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arXiv:2304.13258

On the role of designs in the data-driven approach to quantum statistical inference

Michele Dall’Arno

Published Apr 26, 2023• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2304.13258

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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.
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.
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.
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.
Generalized symmetric informationally complete (SIC) measurements are SIC measurements that are not necessarily rank 1. They are interesting originally because of their connection with rank-1 SICs. Here we reveal several merits of generalized SICs in connection with quantum state tomography and Lie algebra that are interesting in their own right. These properties uniquely characterize generalized SICs among minimal informationally complete (IC) measurements although, on the face of it, they bear little resemblance to the original definition. In particular, we show that in quantum state tomography generalized SICs are optimal among minimal IC measurements with given average purity of measurement outcomes. Besides its significance to the current study, this result may help us to understand tomographic efficiencies of minimal IC measurements under the influence of noise. When minimal IC measurements are taken as bases for the Lie algebra of the unitary group, generalized SICs are uniquely characterized by the antisymmetry of the associated structure constants.
Minimal Informationally Complete quantum measurements, or MICs, illuminate the structure of quantum theory and how it departs from the classical. Central to this capacity is their role as tomographically complete measurements with the fewest possible number of outcomes for a given finite dimension. Despite their advantages, little is known about them. We establish general properties of MICs, explore constructions of several classes of them, and make some developments to the theory of MIC Gram matrices. These Gram matrices turn out to be a rich subject of inquiry, relating linear algebra, number theory and probability. Among our results are some equivalent conditions for unbiased MICs, a characterization of rank-1 MICs through the Hadamard product, several ways in which immediate properties of MICs capture the abandonment of classical phase space intuitions, and a numerical study of MIC Gram matrix spectra. We also present, to our knowledge, the first example of an unbiased rank-1 MIC which is not group covariant. This work provides further context to the discovery that the symmetric informationally complete quantum measurements (SICs) are in many ways optimal among MICs. In a deep sense, the ideal measurements of quantum physics are not orthogonal bases.
Identifying causal order from restricted projective data is generally nontrivial. When two quantum players interact only through an unobserved environment, the available local measurement statistics are typically not tomographically complete, so the underlying process cannot in general be reconstructed exactly from the observed distribution. As a result, causal direction can be statistically identifiable in some cases but fundamentally indistinguishable in others. In this work, we determine necessary and sufficient conditions for deciding when an observed distribution is compatible with a memoryless sequential quantum process in a fixed direction. We show that directional conditional-independence structure and the positivity criterion based on the pseudo-density matrix, as developed in recent work by We show that directional conditional-independence structure and PDM-based positivity criteria are not sufficient by themselves. We then specialize to the two-qubit Pauli setting, where the problem remains non-tomographic but becomes explicitly tractable. In this regime, we characterize when the two sequential directions are statistically indistinguishable, and we show by example that positivity alone does not exclude more general memoryful strategies, whereas the additional algebraic consistency requirement does.