert space dimension. We develop methods for (i) robustly self-testing extremal qubit POVMs (which feature either three or four outcomes), and (ii) certify that an uncharacterised qubit measurement is non-projective, or even a genuine four-outcome POVM. Our methods are robust to noise and thus applicable in practice, as we demonstrate in a photonic experiment. Specifically, we show that our experimental data implies that the implemented measurements are very close to certain ideal three and four outcome qubit POVMs, and hence non-projective. In the latter case, the data certifies a genuine four-outcome qubit POVM. Our results open interesting perspective for strong "black-box" certification of quantum devices.

Sequential random access codes and self-testing of quantum measurement instruments

“…They are primitives for network coding [4], random number generation [5] and quantum key distribution [6]. QRACs are also common in foundational aspects of quantum theory; examples include the comparison of different quantum resources [7,8], dimension witnessing [9], self-testing [10][11][12] and attempts at characterising quantum correlations from information-theoretic principles [13].…”

“…Self-testing is typically studied in Bell experiments where notably methods for self-testing quantum instruments have been developed [15,16]. Recently however, self-testing was introduced in the broad scope of prepare-and-measure scenarios [10], and was further developed using QRACs to robustly self-test both preparations and measurements [10][11][12]. Notably however, prepare-and-measure scenarios do not enable self-tests of general quantum operations.…”

Compounds of symmetric informationally complete measurements and their application in quantum key distribution

“…SIC POVMs are key tools for quantum state tomography [11][12][13], which has motivated their experimental realization in highdimensional Hilbert spaces [14][15][16]. Generally, SICs and SIC POVMs are used in a range of protocols: quantum key distribution (QKD) [17][18][19], entanglement detection [20][21][22], device-independent random number generation [23,24], dimension witnessing [25], and characterization of quantum devices [26][27][28][29][30]. Moreover, SICs have been studied in the context of quantum nonlocality [24,[31][32][33] and they have an interesting foundational role in QBism [34].…”

Weak form of self-testing

“…Self-testing is an active research field and a particularly interesting direction is to explore its powers and limitations by deriving new types of self-testing statements or impossibility results. For instance we have recently learned that one can self-test quantum channels [49], entangled measurements [50,51], and quantum instruments [52], or that one can extend the concept of self-testing to prepare-and-measure scenarios [53][54][55][56][57][58]. In this work we derive a new type of self-testing statement which allows us to certify the state but not the measurements.…”