Device-Independent Certification of a Nonprojective Qubit Measurement

Abstract

Quantum measurements on a two-level system can have more than two independent outcomes, and in this case, the measurement cannot be projective. Measurements of this general type are essential to an operational approach to quantum theory, but so far, the nonprojective character of a measurement could only be verified experimentally by already assuming a specific quantum model of parts of the experimental setup. Here, we overcome this restriction by using a device-independent approach. In an experiment on pairs of polarization-entangled photonic qubits we violate by more than 8 standard deviations a Bell-like correlation inequality which is valid for all sets of two-outcome measurements in any dimension. We combine this with a device-independent verification that the system is best described by two qubits, which therefore constitutes the first device-independent certification of a nonprojective quantum measurement.The qubit is the abstract notion for any system which can be modeled in quantum theory by a two-level system. In such a system, any observable has at most two eigenvalues and hence any projective measurement can have at most two outcomes. Still, a qubit allows for an infinite number of different two-outcome measurements, the value of which, in general, cannot be known to the observer beforehand, but rather follows a binomial distribution. In quantum information theory, additional properties reflecting this binary structure have been revealed, e.g., the information capacity of a qubit is one classical bit, even when using entangled qubits [1]. Nonetheless, the properties of a qubit sometimes break with the binary structure, e.g., transferring the quantum state of a qubit is only possible with the communication of two classical bits and the help of entanglement [2]. Moreover, it is well-known that general quantum measurements can be nonprojective and have more than two irreducible outcomes [3]. The most general quantum measurement with n outcomes is described by positive semidefinite, possibly nonprojective, operators E 1 , E 2 , . . . , E n with E k = 1 1. The number of outcomes is reducible, if it is possible to writen are measurements for each λ, p λ is a probability distribution over λ, and for each λ there is at least one k λ with E (λ) k λ = 0. Nonprojective measurements found first applications in quantum information processing in the context of the discrimination of nonorthogonal quantum states. Ivanovic [4] found that it is possible to discriminate two pure qubit states without error even if the two states are nonorthogonal, but at the cost of allowing a third measurement outcome that indicates a failure of the discrimination procedure. The strategy with the lowest failure probability can be shown to be an irreducible three-outcome measurement [5]. Also recently, nonprojective measurements proved to be essential in purely information theoretical tasks like improving randomness certification [6].A peculiarity of nonprojective qubit measurements with more than two irreducible outcomes is that there ...

Enabling Computation of Correlation Bounds for Finite-Dimensional Quantum Systems via Symmetrization

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“…To this end, we introduce a family of CCPs, prove that they enable self-tests of d-dimensional symmetric informationally complete (SIC) POVMs, then use symmetrised semidefinite relaxations to bound the correlations attainable under projective measurements. This allows us to go beyond previously studied qubit systems [32][33][34][35][36] and robustly certify the nonprojectiveness of SIC-POVMs subject to imperfections.…”

Quantum steering

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“…A possible way to circumvent this problem is to consider a Bell scenario with quantum systems of bounded Hilbert space dimension. In particular, Refs [12,22] recently reported the experimental certification of a non-projective measurement in a Bell experiment assuming qubits. However, these experiments do not represent selftests, as they certify the non-projective character of a measurement, but not how it relates to a specific target POVM.…”