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arXiv:2101.08313

On-State Commutativity of Measurements and Joint Distributions of Their Outcomes

Jan Czajkowski, Alex B. Grilo

Published Jan 20, 2021• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.2101.08313

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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.
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.
There is a contact problem between classical probability and quantum outcomes. Thus, a standard result from classical probability on the existence of joint distributions ultimately implies that all quantum observables must commute. An essential task here is a closer identification of this conflict based on deriving commutativity from the weakest possible assumptions, and showing that stronger assumptions in some of the existing no-go proofs are unnecessary. An example of an unnecessary assumption in such proofs is an entangled system involving nonlocal observables. Another example involves the Kochen-Specker hidden variable model, features of which are also not needed to derive commutativity. A diagram is provided by which user-selected projectors can be easily assembled into many new, graphical no-go proofs.
In the quantum probability theory the 6-field of random events is replaoed by the lattice of orthogonal projectors in a separable infinite dimensional Hilbert space H.A countably additive function from this lattices to the unit interval constitutes a state, the non-commutative analogue of a probability measure.The Theorem of Gleason [4] asserts that every state is of the form n--tr jtT, where jt runs over all projectors and T is a probability operator on H, i.e. a positive linear operator of unit trace.Conversely, every probability operator determines a state by the Gleason formula.From now onwards let S stand for the set of all states, i.e. all probability operators on H.We shall dienote by the space of-? ell nuclear linear operators acting in H with the norm jjïj|.j= tr{TT*) 1 ^2.Of coarse S is e closed and convex subset Of TyIn quantum theory to every physical quantity or observable th^re corresponds e sslf»adjoint not necessarily bounded linear operator on Hs By 0 ws shall denote the set of all obt ;ry£bles-Given he-0 f tac pi'cb^ivility distribution of A at the etato T is defined for ail Eoi'oi subsets E of the real
Chapter 1: On the existence of quantum representations for two dichotomic measurements. Under which conditions do outcome probabilities of measurements possess a quantum-mechanical model? This kind of problem is solved here for the case of two dichotomic von Neumann measurements which can be applied repeatedly to a quantum system with trivial dynamics. The solution uses methods from the theory of operator algebras and the theory of moment problems. The ensuing conditions reveal surprisingly simple relations between certain quantum-mechanical probabilities. It also shown that generally, none of these relations holds in general probabilistic models. This result might facilitate further experimental discrimination between quantum mechanics and other general probabilistic theories. Chapter 2: Possibilistic Physics. I try to outline a framework for fundamental physics where the concept of probability gets replaced by the concept of possibility. Whereas a probabilistic theory assigns a state-dependent probability value to each outcome of each measurement, a possibilistic theory merely assigns one of the state-dependent labels "possible to occur" or "impossible to occur" to each outcome of each measurement. It is argued that Spekkens' combinatorial toy theory of quantum mechanics is inconsistent in a probabilistic framework, but can be regarded as possibilistic. Then, I introduce the concept of possibilistic local hidden variable models and derive a class of possibilistic Bell inequalities which are violated for the possibilistic Popescu-Rohrlich boxes. The chapter ends with a philosophical discussion on possibilistic vs. probabilistic. It can be argued that, due to better falsifiability properties, a possibilistic theory has higher predictive power than a probabilistic one. Chapter 3: The quantum region for von Neumann measurements with postselection. It is determined under which conditions a probability distribution on a finite set can occur as the outcome distribution of a quantum-mechanical von Neumann measurement with hyphenation given that the scalar product between the initial and the final state is known as well as the success probability of the postselection. An intermediate von Neumann measurement can enhance transition probabilities between states such that the error probability shrinks by a factor of up to 2. Chapter 4: A presentation of the category of stochastic matrices. This chapter gives generators and relations for the strict monoidal category of probabilistic maps on finite cardinals (i.e., stochastic matrices). Chapter 5: Convex Spaces: Definition and Examples. We try to promote convex spaces as an abstract concept of convexity which was introduced by Stone as "barycentric calculus''. A convex space is a set where one can take convex combinations in a consistent way. By identifying the corresponding Lawvere theory as the category from chapter 4 and using the results obtained there, we give a different proof of a result of Swirszcz which shows that convex spaces can be identified with algebras of a finitary version of the Giry monad. After giving an extensive list of examples of convex sets as they appear throughout mathematics and theoretical physics, we note that there also exist convex spaces that cannot be embedded into a vector space: semilattices are a class of examples of purely combinatorial type. In an information-theoretic interpretation, convex subsets of vector spaces are probabilistic, while semilattices are possibilistic. Convex spaces unify these two concepts.
We show that the joint behavior of an arbitrary pair of (generally noncommuting) quantum observables can be described by quasi-probabilities, which are an extended version of the standard probabilities used for describing the outcome of measurement for a single observable. The physical situations that require these quasi-probabilities arise when one considers quantum measurement of an observable conditioned by some other variable, with the notable example being the weak measurement employed to obtain Aharonov’s weak value. Specifically, we present a general prescription for the construction of quasi-joint probability (QJP) distributions associated with a given combination of observables. These QJP distributions are introduced in two complementary approaches: one from a bottom-up, strictly operational construction realized by examining the mathematical framework of the conditioned measurement scheme, and the other from a top-down viewpoint realized by applying the results of the spectral theorem for normal operators and their Fourier transforms. It is then revealed that, for a pair of simultaneously measurable observables, the QJP distribution reduces to the unique standard joint probability distribution of the pair, whereas for a noncommuting pair there exists an inherent indefiniteness in the choice of such QJP distributions, admitting a multitude of candidates that may equally be used for describing the joint behavior of the pair. In the course of our argument, we find that the QJP distributions furnish the space of operators in the underlying Hilbert space with their characteristic geometric structures such that the orthogonal projections and inner products of observables can be given statistical interpretations as, respectively, “conditionings” and “correlations”. The weak value AwA_{w}Aw​ for an observable AAA is then given a geometric/statistical interpretation as either the orthogonal projection of AAA onto the subspace generated by another observable BBB, or equivalently, as the conditioning of AAA given BBB with respect to the QJP distribution under consideration.
Violations of Bell's inequalities imply that joint probabilities generated by non-commutative measurements on two (non-identical) quantum particles do not have a single common cause. But joint probabilities generated for such non-identical particles via commutative measurements do have non-trivial common cause variables. We focus on commutative measurements and consider two identical quantum particles, whose density matrices and observables (hermitian operators) are necessarily permutation-symmetric. It is natural to demand that the common cause describing joint probabilities is also permutation symmetric, i.e., it acts symmetrically on both particles. Looking at various ways of defining joint probabilities from the same measurement data, we conclude that either symmetric common causes need not exist (i.e., that the particles can be hiddenly distinguishable), or that symmetric screening variables exist, but they are trivial, i.e., no single common cause can explain all single-measurement correlations.