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arXiv:1907.10999 · Physical Review A

Lüders channels and the existence of symmetric-informationally-complete measurements

John B. DeBrota, Blake C. Stacey

Published Dec 19, 2019• 18 citations
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Abstract

The L\"uders rule provides a way to define a quantum channel given a quantum measurement. Using this construction, we establish an if-and-only-if condition for the existence of a $d$-dimensional symmetric-informationally-complete quantum measurement (a SIC) in terms of a particular depolarizing channel. Moreover, the channel in question satisfies two entropic optimality criteria.

DOI
10.1103/physreva.100.062327

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The L"uders rule provides a way to define a quantum channel given a quantum measurement. Using this construction, we establish an if-and-only-if condition for the existence of a ddd-dimensional symmetric-informationally-complete quantum measurement (a SIC) in terms of a particular depolarizing channel. Moreover, the channel in question satisfies two entropic optimality criteria.
The Luders rule provides a way to define a quantum channel given a quantum measurement. Using this construction, we establish an if-and-only-if condition for the existence of a ddd-dimensional Symmetric Informationally Complete quantum measurement (a SIC) in terms of a particular depolarizing channel. Moreover, the channel in question satisfies two entropic optimality criteria.
Symmetric informationally complete (SIC) positive operator valued measures (POVMs) are a class of quantum measurements which, in addition to being informationally complete, satisfy three conditions: (1) every POVM element is rank one, (2) the Hilbert-Schmidt inner product between any two distinct elements is constant, and (3) the trace of each element is constant. The third condition is often overlooked, since it may give the impression that it follows trivially from the second. We show that this condition cannot be removed, as it leads to two distinct values for the trace of an element of the POVM. This observation has led us to define a broader class of measurements which we call semi-SIC POVMs. In dimension two, we show that semi-SIC POVMs exist, and we construct the entire family. In higher dimensions, we characterize key properties and applications of semi-SIC POVMs, and note that the proof of their existence remains open.
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.
Informationally complete (IC) measurements are a useful class of measurements, as their outcome statistics uniquely determine an unknown quantum state. Hence, they are important for certain tasks such as quantum state tomography, quantum process tomography, etc. In this work, we study the quantification of informational completeness for arbitrary quantum measurements by introducing and characterizing a faithful measure for it. We explicitly evaluate the informational completeness of qubit symmetric informationally complete (SIC) measurements and show that it is an upper bound for all qubit minimal informationally complete measurements. Furthermore, by introducing a faithful measure, we try to quantify and characterize the ability of an arbitrary quantum channel to preserve the informational completeness of any IC measurement when the channel acts on it in the Heisenberg picture. We call this measure informational completeness-preservability (IC preservability) of quantum channels. After studying its properties, we establish its relation to another quantity, namely, the absolute output coherence of a quantum channel, which quantifies the minimum amount of coherence (w.r.t. an arbitrary incoherent basis) that can always be obtained from the output of that channel. Finally, we illustrate our results with an example from light-matter interaction scenarios. Thus, in this work, not only do we try to provide a quantitative framework for studying both the informational completeness of quantum measurements and the ability of quantum channels to preserve it, but we also try to offer key insight into the conceptual relation between informational completeness and quantum coherence.
A broad class of informationally complete symmetric measurements is introduced. It can be understood as a common generalization of symmetric, informationally complete positive operator-valued measures and mutually unbiased bases. Additionally, it provides a natural way to define two more families of mutually unbiased symmetric measurement operators in any finite dimension. We show a general method of their construction, together with an example of an optimal measurement. Finally, we analyze the properties of symmetric measurements and provide applications in entropic relations and entanglement detection.