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 ...
We report on a new class of dimension witnesses, based on quantum random access codes, which are a function of the recorded statistics and that have different bounds for all possible decompositions of a high-dimensional physical system. Thus, it certifies the dimension of the system and has the new distinct feature of identifying whether the high-dimensional system is decomposable in terms of lower dimensional subsystems. To demonstrate the practicability of this technique, we used it to experimentally certify the generation of an irreducible 1024-dimensional photonic quantum state. Therefore, certifying that the state is not multipartite or encoded using noncoupled different degrees of freedom of a single photon. Our protocol should find applications in a broad class of modern quantum information experiments addressing the generation of high-dimensional quantum systems, where quantum tomography may become intractable.
An important problem in quantum information processing is the certification of the dimension of quantum systems without making assumptions about the devices used to prepare and measure them, that is, in a device-independent manner. A crucial question is whether such certification is experimentally feasible for high-dimensional quantum systems. Here we experimentally witness in a device-independent manner the generation of six-dimensional quantum systems encoded in the orbital angular momentum of single photons and show that the same method can be scaled, at least, up to dimension 13. Introduction.-Dimensionality is a fundamental property of physical systems and a key resource in quantum information processing. Phenomena such as contextuality require systems of a certain minimum dimension to occur [1,2]; applications such as quantum secure communication have different levels of security depending on the dimension of the systems [3,4], and methods to characterize quantum states strongly depend on the assumed dimension of the systems [5]. It is therefore of crucial importance to develop methods to certify whether a source produces systems that have at least a certain dimension and to distinguish quantum systems from classical systems of the same dimension. The first theoretical tools for providing lower bounds on the dimension of quantum systems were based on Bell inequalities [6,7] and random access codes [8].
Measurement scenarios containing events with relations of exclusivity represented by pentagons, heptagons, nonagons, etc., or their complements are the only ones in which quantum probabilities cannot be described classically. Interestingly, quantum theory predicts that the maximum values for any of these graphs cannot be achieved in Bell inequality scenarios. With the exception of the pentagon, this prediction remained experimentally unexplored. Here we test the quantum maxima for the heptagon and the complement of the heptagon using three-and five-dimensional quantum states, respectively. In both cases, we adopt two different encodings: linear transverse momentum and orbital angular momentum of single photons. Our results exclude maximally noncontextual hidden-variable theories and are in good agreement with the maxima predicted by quantum theory.
We report a method that exploits a connection between quantum contextuality and graph theory to reveal any form of quantum contextuality in high-precision experiments. We use this technique to identify a graph which corresponds to an extreme form of quantum contextuality unnoticed before and test it using high-dimensional quantum states encoded in the linear transverse momentum of single photons. Our results open the door to the experimental exploration of quantum contextuality in all its forms, including those needed for quantum computation.
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