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Physical Review A

Construction and classification of all equioverlapping measurements in qubit and qutrit systems

Zhihua Guo, Yan Liu, Tsung-Lin Lee, Shunlong Luo

Published Jan 21, 2025• 3 citations
Publisher

Abstract

The overlap between quantum measurement operators is a fundamental quantity, and the notion of an equioverlapping measurement captures a symmetric feature of quantum measurements in terms of overlap. Such equioverlapping measurements, as a special class of quantum measurements, interpolate between von Neumann measurements and symmetric informationally complete positive operator-valued measures (SIC-POVMs), and exhibit a rich and subtle structure. In this work, we classify and completely determine all equioverlapping measurements in qubit and qutrit systems.

DOI
10.1103/physreva.111.012430

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The overlap between quantum measurement operators is a fundamental quantity, and the notion of an equioverlapping measurement captures a symmetric feature of quantum measurements in terms of overlap. Such equioverlapping measurements, as a special class of quantum measurements, interpolate between von Neumann measurements and symmetric informationally complete positive operator-valued measures (SIC-POVMs), and exhibit a rich and subtle structure. In this work, we classify and completely determine all equioverlapping measurements in qubit and qutrit systems.
Equioverlapping measurements, as a natural class of quantum measurements with the appealing property of equal overlap between any pair of measurement operators, generalize both the notions of equiangular tight frames (including von Neumann measurements) and symmetric informationally complete positive operator valued measures (SIC-POVMs). The structures of equioverlapping measurements in general dimensions are rather subtle and complicated. In this work, we reveal some structural properties of equioverlapping measurements which may be useful for constructing and classifying equioverlapping measurements. In particular, we obtain two bounds for the number of measurement operators in an equioverlapping measurement. We further illustrate how equioverlapping measurements go beyond SIC-POVMs in a nontrivial fashion with some illuminating examples in lower dimensions (two, three, and four). Finally, we present the challenging problem of fully classifying equioverlapping measurements and discuss some related perspectives.
Generalized quantum measurements (also known as positive operator-valued measures or POVMs) are of great importance in quantum information and quantum foundations, but often difficult to perform. We present an experimental approach which can in principle be used to perform arbitrary POVMs in a linear-optical context. One of the most interesting POVMs, the symmetric, informationally complete-POVM (or SIC-POVM), is the most compact set of measurements that can be used to fully describe a quantum state. We use our technique to carry out the first experimental characterization of the state of a qutrit using SIC-POVMs. Because of the highly symmetric nature of this measurement, such a representation has the unique property that it permits all other measurement outcomes to be predicted by a simple extension of the classical Bayesian sum rule, making no use of complex amplitudes or Hilbert-space operators. We demonstrate this approach on several qutrit states encoded in single photons.
It is crucial for various quantum information processing tasks that the state of a quantum system can be determined reliably and efficiently from general quantum measurements. One important class of measurements for this purpose is symmetric informationally complete positive operator-valued measurements (SIC-POVMs). SIC-POVMs have the advantage of providing an unbiased estimator for the quantum state with the minimal number of outcomes needed for full tomography. By virtue of Naimark's dilation theorem, any POVM can always be realized with a suitable coupling between the system and an auxiliary system and by performing a projective measurement on the joint system. In practice, finding the appropriate coupling is rather nontrivial. Here we propose an experimental design for directly implementing SIC-POVMs using multiport devices and path-encoded qubits and qutrits, the utility of which has recently been demonstrated by several experimental groups around the world. Furthermore, we describe how these multiports can be attained in practice with an integrated photonic system composed of nested linear optical elements.
Generalized quantum measurements [also known as positive operator-valued measures (POVMs)] are of great importance in quantum information and quantum foundations but are often difficult to perform. We present an experimental approach which can in principle be used to perform arbitrary POVMs in a linear-optical context. One of the most interesting POVMs, the symmetric informationally complete (SIC) POVM, is the most compact set of measurements that can be used to fully describe a quantum state. We use our technique to carry out the first experimental characterization of the state of a qutrit using SIC POVMs. Because of the highly symmetric nature of this measurement, such a representation has the unique property that it permits all other measurement outcomes to be predicted by a simple extension of the classical Bayesian sum rule, making no use of complex amplitudes or Hilbert-space operators. We demonstrate this approach on several qutrit states encoded in single photons.
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.
An unexpected connection exists between compatibility criteria for quantum states and Symmetric Informationally Complete quantum measurements (SIC-POVMs). Beginning with Caves, Fuchs and Schack’s "Conditions for compatibility of quantum state assignments", I show that a qutrit SIC-POVM studied in other contexts enjoys additional interesting properties. Compatibility criteria provide a new way to understand the relationship between SIC-POVMs and mutually unbiased bases, as calculations in the SIC representation of quantum states make clear. This, in turn, illuminates the resources necessary for magic-state quantum computation, and why hidden-variable models fail to capture the vitality of quantum mechanics.