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A Generalization of the Functional Calculus of Observables And A Generalization of the Functional Calculus of Observables and Notion of Joint Measurability to the Case of Non-commuting Observables Richard DeJonghe, Kimberly Frey, and Tom Imbo University of Illinois at Chicago, Department of Physics, 845 W. Taylor St., Chicago, IL 60607 (Dated: October 9, 2018) For any pair of bounded observables A and B with pure point spectra, we construct an associated “joint observable” which gives rise to a notion of a joint (projective) measurement of A and B, and which conforms to the intuition that one can measure non-commuting observables simultaneously, provided one is willing to give up arbitrary precision. As an application, we show how our notion of a joint observable naturally allows for a construction of a “functional calculus,” so that for any 2 pair of observables A and B as above, and any (Borel measurable) function f : R → R, a new “generalized observable” f(A, B) is obtained. Moreover, we show that this new functional calculus has some rather remarkable properties. I. INTRODUCTION or product, respectively. Our generalized functional calculus agrees with the ordinary functional calculus when the observables com- In quantum theory observable quantities are repre- mute. For non-commuting observables A and B, there sented by Hermitian operators, and the orthodox view is associated with each generalized observable f(A, B), is that two observables are simultaneously measurable if a natural family of ordinary observables (parameterized and only if their corresponding Hermitian operators com- by E as in section IV A). Each member f (A, B) of this mute. However, to focus on a commonly used example, E family has the following properties (among others): one also frequently hears that the uncertainty relation ∆x∆p & ~ should be taken to mean that one can si- multaneously measure position and momentum, just not (1) A simultaneous eigenstate of A and B with eigenval- with arbitrary precision. Beginning with Arthurs and ues a,b respectively, is an eigenstate of fE (A, B) with Kelly [1], various researchers have investigated the possi- eigenvalue f(a,b). bility of measuring non-commuting observables, typically (2) Every element of the spectrum of fE (A, B) is of the in the framework of POVMs (i.e. unsharp measurements) form f(a,b), where a,b are in the spectra of A and [2–9], substantiating the above connection between the B, respectively. uncertainty principle and joint measurability. In this paper we will discuss a new notion of a sharp While property (1) above is perhaps expected for any joint measurement of a pair of bounded observables with reasonable construction (and indeed is satisfied for the pure point spectra [10]. This joint measurement also ordinary linear algebraic sum A + B), property (2) is a conforms to the intuition that one can measure non- novel feature of our functional calculus — compare this commuting observables simultaneously, provided one is to the complicated relationship between the spectrum of, willing to give up arbitrary precision. The way in which e.g. A + B and the spectra of A and B when [A, B] 6= 0. we arrive at this notion of joint measurability is via the As such, it is immediately clear that the operations cor- construction of joint observables — indeed, our notion responding to addition in this functional calculus will not of a joint observable generalizes that of Gudder [11] and reproduce the usual linear algebraic sum whenever A and Varadarajan [12] to the case of non-commuting observ- B do not commute. arXiv:1407.7606v2 [quant-ph] 30 Jul 2014 ables. This paper is organized as follows. In sections II-V Additionally, we find that our notion of a joint observ- we restrict our attention to finite-dimensional Hilbert able naturally allows for the construction of a “functional spaces [15]. In section II we review projection lattices, calculus of observables” — i.e. for any (Borel measur- projection-valued measures, and Gudder’s notion of a able) function f : R2 → R, and for any observables A and joint observable in the commuting case. In section III we B as above, we have a natural way of defining an object present our notion of a joint observable as well as demon- f(A, B), which can be thought of as a “generalized ob- strate a schema which realizes the corresponding joint servable” [13]. Of course, when A and B commute, one measurement in the framework of quantum operations, already has recourse to a fully developed functional calcu- and connect our joint observable to the uncertainty prin- lus of observables. In particular, one can define f(A, B) ciple. As an application, in section IV we use this notion either using the spectral decompositions of A and B (see, of a joint observable to construct our functional calculus e.g. [14]) or by using Gudder’s aforementioned notion of and describe its properties. We follow with a simple ex- a joint observable [11, 12]. As may be expected, when the ample of the new functional calculus “in action”. Finally, function f is simply addition or multiplication of num- in section V we conclude with some brief remarks. Proofs bers, applying f to commuting observables in the man- and further technical results in the more general case ner just described yields the usual linear algebraic sum of observables on (possibly) infinite-dimensional Hilbert 2 spaces can be found in the appendix. In what follows (ex- and M is a Boolean σ-algebra of subsets of Ω whose ele- cept in the appendix), H will denote a finite-dimensional ments are called measurable sets). A PVM on (Ω, M) complex Hilbert space. We will further take all functions is a σ-homomorphism from M to LH — i.e. a map 2 f : R → R to be Borel measurable, and all partitions of α : M → LH which satisfies measurable spaces to consist of measurable sets. (i) α(Ω) = I; (ii) if R1, R2 ∈ M are such that R1 ∩ R2 = ∅, then II. BACKGROUND α(R1)⊥ α(R2); ∞ ∞ A. Observables, Spectral Families, and (iii) α(Ri)= α Ri for all R1, R2,... ∈M such Projection-Valued Measures i=1 i=1 X [ that Ri ∩ Rj = ∅ whenever i 6= j. There are many different ways to represent the ob- The set of observables on H are in 1-1 correspondence servables of a quantum system. The usual way is as a R R R self-adjoint linear operator A on H. By the spectral the- with the PVMs on ( , B( )), where B( ) denotes the n Borel subsets of R. Explicitly, for a given self-adjoint op- orem A = i=1 λiPi, where Pi is the projector onto the erator A, the associated PVM αA is given by αA(R) = eigenspace of H associated with the eigenvalue λi of A, Pi (for any R ∈ B(R)), where the Pi’s are the and P ⊥ PP (i.e. P P = P P = 0) when i 6= j. With- λi∈R i j i j j i projectors in the spectral decomposition of A; conversely, out loss of generality, we assume that λi < λi+1 for all P i ∈ {1,...,n}. Another way of representing observables given a PVM α, there is a unique observable determined is in terms of a spectral family — i.e. a one parameter by the spectral family whose elements are defined by Eλ := α (−∞, λ] . In the sequel, we will use the same family {Eλ}λ∈R of projection operators on H such that (i) E ≤ E whenever λ ≤ µ, (ii) lim E = 0, symbol (as well as the term ‘observable’) to refer to any λ µ λ→−∞ λ of these three ways of representing an observable, as this and (iii) limλ→+∞Eλ = I, where 0 is the zero opera- tor and I is the identity operator on H, and ≤ is the standard abuse of notation enables us to streamline the partial order on projectors defined by E ≤ E when- following discussion. For example, for an observable A, λ µ and any S ∈B(R), we will simply write A(S) instead of ever the respective subspaces Σλ, Σµ onto which they αA(S). project satisfy Σλ ⊆ Σµ. (Note that 0 ≤ P ≤ I for every projection operator P .) There is a 1-1 correspon- dence between such spectral families of projectors and B. Coarse-graining of Observables self-adjoint operators on H [16]. Namely, for the self- adjoint operator A given above, and for λ ∈ R, the el- R ements of the corresponding spectral family are given Given an observable A, any set S ∈ B( ) naturally corresponds to a two-outcome measurement associated by Eλ = Pi. Conversely, given a spectral fam- λi≤λ with the projection operators A(S) and A(S)⊥ = A(Sc), ily {E } R, its associated self-adjoint operator is given λ λ∈P where Sc is the set theoretic complement of S. For a by λ∈R λ Eλ − limǫ→0+ Eλ−ǫ . system in the state represented by the density operator Alternatively,P one can describe observables on H in ρ, an unselected measurement of A associated with these terms of projection-valued measures (PVMs), which have two outcomes takes ρ to A(S) ρ A(S) + A(Sc) ρ A(Sc). an elegant formulation using the projection lattice LH Since the projection operator A(S) corresponds to the of H (where LH is the set of all projection operators on H subspace spanned by all states |ψi ∈ H such that one equipped with the partial order ≤ defined above), and can say with certainty that the value of the observable which we describe below after some preliminaries about A is in the range S (i.e.
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