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Zeitschrift für Naturforschung A

When do Projections Commute?

W. Rehder

Published Apr 1, 1980• 15 citations
Publisher

Abstract

Abstract Necessary and sufficient conditions for commutativity of two projections in Hilbert space are given through properties of so-called conditional connectives which are derived from the conditional probability operator PQP. This approach unifies most of the known proofs, provides a few new criteria, and permits certain suggestive interpretations for compound properties of quantum-mechanical systems.

DOI
10.1515/zna-1980-0415

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Abstract Necessary and sufficient conditions for commutativity of two projections in Hilbert space are given through properties of so-called conditional connectives which are derived from the conditional probability operator PQP. This approach unifies most of the known proofs, provides a few new criteria, and permits certain suggestive interpretations for compound properties of quantum-mechanical systems.
Quantum mechanics does not permit joint measurements of non-commuting observables.However, it is possible to measure the weak value of a projection operator, followed by the precise measurement of a different property.The results can be interpreted as complex joint probabilities of the two non-commuting measurement outcomes.Significantly, it is possible to predict the outcome of completely different measurements by combining the joint probabilities of the initial state with complex conditional probabilities relating the new measurement to the possible combinations of measurement outcomes used in the characterization of the quantum state.We can therefore conclude that the complex conditional probabilities observed in weak measurements describe fundamental state-independent relations between non-commuting properties that represent the most fundamental form of universal laws in quantum physics.
In this note, we analyze joint probability distributions that arise from outcomes of sequences of quantum measurements performed on sets of quantum states. First, we identify some properties of these distributions that need to be fulfilled to get a classical behavior. Secondly, we prove that a joint distribution exists iff measurement operators "on-state" permute (permutability is the commutativity of more than two operators). By "on-state" we mean properties of operators that hold only on a subset of states in the Hilbert space. Then, we disprove a conjecture proposed by Carstens, Ebrahimi, Tabia, and Unruh (eprint 2018), which states that the property of partial on-state permutation implies full on-state permutation. We disprove this conjecture with a counterexample where pairwise "on-state" commutativity does not imply on-state permutability, unlike in the case of commutativity for all states in the Hilbert space. Finally, we explore the new concept of on-state commutativity by showing a simple proof that if two projections almost on-state commute, then there is a commuting pair of operators that are on-state close to the originals. This result was originally proven by Hastings (Communications in Mathematical Physics, 2019) for general operators.
The set of bounded observables for a quantum system is represented by the set of bounded self-adjoint operators S(H) on a complex Hilbert space H, and the quantum effects for a physical system can be described by the set E(H) of positive contractive operators on a complex Hilbert space H. In this note, by the techniques of operator block and spectral, we give the simpler representation of A∧P and obtained the new necessary and sufficient conditions for A∨P, for A∊S(H) and P∊P(H), where P(H) is the set of all orthogonal projection operators on H. In particular, we get that if A∨P exists, then A∨P∊E(H) for A∊E(H) and P∊P(H). In addition, we consider the relations between the existence of A∨B, A−∨B−, and A+∨B+, where A+, B+, A−, and B− are the positive and negative parts of A,B∊S(H).
The symmetric logarithmic derivative Cramér-Rao bound (SLDCRB) provides a fundamental limit to the minimum variance with which a set of unknown parameters can be estimated in an unbiased manner. It is known that the SLDCRB can be saturated provided the optimal measurements for the individual parameters commute with one another. However, when this is not the case the SLDCRB cannot be attained in general. In the experimentally relevant setting, where quantum states are measured individually, necessary and sufficient conditions for when the SLDCRB can be saturated are not known. In this setting the SLDCRB is attainable provided the SLD operators can be chosen to commute on an extended Hilbert space. However, beyond this relatively little is known about when the SLD operators can be chosen in this manner. In this paper we present explicit examples which demonstrate novel aspects of this condition. Our examples demonstrate that the SLD operators commuting on any two of the following three spaces: support space, support-kernel space and kernel space, is neither a necessary nor sufficient condition for commutativity on the extended space. We present a simple analytic example showing that the Nagaoka-Hayashi Cramér-Rao bound is not always attainable. Finally, we provide necessary and sufficient conditions for the attainability of the SLDCRB in the case when the kernel space is one-dimensional. These results provide new information on the necessary and sufficient conditions for the attainability of the SLDCRB.