We consider the problem of characterizing the set of input-output correlations that can be generated by an arbitrarily given quantum measurement. Our main result is to provide a closed-form, full characterization of such a set for any qubit measurement, and to discuss its geometrical interpretation. As applications, we further specify our results to the cases of real and complex symmetric, informationally complete measurements and mutually unbiased bases of a qubit, in the presence of isotropic noise. Our results provide the optimal device-independent tests of quantum measurements.
We consider the problem of characterizing the set of input-output correlations that can be generated by an arbitrarily given quantum measurement. Our main result is to provide a closed-form, full characterization of such a set for any qubit measurement, and to discuss its geometrical interpretation. As applications, we further specify our results to the cases of real and complex symmetric, informationally complete measurements and mutually unbiased bases of a qubit, in the presence of isotropic noise. Our results provide the optimal device-independent tests of quantum measurements.
We construct a correspondence between quantum states and the observable input-output correlations they are compatible with. The problem is framed as a game involving an experimenter, claiming to be able to prepare some family of states, and a theoretician, whose aim is to falsify such a claim based on observed correlations only. For any such claim, the optimal strategy consists of providing (i) to the experimenter, all the measurements that generate extremal input-output correlations, and (ii) to the theoretician, the full characterization of such correlations. Comparing the correlations observed in (i) with those predicted by (ii) corresponds to device-independently testing the states. While no assumption is made about the actual states and measurements, we derive the optimal strategy in closed form for the case when the claim consists of qubit states and the performed measurements are tests, and as applications we specify our results to the case of any pair of pure states and to the case of pure states uniformly distributed on the Bloch equatorial plane.
We develop a device-independent framework for testing quantum channels. That is, we falsify a hypothesis about a quantum channel based only on an observed set of input-output correlations. Formally, the problem consists of characterizing the set of input-output correlations compatible with any arbitrary given quantum channel. For binary (i.e. two input symbols, two output symbols) correlations, we show that extremal correlations are always achieved by orthogonal encodings and measurements, irrespective of whether or not the channel preserves commutativity. We further provide a full, closed-form characterization of the sets of binary correlations in the case of: (i) any dihedrally covariant qubit channel (such as any Pauli and amplitude-damping channels) and (ii) any universally-covariant commutativity-preserving channel in an arbitrary dimension (such as any erasure, depolarizing, universal cloning and universal transposition channels).
Recent theoretical developments are reviewed concentrating on the description of optimal generalised quantum measurement processes in physical systems with finite dimensional Hilbert spaces. Based on a self contained presentation of the required underlying quantum theoretical concepts this review particularly addresses recently explored basic theoretical properties and possible applications of informationally complete symmetric generalised measurements, a concept unifying the description of numerous so far largely separately explored measurement scenarios, such as measurements based on complete systems of mutually unbiased bases or symmetric informationally complete measurements. The recent results concerning the construction and existence of these informationally complete symmetric generalised measurements may also shed new light on still open questions concerning some of their special cases, such as the existence of complete sets of mutually unbiased bases in Hilbert spaces of non-prime power dimensions.
Starting from a simple estimation problem, here we propose a general approach for decoding quantum measurements from the perspective of information extraction.By virtue of the estimation fidelity only, we provide surprisingly simple characterizations of rank-1 projective measurements, mutually unbiased measurements, and symmetric informationally complete measurements.Notably, our conclusions do not rely on any assumption on the rank, purity, or the number of measurement outcomes, and we do not need bases to start with.Our work demonstrates that all these elementary quantum measurements are uniquely determined by their information-extraction capabilities, which are not even anticipated before.In addition, we offer a new perspective for understanding noncommutativity and incompatibility from tomographic performances, which also leads to a universal criterion for detecting quantum incompatibility.Furthermore, we show that the estimation fidelity can be used to distinguish inequivalent mutually unbiased bases and symmetric informationally complete measurements.In the course of study, we introduce the concept of (weighted complex projective) 1/2-designs and show that all 1/2-designs are tied to symmetric informationally complete measurements, and vice versa.
The demonstration and use of Bell-nonlocality, a concept that is fundamentally striking and is at the core of applications in device independent quantum information processing, relies heavily on the assumption of measurement independence, also called the assumption of free choice. The latter cannot be verified or guaranteed. In this paper, we consider a relaxation of the measurement independence assumption. We briefly review the results of Phys. Rev. Lett. 113, 190402 (2014), which show that with our relaxation, the set of so-called measurement dependent local (MDL) correlations is a polytope, i.e. it can be fully described using a finite set of linear inequalities. Here we analyze this polytope, first in the simplest case of 2 parties with binary inputs and outputs, for which we give a full characterization. We show that partially entangled states are preferable to the maximally entangled state when dealing with measurement dependence in this scenario. We further present a method which transforms any Bell-inequality into an MDL inequality and give valid inequalities for the case of arbitrary number of parties as well as one for arbitrary number of inputs. We introduce the assumption of independent sources in the measurement dependence scenario and give a full analysis for the bipartite scenario with binary inputs and outputs. Finally, we establish a link between measurement dependence and another strong hindrance in certifying nonlocal correlations: nondetection events.
Symmetric, informationally complete, positive operator-valued measurements (SIC-POVM) are fundamental in quantum information and foundations of quantum theory[1]. For example, SIC-POVMs are crucial for optimal quantum state tomography, quantum key distribution with optimal trade-off between security and key rate, device-independent certification of optimal randomness from one bit of entanglement, and information-based interpretations and operational reconstructions of quantum theory. Experimentally, photonic experiments of qubit and qutrit tomography, and quantum key distribution[2] have aimed for SIC-POVMs. However, these experiments rely on specific models for parts of the experimental setups and do not allow us to test whether the target measurements are truly SIC-POVMs in a "device independent" (DI) way that is, without relying on any modelling of the devices.