Performing perfect/conclusive quantum state exclusion means to be able to discard with certainty at least one out of n possible quantum state preparations by performing a measurement of the resulting state. This task of state exclusion has recently been studied at length in \cite{bandyopadhyay2014conclusive}, and it is at the heart of the celebrated PBR thought experiment \cite{pusey2012reality}. When all the preparations correspond to pure states and there are no more of them than their common dimension, it is an open problem whether POVMs give any additional power for this task with respect to projective measurements. This is the case even for the simple case of three states in three dimensions, which is mentioned in \cite{caves2002conditions} as unsuccessfully tackled. In this paper, we give an analytical proof that in this case considering POVMs does indeed not give any additional power with respect to projective measurements. To do so, we first make without loss of generality some assumptions about the structure of an optimal POVM. The justification of these assumptions involves arguments based on convexity, rank and symmetry properties. We show then that any pure states perfectly excluded by such a POVM meet the conditions identified in \cite{caves2002conditions} for perfect exclusion by a projective measurement of three pure states in three dimensions. We also discuss possible generalizations of our work, including an application of Quadratically Constrained Quadratic Programming that might be of special interest.
Performing perfect/conclusive quantum state exclusion means to be able to discard with certainty at least one out of n possible quantum state preparations by performing a measurement of the resulting state. This task of state exclusion has recently been studied at length in \cite{bandyopadhyay2014conclusive}, and it is at the heart of the celebrated PBR thought experiment \cite{pusey2012reality}. When all the preparations correspond to pure states and there are no more of them than their common dimension, it is an open problem whether POVMs give any additional power for this task with respect to projective measurements. This is the case even for the simple case of three states in three dimensions, which is mentioned in \cite{caves2002conditions} as unsuccessfully tackled. In this paper, we give an analytical proof that in this case considering POVMs does indeed not give any additional power with respect to projective measurements. To do so, we first make without loss of generality some assumptions about the structure of an optimal POVM. The justification of these assumptions involves arguments based on convexity, rank and symmetry properties. We show then that any pure states perfectly excluded by such a POVM meet the conditions identified in \cite{caves2002conditions} for perfect exclusion by a projective measurement of three pure states in three dimensions. We also discuss possible generalizations of our work, including an application of Quadratically Constrained Quadratic Programming that might be of special interest.
In the task of quantum state exclusion, we consider a quantum system prepared in a state chosen from a known set. The aim is to perform a measurement on the system which can conclusively rule that a subset of the possible preparation procedures cannot have taken place. We ask what conditions the set of states must obey in order for this to be possible and how well we can complete the task when it is not. The task of quantum state discrimination forms a subclass of this set of problems. Within this paper, we formulate the general problem as a semidefinite program (SDP), enabling us to derive sufficient and necessary conditions for a measurement to be optimal. Furthermore, we obtain a necessary condition on the set of states for exclusion to be achievable with certainty, and we give a construction for a lower bound on the probability of error. This task of conclusively excluding states has gained importance in the context of the foundations of quantum mechanics due to a result from Pusey, Barrett, and Rudolph (PBR). Motivated by this, we use our SDP to derive a bound on how well a class of hidden variable models can perform at a particular task, proving an analog of Tsirelson's bound for the PBR experiment and the optimality of a measurement given by PBR in the process. We also introduce variations of conclusive exclusion, including unambiguous state exclusion, and state exclusion with worst-case error.
Quantum state exclusion is the task of identifying at least one state from a known set that was not used in the preparation of a quantum system. A set of quantum states is said to admit state exclusion if there exists a measurement whose outcomes can be put in one-to-one correspondence with the states in the set, such that each outcome rules out its corresponding state with certainty (while possibly also ruling out other states), and each outcome occurs with nonzero probability for at least one state in the set. State exclusion, however, is not always possible in the single-copy setting. In this paper, we investigate whether access to multiple identical copies of the system enables state exclusion. We prove that for any set of three or more pure states, state exclusion becomes possible with a finite number of copies. Moreover, we show that the number of copies required may be arbitrarily large: in particular, for every natural number N, we construct sets of states for which state exclusion remains impossible with N or fewer copies.
Determining the conditions under which positive operator-valued measures (POVMs), the most general class of quantum measurements, outperform projective measurements remains a challenging and largely unresolved problem. Of particular interest are projectively simulable POVMs, which can be realized through probabilistic mixtures of projective measurements, and therefore offer no advantage over projective schemes. Characterizing the boundary between simulable and non-simulable POVMs is, however, a difficult task, and existing tools either fail to scale efficiently, provide limited experimental feasibility or work only for specific POVMs. Here, we introduce and demonstrate a general method to certify non-simulability of a POVM by introducing a complete hierarchy of semidefinite programs. It provides upper bounds on the non-simulability measure of critical visibility of arbitrary POVMs which are tight in many cases and outperform previously known criteria. We experimentally certify the non-simulability of two- and three-dimensional POVMs using a trapped-ion qudit quantum processor by constructing non-simulability witnesses and introduce a modification of our framework that makes them robust against state preparation errors. Finally, we extend our results to the setting where an additional ancilla system is available.
Quantum state exclusion is the task of determining which states from a given set a system was not prepared in. We provide a complete solution to optimal quantum state exclusion for arbitrary sets of pure states generated by finite groups, establishing necessary and sufficient conditions for perfect (zero-error conclusive) exclusion. When perfect exclusion is impossible, we introduce two natural extensions: minimum-error and unambiguous exclusion. For both, we derive the optimal protocols and present analytical expressions for the corresponding failure probabilities and measurements, providing additional insight into how quantum states encode information.
The optimization conditions for minimum error discrimination of linearly independent pure states comprise of two kinds: stationary conditions over the space of rank one projective measurements and the global maximization conditions. A discrete number of projective measurments will solve th former of which a unique one will solve the latter. In the case of three real linearly independent pure states we show that the stationary conditions translate to a system of simultaneous polynomial (non linear) equations in three variabes thus explaining why it's so difficult to obtain a closed-form solution for the optimal POVM. Additionally, our method suggests that as an ensemble of LI pure states is varied as a smooth function of some independent parameters, the optimal POVM will also vary smoothly as a function of the same parameters. By employing the implicit functions theorem we exploit this fact to obtain a technique to find the solution of MED of LI pure states by dragging the solution from a known example (say, pure orthogonal states) to any general linearly indepenent ensemble of pure states in the same Hilbert space. By employing RK4 to solve the first order coupled non-linear differential equations find that the resulting error is within the RK4 error performance.
In the heart of every physical theory lies measurement theory. While classical measurements are passive probes, QMs can be viewed as active physical processes that shape the available information. Depending on the measurement type, different aspects of the information about a system are retrieved. All possible measurements in quantum physics are described by mathematical objects called Positive Operator-Valued Measures (POVMs). The most familiar standard measurements are called Projective-Valued Measures (PVMs) and make up a subset of POVMs. The complementary subset of POVMs is non-projective. However, some non-projective POVMs can be expressed as a probabilistic mixture of PVMs. Such POVMs are called projectively simulable, while non-simulable POVMs are genuinely non-projective. These can be made projectively simulable by adding noise to the measurements. If a genuinely non-projective POVM requires a large amount of noise before it becomes projectively simulable, it is more non-projective than a POVM that requires less noise. Hence, the degree of non-projectiveness is determined by the required noise strength. The act of measuring a quantum system changes the quantum state. While quantum measurements only take into account the readable classical output value, quantum instruments account for both the classical outcome and the post-measurement state. Hence, instruments are associated with measurements, and in an analogous way, projective and projectively simulable instruments can be defined. For many purposes, non-projective POVMs are more useful than PVMs, and the more non-projective a POVM is, the greater the advantage of using them over PVMs. Therefore, the aim of the thesis is to investigate the genuinely non-projective POVMs. More precisely, the degree of non-projectiveness in qubit POVMs is studied. This is done by convex optimization, specifically by solving semidefinite programs. Furthermore, the thesis addresses the open problem of finding the most non-projective qubit quantum instrument by a witness-based see-saw algorithm.