The range of a quantum measurement is the set of outcome probability distributions that can be produced by varying the input state. We introduce data-driven inference as a protocol that, given a set of experimental data as a collection of outcome distributions, infers the quantum measurement which is (i) consistent with the data, in the sense that its range contains all the distributions observed, and (ii) maximally noncommittal, in the sense that its range is of minimum volume in the space of outcome distributions. We show that data-driven inference is able to return a unique measurement for any data set if and only if the inference adopts a (hyper)spherical state space (for example, the classical or the quantum bit). In analogy to informational completeness for quantum tomography, we define observational completeness as the property of any set of states that, when fed into any given measurement, produces a set of outcome distributions allowing for the correct reconstruction of the measurement via data-driven inference. We show that observational completeness is strictly stronger than informational completeness, in the sense that not all informationally complete sets are also observationally complete. Moreover, we show that for systems with a (hyper)spherical state space, the only observationally complete simplex is the regular one, namely, the symmetric informationally complete set.
The range of a quantum measurement is the set of outcome probability distributions that can be produced by varying the input state. We introduce data-driven inference as a protocol that, given a set of experimental data as a collection of outcome distributions, infers the quantum measurement which is (i) consistent with the data, in the sense that its range contains all the distributions observed, and (ii) maximally noncommittal, in the sense that its range is of minimum volume in the space of outcome distributions. We show that data-driven inference is able to return a unique measurement for any data set if and only if the inference adopts a (hyper)spherical state space (for example, the classical or the quantum bit). In analogy to informational completeness for quantum tomography, we define observational completeness as the property of any set of states that, when fed into any given measurement, produces a set of outcome distributions allowing for the correct reconstruction of the measurement via data-driven inference. We show that observational completeness is strictly stronger than informational completeness, in the sense that not all informationally complete sets are also observationally complete. Moreover, we show that for systems with a (hyper)spherical state space, the only observationally complete simplex is the regular one, namely, the symmetric informationally complete set.
Data-driven inference was recently introduced as a protocol that, upon the input of a set of data, outputs a mathematical description for a physical device able to explain the data. The device so inferred is automatically self-consistent, that is, capable of generating all given data, and least committal, that is, consistent with a minimal superset of the given dataset. When applied to the inference of an unknown device, data-driven inference has been shown to output always the "true" device whenever the dataset has been produced by means of an observationally complete setup, which plays here the same role played by informationally complete setups in conventional quantum tomography. In this paper we develop a unified formalism for the data-driven inference of states and measurements. In the case of qubits, in particular, we provide an explicit implementation of the inference protocol as a convex programming algorithm for the machine learning of states and measurements. We also derive a complete characterization of observational completeness for general systems, from which it follows that only spherical 2-designs achieve observational completeness for qubit systems. This result provides symmetric informationally complete sets and mutually unbiased bases with a new theoretical and operational justification.
Abstract Given a physical device as a black box, one can in principle fully reconstruct its input–output transfer function by repeatedly feeding different input probes through the device and performing different measurements on the corresponding outputs. However, for such a complete tomographic reconstruction to work, full knowledge of both input probes and output measurements is required. Such an assumption is not only experimentally demanding, but also logically questionable, as it produces a circular argument in which the characterization of unknown devices appears to require other devices to have been already characterized beforehand. Here, we introduce a method to overcome such limitations present in usual tomographic techniques. We show that, even without any knowledge about the tomographic apparatus, it is still possible to infer the unknown device to a high degree of precision, solely relying on the observed data. This is achieved by employing a criterion that singles out the minimal explanation compatible with the observed data. Our method, that can be seen as a data-driven analog of tomography, is solved analytically and implemented as an algorithm for the learning of qubit channels.
Designs, and in particular symmetric, informationally complete (SIC) structures, play an important role in the quantum tomographic reconstruction process and, by extension, in certain interpretations of quantum theory focusing on such a process. This fact is due to the symmetry of the reconstruction formula that designs lead to. However, it is also known that the same tomographic task, albeit with a less symmetric formula, can be accomplished by any informationally complete (non necessarily symmetric) structure. Here we show that, if the tomographic task is replaced by a data-driven inferential approach, the reconstruction, while possible with designs, cannot by accomplished anymore by an arbitrary informationally complete structure. Hence, we propose the data-driven inference as the arena in which the role of designs naturally emerges. Our inferential approach is based on a minimality principle according to which, among all the possible inferences consistent with the data, the weakest should be preferred, in the sense of majorization theory and statistical comparison.
Quantum computation has been growing rapidly in both theory and experiments.In particular, quantum computing devices with a large number of qubits have been developed by IBM, Google, IonQ, and others.The current quantum computing devices are noisy intermediate-scale quantum devices, and so approaches to validate quantum processing on these quantum devices are needed.One of the most common ways of validation for an n-qubit quantum system is quantum tomography, which tries to reconstruct a quantum system's density matrix by a complete set of observables.However, the inherent noise in the quantum systems and the intrinsic limitations pose a critical challenge to precisely know the actual measurement operators that make quantum tomography impractical in experiments.Here, we propose an alternative approach to quantum tomography, based on the maximal information entropy, that can predict the values of unknown observables based on the available mean measurement data.This can then be used to reconstruct the density matrix with high fidelity even though the results for some observables are missing.Of additional contexts, a practical approach to the inference of the quantum mechanical state using only partial information is also needed.
The concept of a quantum state represents one of the most fundamental pillars of the paradigm of quantum theory.'3 Contrary to its mathematical elegance and convenience in calculations, the physical interpretation of a quantum state is not so transparent. The problem is that the quantum state (described either by a state vector, or density operator or a phase-space probability density distribution) does not have a well defined objective status, i.e. a state vector is not an objective property of a particle. According to Peres (see,1 p. 374): "There is no physical evidence whatsoever that every physical system has at every instant a well defined state. .. In strict interpretation of quantum theory these mathematical symbols [i.e., state vectorsi represent statistical information enabling us to compute the probabilities of occurrence of specific events." Once this point of view is adopted then it becomes clear that any "measurement" or a reconstruction of a density operator (or its mathematical equivalent) can be understood exclusively as an expression of our knowledge about the quantum mechanical state based on a certain set of measured data. To be more specific, any quantum-mechanical reconstruction scheme is nothing more than an a posteriori estimation of the density operator of a quantum-mechanical (microscopic) system based on data obtained with the help of a macroscopic measurement apparatus.3 The quality of the reconstruction depends on the "quality" of the measured data and the efficiency of the reconstruction procedure with the help of which the data analysis is performed. In particular, we can specify three different situations. Firstly, when all system observables are precisely measured. In this case the complete reconstruction of an initially unknown state can be performed (we will call this the reconstruction on the complete observation level) . Secondly, when just part of the system observables is precisely measured then one cannot perform a complete reconstruction of the measured state. Nevertheless, the reconstructed density operator still uniquely determines mean values of the measured observables (we will denote this scheme as reconstruction on incomplete observation levels) . Finally, when measurement does not provide us with sufficient information to specify the exact mean values (or probability distributions) but only the frequencies of appearances of eigenstates of the measured observables, then one can perform an estimation (e.g. reconstruction based on quantum Bayesian inference) which is the "best" with respect to the given measured data and the a priori knowledge about the state of the measured system.
Quantum state tomography is the task of inferring the state of a quantum system by appropriate measurements. Since the frequency distributions of the outcomes of any finite number of measurements will generally deviate from their asymptotic limits, the estimates computed by standard methods do not in general coincide with the true state and, therefore, have no operational significance unless their accuracy is defined in terms of error bounds. Here we show that quantum state tomography, together with an appropriate data analysis procedure, yields reliable and tight error bounds, specified in terms of confidence regions-a concept originating from classical statistics. Confidence regions are subsets of the state space in which the true state lies with high probability, independently of any prior assumption on the distribution of the possible states. Our method for computing confidence regions can be applied to arbitrary measurements including fully coherent ones; it is practical and particularly well suited for tomography on systems consisting of a small number of qubits, which are currently in the focus of interest in experimental quantum information science.