arXiv:1407.7606v2 [quant-ph] 30 Jul 2014 n lofeunl er htteucranyrelation uncertainty the example, that used hears commonly ∆ frequently a on also focus one to com- However, operators Hermitian mute. if corresponding view their measurable if simultaneously orthodox only are and the observables two and that operators, is Hermitian by sented ntefaeoko OM (i.e. of framework the typically in observables, non-commuting possi- measuring the of investigated bility have researchers various [1], Kelly precision arbitrary momentum, with and position multaneously 29,sbtnitn h bv oncinbtenthe measurability. between joint and connection principle above uncertainty the substantiating [2–9], aclso bevbe”—ie o n Brlmeasur- (Borel any for i.e. function — able) “functional observables” a of of construction calculus the for allows naturally able observ- non-commuting of and case [11] the ables. Gudder of to notion that [12] our generalizes the Varadarajan indeed, observable via joint — is a measurability observables of joint joint which of in of notion way construction this The non- at precision. arrive measure arbitrary we up can is give also one one to measurement provided willing that simultaneously, joint intuition observables This commuting the to [10]. with conforms spectra observables bounded point of pure pair a of measurement joint evbe 1] fcus,when course, Of [13]. servable” f B es applying bers, e utdsrbdyed h sa ieragbacsum algebraic linear usual the yields described just ner on bevbe[1 2.A a eepce,we the when expected, be may of As notion function 12]. aforementioned [11, Gudder’s observable using joint a by or [14]) of e.g. decompositions spectral the using either u fosrals npriua,oecndefine can one particular, In observables. calcu- functional of developed lus fully a to recourse has already ( x ,B A, nqatmter bevbeqatte r repre- are quantities observable theory quantum In nti ae ewl ics e oino a of notion new a discuss will we paper this In diinly efidta u oino on observ- joint a of notion our that find we Additionally, saoe ehv aua a fdfiiga object an defining of way natural a have we above, as ∆ eeaiaino h ucinlCluu fObservables of Calculus Functional the of Generalization A p ,wihcnb huh fa gnrlzdob- “generalized a as of thought be can which ), & f ~ ssml diino utpiaino num- of multiplication or addition simply is gnrlzdobservable” “generalized fajitosral aual losfracntuto of construction an a As for observables precision. allows of arbitrary naturally pair observable up non-com joint give measure a to can of willing one is that one intuition provided the to conforms which jitosral”wihgvsrs oanto fajit(pr joint a of notion a to rise gives which observable” “joint a oerte eakbeproperties. remarkable rather some has hudb ae oma htone that mean to taken be should nvriyo liosa hcg,Dprmn fPyis 8 Physics, of Department Chicago, at Illinois of University f For f .INTRODUCTION I. : ocmuigosralsi h man- the in observables commuting to R any esrblt oteCs fNncmuigObservables Non-commuting of Case the to Measurability 2 → aro one observables bounded of pair R einn ihAtusand Arthurs with Beginning . n o n observables any for and , A ihr eoge ibryFe,adTmImbo Tom and Frey, Kimberly DeJonghe, Richard A unsharp and and f ( B ,B A, B saoe n n Brlmaual)function measurable) (Borel any and above, as measurements) sotie.Mroe,w hwta hsnwfntoa cal functional new this that show we Moreover, obtained. is ) omt,one commute, A and Dtd coe ,2018) 9, October (Dated: A f utnot just can B ( A and ,B A, sharp (see, and si- B ) ihpr on pcr,w osrc nassociated an construct we spectra, point pure with erdc h sa ieragbacsmwhenever sum algebraic not linear cor- will usual calculus operations the functional the reproduce this that in addition clear to immediately responding is it such, As B otecmlctdrltosi ewe h pcrmof, this spectrum compare the e.g. between — relationship calculus complicated the functional to our of feature novel esnbecntuto adide sstse o the any for sum for satisfied algebraic is expected linear indeed perhaps ordinary (and is construction above reasonable (1) property While aiyhstefloigpoete aogothers): (among properties following the has family n ute ehia eut ntemr eea case general Hilbert more infinite-dimensional the (possibly) in on observables results of technical further and ml ftenwfntoa acls“nato” Finally, ex- action”. simple “in section a calculus in with functional new follow the We of calculus ample properties. functional its our construct describe to and observable joint section a in of prin- application, uncertainty an the As to operations, ciple. observable joint quantum joint our of corresponding connect framework and the the realizes in demon- which measurement as schema well as a observable joint strate a of notion our present on bevbei h omtn ae nsection a In of case. commuting notion the in Gudder’s observable joint and Hilbert measures, section finite-dimensional projection-valued In to attention [15]. spaces our restrict we (2) (1) aua aiyo riayosrals(parameterized observables ordinary by of family natural a ue o o-omtn observables com- non-commuting observables For the when mute. calculus functional ordinary respectively. product, or sascae ihec eeaie observable generalized each with associated is hsppri raie sflos nsections In follows. as organized is paper This u eeaie ucinlcluu geswt the with agrees calculus functional generalized Our ontcommute. not do E ues B eigenvalue vr lmn ftesetu of spectrum the of element Every of eigenstate simultaneous A form A si section in as respectively. , + fntoa acls”s htfrany for that so calculus,” “functional a jcie esrmn of measurement ojective) ,b a, 5W alrS. hcg,I 60607 IL Chicago, St., Taylor W. 45 B plcto,w hwhwornotion our how show we application, f V ( uigosralssimultaneously, observables muting n h pcr of spectra the and epciey sa iesaeof eigenstate an is respectively, ,b a, ecnld ihsm re eak.Proofs remarks. brief some with conclude we f ,where ), ( ,b a, VA IV ). .Ec member Each ). II ,b a, f n oino Joint of Notion and erve rjcinlattices, projection review we : r ntesetaof spectra the in are R A 2 A A → and + A and R IV B and new a , B ,poet 2 sa is (2) property ), B eueti notion this use we f culus and , E hn[ when B A f ( E ,B A, iheigenval- with f ( and ,B A, E ( ,B A, ,B A, so the of is ) B f fthis of ) ( there , A ,B A, A with ) III ] 0. 6= II and and we - V ), 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 . 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. a measurement of A yields a c the projection lattice. LH is a complete lattice, which is value in S with probability 1), and similarly for A(S ), c to say that for any subset of projectors {Pj }j∈J ⊆ LH, we see that one can think of A(S) and A(S ) together there exists both a least upper bound and greatest lower as a “coarse-graining” of A whereby one only determines bound in LH, which we denote by j∈J Pj and j∈J Pj , which region A takes its value in upon measurement (ei- respectively [17]. (For pairs of projectors P and P , ther S or Sc), but not the specific value. In fact, since W 1 V 2 we use P1 ∨ P2 and P1 ∧ P2 to denote their least upper the set of “outcomes” of this coarse-grained measurement and greatest lower bounds.) Furthermore, to each projec- consists of P := {S,Sc}, one can define a course-grained tor P there corresponds a unique projector P ⊥ which sat- observable A˜ associated with A more precisely as a PVM isfies P ⊥ P ⊥ (and hence P ∧ P ⊥ = 0) and P ∨ P ⊥ = I, on the measurable space (P, 2P ), where 2P is the set of all ⊥ ∞ ˜ namely P = I − P . Finally, for {Pi}i=1 a set of pairwise subsets of P. In particular, we define A(Q) := A( Q), ∞ ∞ P orthogonal projectors, we have i=1 Pi = i=1 Pi, and where Q ∈ 2 (and X := Z∈X Z for any set X whose for commuting projectors P1 and P2, we have P1 ∧ P2 = elements Z are, themselves, sets). This procedure canS be W P P1P2. generalized, allowingSA to beS a PVM on any measurable We are now in a position to define a PVM on (Ω, M), space (Ω, M), and P to be any partition of Ω — each where (Ω, M) is a measurable space (that is, Ω is a set such partition will be associated with a coarse-grained 3

−1 PVM A˜ [18]. Since A˜ is determined uniquely by A and of the joint observable by f(A, B) := JAB ◦ f [11], P, any observable contains complete information about where f −1 : B(R) →B(R2) is the map which takes each all its possible coarse-grainings. Additionally, note that X ∈B(R) to its pre-image under f. As mentioned previ- since unselected measurements are a special case of ously, when the function f is simply addition or multipli- preserving quantum operations [19] and each partition of cation of numbers, applying f to commuting observables the outcome space is associated with an unselected mea- in the manner just described yields the usual linear alge- surement of the (course-grained) observable, we see that braic sum or product, respectively. the description of measurements of these observables fits naturally into the more general framework of quantum operations. D. Example: JAB when dim H = 2 and [A, B] = 0

In what follows, we take H to be a two-dimensional C. Joint Observables in the Commuting Case Hilbert space. Also, let A = αI +a·σ and B = βI +b·σ, where α, β ∈ R, a, b ∈ R3, I is the identity , and If A and B are observables on H, a PVM σ is the 3-vector consisting of the Pauli matrices. Note 2 2 K : B(R ) → LH (where B(R ) denotes the Borel subsets that the eigenvalues of A and B are a± := α ± |a| and 2 of R ) which satisfies K(R1 ×R2)= A(R1)∧B(R2) for all b± := β ±|b|, respectively, and assume [A, B] = 0 (which R1, R2 ∈B(R) is said to be a joint observable for A and is equivalent to a and b being co-linear). B. It follows immediately that any joint observable K In the case in which both A and B each have a single for A and B has the property that K(R1 × R)= A(R1) eigenvalue (i.e. A = αI and B = βI), we have that for 2 and K(R × R2) = B(R2), which is to say that K has any Q ∈B(R ), the expected margins. If we suppose that A and B commute, it is straightforward to show that the map I if (α, β) ∈ Q R2 R2 JAB(Q)= (3) JAB : B( ) → LH defined by (for Q ∈B( )) ( 0 if (α, β) ∈/ Q. JAB(Q) := A(R1) ∧ B(R2) (1) The next case to consider is when only one of the ob- R1×R2⊆Q servables, say A, has two distinct eigenvalues. In this R ×R_∈B(R2)   1 2 case, the projector onto the eigenspace of A with eigen- 1 σ is a joint observable. Gudder [11] was the first to show value a+ is given by P+ := 2 (I + aˆ · ), where aˆ = a/|a|. that a joint observable for A and B exists if and only if Similarly the projector on to the eigenspace of A with 1 σ ⊥ [A, B] = 0, and moreover, that this joint observable is eigenvalue a− is given by P− := 2 (I − aˆ · ) = P+ . unique. (However, as far as we can tell, expression (1) Then, for any Q ∈B(R2), it is easy to see that above for this unique joint observable has not previously appeared in the literature.) I if both (a+,β), (a−,β) ∈ Q R2 For any R1 × R2 ∈ B( ) and for any observables P if (a ,β) ∈ Q and (a ,β) ∈/ Q J (Q)=  + + − (4) A and B, we have that A(R1) ∧ B(R2) is the projec- AB  P− if (a+,β) ∈/ Q and (a−,β) ∈ Q tion operator onto the subspace spanned by the set of all  0 if both (a ,β), (a ,β) ∈/ Q. states |ψi ∈ H such that if the system is initially in the + −  state |ψi, a measurement of the observable A yields (with Moving on to the case in which both A and B have certainty) an outcome in R1 and a measurement of the two distinct eigenvalues, there are two possibilities corre- observable B yields (with certainty) an outcome in R . 2 sponding to whether the eigenstate |a+i of A with eigen- Given this, we see that when A and B commute, J AB value a+ is an eigenstate of B with eigenvalue b+ or b−. encodes all information about a simultaneous measure- We proceed assuming B|a+i = b+|a+i; analysis of the ment of A and B, as well as information about all possi- other possibility proceeds analogously. Notice that in ble “coarse-grainings” of this simultaneous measurement this case P , as defined above, is also the projector onto R2 + associated with partitions of the outcome space . This the eigenspace of B with eigenvalue b . A straightfor- R2 + is to say that for any partition {Q1,...,Qn} of with ward computation then shows (for Q ∈B(R2)) R2 n Qi ∈ B( ) for all i, we have that i=1 JAB(Qi) = I, and the (trace preserving) P n I if both (a+,b+), (a−,b−) ∈ Q ρ → JAB(Qi) ρJAB(Qi) (2) P+ if (a+,b+) ∈ Q and (a−,b−) ∈/ Q JAB(Q)=  i=1  P− if (a+,b+) ∈/ Q and (a−,b−) ∈ Q X  describes an (unselected and course-grained) simultane- 0 if both (a+,b+), (a−,b−) ∈/ Q. ous measurement of A and B.  (5) 2  Further, for any function f : R → R and pair of Note that (as expected) JAB is a PVM in each of the commuting observables A and B, one obtains the afore- three cases discussed above. Also, it is easy to see that mentioned of (commuting) observ- JAB(R1 × R2)= A(R1) ∧ B(R2) for any R1, R2 ∈B(R), ables by taking the PVM f(A, B) to be defined in terms so that JAB is, in fact, a joint observable. 4

III. JOINT OBSERVABLES IN THE U on H ⊗ A satisfying NON-COMMUTING CASE n 0 A. Generalized Joint Observables and Joint U |ψi⊗|0i = JAB|ψi⊗|0i+ JAB(Qi)|ψi ⊗|ii. (8) i=1 Measurability  X  Starting with an initial state ρ on H, form the state We will now demonstrate that for any pair of observ- ρ′ := ρ ⊗ |0ih0| on H ⊗ A. Then evolve the state as ables A and B, the expression for JAB in (1) above, which ρ′ 7→ Uρ′U †, and, following this, projectively measure is still well-defined when [A, B] 6= 0, has a natural inter- n the operator I ⊗ i=1 |iihi|, selecting for the +1 eigen- pretation in terms of measurement even though it is no value. Finally, trace over A. It is straightforward to see longer a PVM in this case. First note that JAB still that this gives theP evolution in equation (2), but now has the correct margins, even when A and B don’t com- n where i=1 JAB(Qi) 6= I in general. mute. It is also straightforward to show that properties (i) and (ii) of PVMs (defined in section II A) still hold. P (See Theorem 1 in the appendix for a proof.) However, while property (iii), also known as countable additivity, B. Connection to the Uncertainty Principle need not hold in general, JAB does satisfy countable sub- 2 additivity, which is to say that for all R1, R2,... ∈B(R ) The extent to which the above quantum operation such that Ri ∩ Rj = ∅ (i 6= j), we have manages to be trace preserving increases, in general, with more coarse-graining of our partitions, due to the sub- ∞ ∞ additivity of JAB. We can interpret this as a manifesta- JAB(Ri) ≤ JAB Ri . (6) tion of the uncertainty principle with regard to our joint i=1 i=1 X  [  measurements — the essential feature is that as we de- crease the resolution of the measurement, it becomes eas- 2 Notice that for any partition {Q1,...,Qn} of R , ier to find states for which we can say that the values of n n i=1 JAB(Qi) = i=1 JAB(Qi) holds as a consequence A and B for that state are constrained to lie in any fixed of property (ii), but that this sum of orthogonal pro- region of the plane. In fact, for any state |ψi ∈ H, we Wjectors need notP equal the identity operator on H can make a direct quantitative connection between the due to the failure of countable additivity (although uncertainty principle and any convex rectangular region n R2 i=1 JAB(Qi) ≤ I by equation (6) above). Hence, to any R1 × R2 ∈B( ) for which JAB (R1 × R2)|ψi = |ψi. R2 n partition {Q1,...,Qn} of with i=1 JAB(Qi) 6= I, As usually stated, for a system in the state |ψi ∈ H, the Pthere corresponds an unselected measurement in which uncertainty principle puts a lower bound on the product there is a chance that we do not obtainP any of our mea- of the (square roots of the) variances of the outcomes surement outcomes — that is, the corresponding quan- of any pair of observables. For example, the Robertson tum operation (as in equation (2)) is not trace preserving. relation [20] for the observables A and B is It is this quantum operation which we will refer to as a simultaneous (or joint) measurement of A and B (inde- 1 n ∆A∆B ≥ |h[A, B]i|, (9) pendent of whether or not i=1 JAB(Qi)= I for the par- 2 2 tition {Q1,...,Qn} of R , and independent of whether or not [A, B] = 0). NoteP also that such measurements where for any self-adjoint operator Z on H, we have that 2 2 are sharp (albeit course-grained) since the JAB (Qi)’s are hZi := hψ|Z|ψi, as well as that ∆Z := hZ i − hZi . pairwise orthogonal projection operators. We will refer Now, for a given state |ψi ∈ H and any convex region 2 p to JAB above as a generalized joint observable. R1 × R2 ⊆B(R ) such that JAB(R1 × R2)|ψi = |ψi, we We now give an explicit realization of such an unse- have lected measurement (as a combination of unitary evolu- 1 tion and selected measurement) using an ancilla system. Area(R1 × R2) ≥ ∆A∆B (10) 2 Let {Q1,...,Qn} be a partition of R . The unselected 2 simultaneous measurement of A and B associated with this partition is given by the following schema. First, (see Theorem 2 in the appendix). The minimal such value define of Area(R1 × R2) can thus be thought of as a measure of how “incompatible” A and B are, or of how uncertain a n joint measurement of A and B is, in the state |ψi. In- 0 JAB := I − JAB(Qi), (7) terestingly, various investigations of joint measurements i=1 of non-commuting observables in the unsharp (POVM) X case also find that their natural measures of the uncer- and let A be an n + 1 dimensional Hilbert space with or- tainty of the joint measurement (the analog of our mini- n thonormal basis {|ii}i=0. From equation (6), it is easy to mal Area(R1 × R2) above) are bounded below exactly as see that for any |ψi ∈ H, there exists a in inequality (10) [1, 2, 4, 6]. 5

C. Generalized Projection-Valued Measures when A and B commute. Although generalized joint ob- servables on two-dimensional Hilbert spaces are relatively

Although JAB is not a PVM when A and B do not simple, they suffice to illustrate the differences between commute, it comes “close” in the sense that it satis- JAB in the commuting and non-commuting cases, as well fies properties (i) and (ii) and is countably sub-additive as some of the basic features of generalized joint observ- (as noted previously). In the sequel, for any measur- ables. able space (Ω, M), a map α : M → LH which satisfies properties (i) and (ii) of PVMs, along with countable sub-additivity E. Coarse-graining and PVMs Associated with JAB

∞ ∞ ′ Another interesting property of J is that it is well- (iii ) α(R ) ≤ α R for all R , R ,... ∈M AB i i 1 2 behaved with regard to the procedure of coarse-graining. i=1 i=1 X  [  In particular, given partitions PA and PB of R associated such that Ri ∩ Rj = ∅ whenever i 6= j with coarse-grainings A˜ and B˜ of observables A and B, respectively, there is a natural partition P of R2 which will be called a generalized projection-valued measure AB allows us to define a coarse-graining J˜ of J in the (gPVM) on (Ω, M). Notice that any gPVM α on (Ω, M) AB AB same manner as for PVMs. It is straightforward to show also satisfies α(∅) = 0, and that if Q,S ∈ M are such that that Q ⊆ S, then α(Q) ≤ α(S) — that is, gPVMs are monotonic and increasing. We now proceed to investi- ˜ JAB = JA˜B˜ . (12) gate further properties of the gPVM JAB.

That is, the construction of our JAB commutes with the operation of coarse-graining [21]. Moreover, when D. Example: JAB when dim H = 2 and [A, B] 6= 0 the coarse-graining is “coarse enough” (specifically, when there exists {Q1,Q2,...} ⊆ PAB satisfying As in section II D we take dim H = 2, and let JAB(Qi) = I), the coarse-grained joint observable a σ b σ i A = αI + · and B = βI + · , with eigenvalues a± J˜ is in fact a PVM, not just a gPVM (see Theorem 3). and b , respectively. PAB ± There is another method by which we can construct We begin by noting that when [A, B] 6= 0, A and B a PVM from J . As we show in the appendix (Theo- each have two distinct eigenvalues. We retain the defini- AB rem 7), any gPVM J on (Ω, M), along with a generating tion of P = 1 (I ± aˆ · σ) (from section II D), and also ± 2 chain E for M (i.e. E ⊆M generates M as a Boolean 1 ˆ σ define Q± := 2 (I ± b · ), so that Q± is the projector σ-algebra, and the elements of E are totally ordered un- onto the eigenspace of B with eigenvalue b±. The com- der inclusion), can be used to construct a unique PVM R2 putation of JAB yields, for any Q ∈ B( ) (where σ(A) on (Ω, M) which agrees with J on the elements of E [22]. denotes the spectrum of A) In the case (Ω, M) = (R2, B(R2)), E a generating chain R2 E for B( ), and J = JAB, we denote this PVM by JAB. J AB(Q)= E (Of course, since JAB is a PVM, it is naturally associated I if at least 3 elements of σ(A) × σ(B) are in Q (for any partition of R2) with a trace preserving quan- tum operation.) When [A, B] = 0, we have J E = J Q+ if both (a±,b+) ∈ Q, and both (a±,b−) ∈/ Q AB AB  for any generating chain E for B(R2). When [A, B] 6= 0  Q− if both (a±,b+) ∈/ Q, and both (a±,b−) ∈ Q  this is not true, and no J E is a joint observable for A  P if both (a ,b ) ∈ Q, and both (a ,b ) ∈/ Q AB  + + ± − ± and B since in this case no joint observable exists. The  E P− if both (a+,b±) ∈/ Q, and both (a−,b±) ∈ Q physical meaning of the JAB’s when [A, B] 6= 0 remains  0 otherwise. obscure.   (11)  For all of the cases considered in section II D (i.e. when F. Other Characterizations of JAB [A, B] = 0), the value of JAB is determined exactly by 2 its action on single points in the space R , but this is no In addition to all of the aforementioned properties of longer true when [A, B] 6= 0. In particular, JAB({p})=0 JAB, we have the following independent characterization R2 for any p ∈ , and so JAB is clearly not a PVM in the of our generalized joint observable. For any PVMs A 2 non-commuting case considered here. It is straightfor- and B, and any set map J : B(R ) → LH, we have ward to see, however, that JAB is a gPVM. Also, from that J = JAB if and only if J satisfies the following equation (11) it is easy to see that JAB has the correct two conditions for all Q ∈ B(R2) and all |ψi ∈ H (see margins. Finally, since H is two-dimensional, all of the Theorem 5 in the appendix): projectors which are in the image of the map JAB oc- cur in the images of the PVMs A and B — this is no (1) If there exist R1, R2 ∈ B(R) with R1 × R2 ⊆ Q and longer generically true in three or higher dimensions, even A(R1) ∧ B(R2)|ψi = |ψi, then J(Q)|ψi = |ψi. 6

(2) If for every R1, R2 ∈B(R) with R1 ×R2 ⊆ Q we have tor U, the generalized observable f(A, B) has the intu- A(R1) ∧ B(R2)|ψi = 0, then J(Q)|ψi = 0. itive property

Qualitatively speaking, property (1) above states the fol- f(UAU †,UBU †)= Uf(A, B)U †, (15) lowing intuitive requirement on the generalized joint ob- which follows directly from Lemma 9 in the appendix. servable JAB: for a given state |ψi, if the system has Finally, just as an unselected measurement of J corre- the value of A in R1 and the value of B in R2, and AB sponds to a non-trace preserving quantum operation, so R1 × R2 ⊆ Q, then the value of the generalized joint too does an unselected measurement of f(A, B). observable JAB is in the range Q. Similarly, property (2) above states that if one can never (i.e. with probability Now, for any pair of observables A and B, and any function f : R2 → R, it turns out that f(A, B) has a fam- zero) measure the value of A in R1 and B in R2 for ev- ily of PVMs associated with it (just as the gPVM J has ery R1 × R2 ⊆ Q, then the value of the generalized joint AB an associated family of PVMs). In particular, for each observable JAB is never in the range Q. generating chain E of B(R), there exists a unique PVM Alternatively, JAB has a characterization related to |ψi f (A, B) agreeing with f(A, B) on all E ∈E (see Theo- possible measurement outcomes. Let M denote the E A rem 7 and Corollary 2 in the appendix). Of course, when set of possible measurement outcomes associated with A [A, B] = 0, f(A, B) is a PVM, and f (A, B)= f(A, B) when the system is in the state |ψi, i.e. E for any generating chain E. As an example, consider arbitrary observables A and |ψi R MA := {λ ∈ : A({λ})|ψi=0 6 }. (13) B, along with the generating chain

R2 |ψi |ψi E⋆ := {(−∞, λ]: λ ∈ R}, (16) Then, for any Q ∈B( ) for which MA × MB ⊆ Q, we have that JAB(Q)|ψi = |ψi, which is to say that if the system is in the state |ψi, a joint measurement of A and B and define is guaranteed to yield an outcome in Q. While the reverse −1 Eλ := JAB ◦ f (−∞, λ] . (17) implication holds for rectangular sets Q = R1 × R2, it does not hold in general. (That this is so follows from the Then {Eλ}λ∈R is a spectral family of projectors on H, |ψi |ψi fact that {|ψi ∈ H : MA ×MB ⊆ Q} is not a subspace which corresponds to an observable in the standard way, of H unless Q is rectangular.) Despite this, JAB(Q) is the and this observable is fE⋆ (A, B). Now, since H is fi- projector onto the span of all the states |ψi ∈ H whose nite dimensional, fE⋆ (A, B) will have a finite number possible measurement outcomes associated with A and B of distinct eigenvalues λ1 < λ2 < ··· < λn. As such, |ψi |ψi −1 are contained in Q (i.e. MA × MB ⊆ Q). So, another since R1 × R2 ⊆ f ((−∞, λ]) exactly when f(a,b) ≤ λ way of thinking of JAB is as the minimal (with respect for every a ∈ R1 and b ∈ R2, it is straightforward to 2 to the partial order ≤ on LH) set map from B(R ) → LH show that a state |ψi ∈ H is an eigenstate of fE⋆ (A, B) satisfying property (1) above. with eigenvalue λi exactly when |ψi is in the span of all states for which any possible measurement outcomes a and b of A and B, respectively, satisfy f(a,b) ≤ λi, but IV. FUNCTIONAL CALCULUS |ψi is orthogonal to any state whose possible measure- ment outcomes a′ and b′ of A and B, respectively, satisfy ′ ′ f(a ,b ) ≤ λ [23]. We will return to f ⋆ (A, B), for A. Basic Properties i−1 E some specific choices of f, shortly. For any generating chain E for B(R), the observable Given any observables A and B, along with a function fE (A, B) satisfies the following nice properties (where, R2 R R f : → , we define the map f(A, B): B( ) → LH by as before, σ(A) denotes the spectrum of A): (for all Q ∈B(R)) (1) σ fE (A, B) ⊆ f σ(A), σ(B) ; −1 f(A, B)(Q) := JAB ◦ f (Q), (14) † † † (2) fE(UAU ,UBU ) = UfE (A, B)U for any unitary  operator U on H; just as in the case [A, B] = 0. Using the fact that JAB is a gPVM, it is straightforward to show that f(A, B) is also (3) If A|ψi = a|ψi and B|ψi = b|ψi for some |ψi ∈ H, a gPVM (see Theorem 6 in the appendix). Unlike poly- then fE (A, B)|ψi = f(a,b)|ψi. nomials in the ordinary linear algebraic sum and product, there are no ambiguities in defining f(A, B) — as an ex- These are proved in the appendix (Theorem 10). Addi- ample, for the two-variable polynomials p(x, y) = xy2x tionally, it follows from property (1) that for any subset and q(x, y) = yx2y (which both represent the same S of R which contains all eigenvalues of both A and B, function from R2 to R), we do not generically have and for any f : R2 → R such that f(x, y) ∈ S for all 2 2 AB A = BA B (where juxtaposition of operators de- x, y ∈ S, we have that all eigenvalues of fE (A, B) are notes the usual linear algebraic product), but we do have also in S. For example, if the eigenvalues of both A and p(A, B) = q(A, B). Additionally, for any unitary opera- B are positive integers and f : R2 → R is addition, then 7 all of the eigenvalues of the “sum” fE (A, B) will also be B. Example: f(A, B) when dim H = 2 positive integers. We now present an equality which illustrates the nat- In what follows, we take H to be a two-dimensional urality of our functional calculus. We begin with some Hilbert space, and f : R2 → R to be addition. While R2 R definitions. Let f,g : → be addition and mul- this example is somewhat trivial, it illustrates some in- R R tiplication of numbers respectively, and let e : → teresting points. As in section II D, we take A = αI +a·σ be the exponential function. For any observables A and B = βI + b · σ, where the eigenvalues are given by R and B,. and for any fixed generating. chain E of B( ), a± := α ± |a| and b± := β ± |b|, respectively. Without let A + B := fE (A, B) and A × B := gE (A, B). If loss of generality, we assume that |a| ≥ |b|. e−1(E) ∈E for any E ∈E (as is the case for E⋆ in (16)), First, recall that when [A, B] = 0, we have that then f(A, B) = A + B. However, when [A, B] 6= 0, then . . f(A, B) is only a gPVM (i.e. it is not a PVM). Of course, A B A+B e × e = e . (18) in this case A and B must each have two distinct eigen- values. One can clearly see that f(A, B) is not a PVM This statement follows directly from Lemmas 10 and 11 by computing in the appendix. Comparing (18) above to the Baker-

Campbell-Hausdorff formula involving the ordinary lin- f(A, B) (−∞,a− + b−] =0 ear algebraic sum and product of non-commuting observ- ables f(A, B) (a−+ b−,a+ + b+) =0

1 1 1 f(A, B) [a+ + b+, ∞) =0, (23) eAeB = eA+B+ 2 [A,B]+ 12 [A,[A,B]]− 12 [B,[A,B]]+···, (19) and noting that if f(A, B) were a PVM, the above three one can clearly see the elegance and simplicity of the new terms would need to sum to I. functional calculus. Our addition and multiplication also As. discussed above, we can form an observable satisfy other nice properties, such as commutativity A + B := fE (A, B) by choosing a generating chain E for . . . . B(R). A natural choice is to take E = E⋆ from (16), in . . A × B = B × A and A + B = B + A (20) A+B which case the spectral family {Eλ }λ∈R for A + B is for any A and B (even when [A, B] 6= 0) and any E. given by However, in some ways the behavior is not as natural — . . . A+B −1 A B E = JAB ◦ f (−∞, λ] = E ∧ E , (24) for example, our .+ and × are not, in general,. associative, λ η λ−η η∈R and generically × does not distribute over +.  _ Finally, we note another natural property of our func- A B tional calculus. For observables A and B we write A ⊑ B where {Eλ }λ∈R and {Eλ }λ∈R are the spectral families if for A and B, respectively. A straightforward calculation then yields (whenever A((−∞, λ]) ≤ B((−∞, λ]) ∀λ ∈ R, (21) [A, B] 6= 0) . which is referred to as the spectral order (see e.g., [24]) A + B = (α + β)I + (|a| − |b|)aˆ · σ, (25) and differs from the usual order on Hermitian operators a a a (which is defined by A ≤ B if hψ|A|ψi ≤ hψ|B|ψi for all where ˆ .= /| |. Note that the eigenvalues of the observ- |ψi ∈ H). In general, we have that if A ⊑ B, then A ≤ B, able A + B are a+ + b− and a− + b+. Compare this to but not conversely; however, these orderings agree when the eigenvalues of the linear algebraic sum A + B, which

A and B are projection operators,. as well as when A are given by (α + β) ± |a + b|, and are clearly not of the and B commute. Now, for + defined as above and with above form. . respect to the generating chain E⋆ in (16), if A and B are We can easily see from the above results that + is not σ observables such that A ⊑ B, then (for any observable C) an associative operation. Let. α = β. = 0 and let C = c· . . . Then the eigenvalues of (A + B) + C are A + C ⊑ B + C, (22) ± ( |a| − |b| − |c|) (26) i.e. the addition of observables defined with respect to . . E⋆ respects the spectral order on the observables. (By while the eigenvalues of A + (B + C) are contrast, the ordinary linear algebraic sum does not re- a b c spect the spectral. order [24].) A similar statement holds ± (| |− | | − | | ). (27) for the operation × defined with respect to E⋆ when the eigenvalues of the observables involved are non-negative The simple example above explicitly illustrates some of numbers. the general features of our functional calculus discussed We now present a simple worked-out example of our in section IV A. functional calculus. 8

V. CONCLUSION VI. TECHNICAL APPENDIX

In the context of finite-dimensional Hilbert spaces, we In this section we prove all of our previous results; have given an explicit construction of a generalized joint moreover, we do this in a more general context than observable JAB for an arbitrary pair of observables A and stated originally. In what follows H will denote a fixed B, as well as described a realization of the corresponding separable complex Hilbert space with projection lattice joint measurement in the framework of quantum opera- LH. Additionally, we assume the reader is conversant tions, both of which agree with their standard definitions with the standard terminology and basic results used in when [A, B] = 0. Further, we have noted that the fail- (see e.g., [14]). Finally, we will take ure of generalized joint observables to be ordinary joint N = {1, 2,...} to denote the positive integers. observables is characterized by their lack of countable additivity (as they are only countably sub-additive), and have demonstrated how this failure can be interpreted A. Basic Definitions and Properties as a manifestation of the uncertainty principle. We then went on to describe the functional calculus of observ- Definition 1. Let (Ω, M) be a measurable space, and ables which arises from our notion of a generalized joint let A : M → LH be such that observable, as well as described some of its remarkable properties. (1) A(Ω) = I; Although the results presented in sections III and IV (2) A(R) ⊥ A(S) for all disjoint R,S ∈M; are for observables on finite-dimensional Hilbert spaces, the appendix extends these results to bounded observ- ∞ ∞ ables with pure point spectra on any separable complex (3) A(Ri) = A Ri for all R1, R2,... ∈ M such i=1 i=1 Hilbert space. Also, although we have chosen (for nota- X  [  tional simplicity) to present our results for pairs of ob- that Ri ∩ Rj = ∅ whenever i 6= j. servables, they can all be extended in a straightforward Then A is called a PVM on (Ω, M), and if furthermore fashion to sets of n observables. In particular, equation Ω = R and M = B(R) (the Boolean σ-algebra of Borel (1) can be generalized to (for observables A1,...,An and subsets of R), then we call A a standard PVM. Q ∈B(Rn)) The standard PVMs are in 1-1 correspondence with

JA1,...,An (Q) := A1(R1) ∧ ... ∧ An(Rn) , (28) (not necessarily bounded) self-adjoint operators on H. R1×...×Rn⊆Q R n Using the spectral family {Eλ}λ∈ defined by a stan- R1×...×_Rn∈B(R )  dard PVM (i.e. A((−∞, λ]) = Eλ for all λ ∈ R), the cor- and one obtains the functional calculus in this case for responding self-adjoint operator A on H is obtained by (Borel measurable) functions f : Rn → R by defining the Riemann-Stieltjes −1 f(A1,...,An) := JA1,...,An ◦ f , which still has all of +∞ the interesting properties discussed in section IV A. A = λdEλ, (29) This work opens up many directions for further re- Z−∞ search, perhaps the most pressing of which is to find in- teresting problems which are more naturally formulated which is defined to converge in the strong operator topol- in terms of our functional calculus of observables instead ogy. Conversely, given a self-adjoint operator whose (right continuous) spectral family is denoted by {Eλ}λ∈R, of the ordinary linear algebraic sum and product, and for R which the new calculus provides novel physical insight. the corresponding standard PVM A : B( ) → LH is defined to be the unique PVM which satisfies (for all Additionally, there are questions which are technical in R nature that we would like to address — e.g. we would α, β ∈ with α ≤ β) like to extend the notions of generalized joint observ- β ables and joint measurability presented here to observ- A((α, β]) = dEλ. (30) ables with continuous portions to their spectra, as well Zα as further explore properties of the families of PVMs as- Definition 2. Let A be a PVM. Then we say A is diag- sociated with gPVMs. Finally, it would be interesting onalizable if A is standard and the self-adjoint operator to design simple experiments in which non-commuting corresponding to A is bounded and has a set of eigenvec- observables are simultaneously measured in the manner tors which forms a (Schauder) basis for H. outlined in section IIIA. Note that A diagonalizable is equivalent to the self- adjoint operator corresponding to A being bounded ACKNOWLEDGMENTS with pure point spectrum. Moreover, for A diagonal- izable, we define σp(A) to be the set of eigenvalues We thank Randall Espinoza, Nick Huggett, Mark of the self-adjoint operator corresponding to A, and Mueller, and Josh Norton for many useful discussions. σ(A) to be the spectrum of this operator. Then we 9 have that (i) σ(A) is compact, (ii) σp(A) is count- in property 3. The other expression in property 3 above able, (iii) σ(A) is the closure of σp(A), and finally (iv) is obtained in a similar fashion. A(R)= A(R ∩ σ(A)) = A(R ∩ σp(A)) for any R ∈B(R) Finally, to see that property 4 holds, let Ri ∈ M for N ∞ N [14]. i ∈ . Since Ri ⊆ i=1 Ri for each i ∈ , property ∞ N As mentioned previously, our generalized joint observ- 2 above gives that J(Ri) ≤ J( i=1 Ri) for each i ∈ . ables as defined in section III above are not quite PVMs Thus, S S — instead, they are generalized projection-valued mea- ∞ ∞ sures (or gPVMs) as defined below. We note that the J(R ) ≤ J( R ). (32) properties of gPVMs actually make them analogous to i i i=1 i=1 the notion of an inner measure, rather than a typical _ [ measure [25]. The other expression in property 4 above is obtained in a similar fashion. Definition 3. Let (Ω, M) be a measurable space. A gen- eralized projection-valued measure, or gPVM, on (Ω, M) (or just Ω, if M is clear from the context) is a map The following characterization of gPVMs will also J : M → LH such that prove useful. (1) J(Ω) = I; Lemma 2. Let (Ω, M) be a measurable space, and let J : M → LH satisfy (2) J(R) ⊥ J(S) for all disjoint R,S ∈M; (1) J(Ω) = I; ∞ ∞ (3) J(Ri) ≤ J Ri for all R1, R2,... ∈ M such (2) J(R) ⊥ J(S) for all disjoint R,S ∈M. i=1 i=1 X  [  Then J is a gPVM iff that Ri ∩ Rj = ∅ whenever i 6= j. Note that every PVM is also trivially a gPVM. We now J(R) ≤ J(S) for all R,S ∈M with R ⊆ S. (33) prove some useful properties of gPVMs. Proof. If J is a gPVM, then by (2) in lemma 1, equation Lemma 1. Let (Ω, M) be a measurable space, and let (33) is satisfied. Conversely, if we have R1, R2,... ∈ M ∞ N J be a gPVM on (Ω, M). Then pairwise disjoint, then J(Ri) ≤ J i=1 Ri for all i ∈ , since J satisfies equation (33), and hence we have (1) J(∅) = 0; S  ∞ ∞ ∞ (2) J(R) ≤ J(S) whenever R,S ∈M with R ⊆ S; J(Ri)= J(Ri) ≤ J Ri (34) i=1 i=1 1=1 n n X _  [  (3) J(Ri) = J(Rn) and J(Ri) = J(R1) for where the first equality is due to the fact that the Ri’s i=1 i=1 are pairwise disjoint. _ ^ R1,...,Rn ∈M with R1 ⊆···⊆ Rn; Given two gPVMs A and B on a measurable space ∞ ∞ ∞ ∞ 2 (Ω, M), we define their joint observable JAB : M → LH (4) J(Ri) ≤ J Ri and J(Ri) ≥ J Ri by i=1 i=1 i=1 i=1 _  [  ^  \  for any R1, R2,... ∈M. JAB(Q) := A(R1) ∧ B(R2) ∀Q ∈M, (35)

R1×R2⊆Q Proof. Regarding property 1, we clearly have Ω ∩∅ = ∅, R1,R_2∈M so property 2 of gPVMs gives that J(∅)⊥J(Ω) = I, from which it follows that J(∅) = 0. where M2 denotes the product σ-algebra of M with it- Next, if R ⊆ S, then R ∩ (S − R) = ∅, so that by self, i.e. the smallest σ-algebra over Ω2 which contains sub-additivity (i.e. property 3 of gPVMs) we have the Cartesian product M×M. As can easily be seen in the case where A, B are PVMs corresponding to non- J(R) ≤ J(R)+J(S −R) ≤ J(R∪(S −R)) = J(S), (31) commuting observables, JAB so defined is not in general a PVM, but only a gPVM. which shows that property 2 holds. To see that property 3 holds, let Ri ∈ M for i ∈ Theorem 1. Given any two gPVMs A and B on a mea- {1,...,n} be such that R1 ⊆ ... ⊆ Rn, and note that surable space (Ω, M), JAB defined above is a gPVM on n Ω2. J(Ri) ≤ i=1 J(Ri) for each i ∈{1,...,n} — in partic- ular, J(R ) ≤ n J(R ). Now, since R ⊆ n R n i=1 i i i=1 i Proof. First, we have that for each Wi ∈ {1,...,n}, property 2 above gives that n W S J(Ri) ≤ J( Ri)= J(Rn) for each i ∈ {1,...,n}, so 2 n i=1 n JAB(Ω )= A(R1)∧B(R2) ≥ A(Ω)∧B(Ω) = I. (36) that i=1 J(Ri) ≤ J( i=1 Ri)= J(Rn). This inequal- 2 S R1×R2⊆Ω ity, along with that above, establishes the first equality R ,R_∈M W S 1 2 10

Next, we note that if Q∩R = ∅, then clearly Q0 ∩R0 = ∅ C. Coarse-graining for all Q0 ⊆ Q and R0 ⊆ R. Hence we have Lemma 3. Let J be a gPVM on the measurable space A(Q1) ∧ B(Q2) ⊥ A(R1) ∧ B(R2) (37) (Ω, M), and let Ei ∈ M (for each i ∈ N), satisfy Ei ∩ for all Q1 × Q2 ⊆ Q and R1 × R2 ⊆ R. Taking the join Ej = ∅ whenever i 6= j, and also over each side of expression (37) gives JAB(Q) ⊥ JAB(R). ∞ We also have that equation (33) is satisfied since when J(Ei)= I. (44) Q ⊆ R, we have that any R1 × R2 ⊆ Q also satisfies i=1 R1 × R2 ⊆ R. Hence JAB is a gPVM by Lemma 2. _ Then for any subset S ⊆ N, we have that

B. Uncertainty Relation J(Ei)= J Ei . (45) i_∈S  i[∈S  |ψi Proof. We prove this by contradiction, so assume that We begin by extending our definition of MA (equation (13) in section IIIF) for a given diagonalizable PVM A equation (45) does not hold. Since J is a gPVM (so and state |ψi ∈ H — namely that, in particular, property 2 of Lemma 1 holds), this means that |ψi R MA := {λ ∈ : A({λ})|ψi=0 6 }. (38) J(Ei)

Proof. First, since J is a gPVM, by property 2 in Lemma coarse-graining of any gPVM with respect to this parti- 1 we have tion — namely define, for any Q˜ ∈MP ,

∞ ∞ J˜(Q˜) := J Q˜ . (60) J(Qi) ≤ J Qi . (53) i=1 i=1  [  _  [  We then find that our gPVMs behave naturally with It remains to show the other inequality. Define respect to this notion of coarse-graining, provided the coarse-graining is “well-behaved” with respect to the ∞ c measurable sets, as illustrated in the following two theo- E0 = Ei , (54) rems. i=1  [  Theorem 3. Let J be a gPVM on the measurable space ∞ and note that E0 ∈ M, as well as that {Ei}i=0 is a (Ω, M), and let P ⊆M be a partition of Ω. Then J˜ partition of Ω, and also that J(E ) = 0. Next, for j ∈ N, 0 is a gPVM on the measurable space (P, MP ). If, fur- define thermore, there is some countable subset P0 ⊆ P such that Ej := {Ei : Ei ⊆ Qj , i ∈ N}∪{E0}, (55) J(E)= I, (61) and note that Q ⊆ E by equation (51), as well as j j E∈P0 that (recalling that X := Z for any set X whose X S Z∈X elements Z are, themselves, sets) then J˜ is a PVM. S S Proof. First we show that J˜ is a gPVM whenever P is J(E)= J(E) ≤ J(Qj ) a partition of Ω. Clearly, J˜ is a map from M to L . E∈Ej E∈Ej P H _ E_6=E0 Property 1 of gPVMs holds, since ≤ J E = J(E). (56) j J˜(P)= J P = J(Ω) = I, (62)  [  E_∈Ej  [  The first equality in the above expression holds since using that J is a gPVM. For property 2, consider R,˜ S˜ ⊆ ˜ ˜ J(E0) = 0, and the following inequalities follow from the P such that R ∩ S = ∅. It is easy to see that this implies ˜ ˜ fact that if Ei ∈ Ej with i 6= 0, then Ei ⊆ Qj and also that R ∩ S = ∅ (since any two elements of P are that Qj ⊆ Ej (along with the fact that J is a gPVM); either equal or disjoint as sets), and so we have the final equality then follows from Lemma 3. Hence, we S S have S J˜(R˜)= J R˜ ⊥ J S˜ = J˜(S˜), (63)  [   [  J(Qj )= J(E). (57) again using that J is a gPVM. Finally, we see that equa- E∈Ej ˜ ˜ _ tion (33) is satisfies, since for R ⊆ S ⊆ P, we clearly ˜ ˜ ∞ have that R ⊆ S, so that Further, define E = j=1 Ej . Then (again since J is a gPVM), we have ˜S˜ S ˜ ˜ ˜ ˜ S J(R)= J R ⊆ J S = J(S) (64) ∞  [   [  J Qi ≤ J E = J(E) since J is a gPVM, and so J˜ is a gPVM by Lemma 2. j=1 E∈E Next, we assume that there is a countable subset  [   [  _ ∞ ∞ P0 ∈MP such that equation (61) holds. Consider some ˜ ∞ ˜ = J(E) = J(Qj ), (58) countable set {Qi}i=1 ⊆ MP . Defining Qi = Qi (for j=1 E∈Ej j=1 all i ∈ N), note that for each E ∈ P0 and each Qi, we _  _  _ S clearly have Qi ∩ E = E or Qi ∩ E = ∅. Then we see where we have again used Lemma 3 to arrive at the sec- immediately (using Lemma 4) that ond equality. The desired result then follows from the inequalities (53) and (58). ∞ ∞ ∞ ∞ J˜(Q˜i)= J(Qi)= J Qi = J˜ Q˜i , (65) i=1 i=1 i=1 i=1 Definition 4. Let (Ω, M) be a measurable space, and _ _  [   [  let P be a partition of Ω. If P⊆M, we define which shows that J˜ is a PVM.

MP := {Q ⊆P : Q ∈ M}. (59) Corollary 1. If J above is a diagonalizable PVM, then ˜ [ J is a diagonalizable PVM. It is easy to see that MP as defined in (59) is indeed a (Boolean) σ-algebra. We can then naturally define a Proof. Take P0 = {E ∈P : E ∩ σp(J) 6= ∅}. 12

Definition 5. Let (Ω, M) be a measurable space, let P we will show that R˜1 ×R˜2 ⊆ Q˜, so consider some X ∈ R˜1 be a partition of Ω such that P⊆M, and let MP be as and Y ∈ R˜2. Then we must have (a,b) ∈ R1 × R2 for in Definition 4 above. If some a ∈ X and b ∈ Y . Since R1 × R2 ⊆ Q˜, we must have (a,b) ∈ Q˜, so that there exists some T ∈ Q˜ with {X ∈P : E ∩ X 6= ∅}∈MP (66) S (a,b) ∈ T . Then, since Q˜ ∈ MP , we must have that ˜ S whenever E ∈M, then we call P an appropriate partition Q ⊆ P , and hence that T ∈ P . But also, X × Y ∈ P , of Ω. and since (a,b) is a common element of X × Y and T , and P is a partition, we must have T = X × Y , so that Note that whenever P is a countable set, it is neces- X × Y ∈ Q˜, and hence that R˜1 × R˜2 ⊆ Q˜. ˜ sarily an appropriate partition. Now, we also have that R1 ∈ MPA by equation (66), and similarly R˜ ∈M , so that R˜ × R˜ ∈M . Then Theorem 4. Let A, B be gPVMs on a measurable space 2 PB 1 2 P we have that A˜(R˜ ) ∧ B˜(R˜ ) is in the join of equation (Ω, M), and let P and P be appropriate partitions of 1 2 A B (70), and also that Ω. Further let P be the partition of Ω2 given by P := {p × q : p ∈ PA, q ∈ PB}. Then A(R1) ∧ B(R2) ≤ A R˜1 ∧ B R˜2

J ˜ ˜ = J˜AB, (67)     AB =[A˜(R˜1) ∧ B˜(R˜[2), (72) where A˜, B˜, and J˜AB are defined (via equation (60)) with since Ri ⊆ R˜i for i = 1, 2. Hence we have established respect to the partitions PA, PB, and P , respectively. J ˜ ˜ (Q˜) ≥ J˜ (Q˜), since each element in the join of AB SAB Proof. For this proof we will need two simple results equation (69) is greater than an element in the join in whose proofs we omit since they use only elementary set equation (70). theory — first, for any sets X and Y whose elements are sets, and such that X ⊆ Y , we have X ⊆ Y . Also, D. Characterization of JAB if we have two sets R ⊆ PA and S ⊆ PB , then we have S S (R × S)= R × S . (68) Before proving our main results, we will need two use- ful lemmas. First, recall that for a diagonalizable PVM [  [   [  ˜ A, the spectrum σ(A) is a compact set which is the clo- Consider any Q ∈MP . We will prove the above result sure of the set of eigenvalues σ (A), and moreover, there by showing that J˜ (Q˜) ≤ J (Q˜) and also J (Q˜) ≤ p AB A˜B˜ A˜B˜ exists an orthonormal basis for H consisting of eigenvec- ˜ ˜ JAB(Q). Now, we have both that tors of A. For such an A, and for any R ∈B(R), we have that A(R) = A(R ∩ σp(A)). From this fact we immedi- J˜AB (Q˜)= JAB Q˜ = A(R1) ∧ B(R2), ately deduce the following lemma. ˜  [  R1×R_2⊆ Q R1,R2∈M Lemma 5. Let A, B be diagonalizable PVMs. Then S (69) 2 as well as JAB (Q)= A(R1) ∧ B(R2) ∀Q ∈B(R ), (73)

R1×R2⊆Q R ,R_∈B(R) J ˜ ˜ (Q˜)= A˜(R˜1) ∧ B˜(R˜2). (70) 1 2 AB R1⊆σ(A),R2⊆σ(B) ˜ ˜ ˜ R1×_R2⊆Q R˜1×R˜2∈MP 2 and JAB Q ∩ σ(A) × σ(B) = JAB(Q) ∀Q ∈B(R ). Considering any element in the join of equation (70), Now, we also have the following nice behavior of our ˜ ˜ ˜ we see that since R1 × R2 ⊆ Q, we must have joint observable with respect to common eigenvectors. Lemma 6. Let A, B be diagonalizable PVMs, and let R˜1 × R˜2 = (R˜1 × R˜2) ⊆ Q,˜ (71) |ψi ∈ H be such that A|ψi = a|ψi and B|ψi = b|ψi. [ [ [ [ Then for any Q ∈B(R2), we have that and moreover, by the definition of MP , we must have ˜ ˜ 2 ˜ ˜ R1 × R2 ∈M , so that both R1, R2 ∈M. From (1) (a,b) ∈ Q implies that J (Q)|ψi = |ψi; ˜ AB this we immediately conclude that indeed JA˜B˜ (Q) ≤ S S S S c J˜AB(Q˜), since each element in the join of equation (70) (2) (a,b) ∈ Q implies that JAB(Q)|ψi = 0. ˜ occurs in the join in equation (69) (taking Ri = Ri for Proof. For the first statement, note that by our assump- i =1, 2). S tion we have A({a})|ψi = |ψi and also B({b})|ψi = |ψi, For the other inequality, consider some term in the and this means that A({a}) ∧ B({b})|ψi = |ψi. Then, if join of equation (69), and then for the R , R occurring 1 2 (a,b) ∈ Q, since JAB is a gPVM, we have that in this term define R˜1 to be the set of all X ∈ PA such that x ∈ R1 for some x ∈ X, and similarly for R˜2. First, JAB(Q) ≥ JAB({(a,b)})= A({a}) ∧ B({b}), (74) 13 establishing (1) above. If, on the other hand, we assume Proof. Since f is measurable, we have that f −1(Q) ∈M c c ′ that (a,b) ∈ Q , then we have that JAB(Q )|ψi = |ψi whenever Q ∈ N , so that J : N → LH. Then we have c by (1), and JAB(Q) ⊥ JAB (Q ) since JAB is a gPVM, J ′(Ω′)= J ◦ f −1(Ω′)= J(Ω) = I. (76) which gives that JAB(Q)|ψi = 0. We now present the characterization theorem for our Next, if R,S ∈ N are disjoint, we have that generalized joint observables. f −1(R) ∩ f −1(S)= f −1(R ∩ S)= f −1(∅)= ∅ (77) Theorem 5. Let A, B be diagonalizable PVMs, and let and since J is a gPVM, this means that J ′(R) ⊥ J ′(S). J : B(R2) → L be a set map. Then J = J if H AB Finally, for R,S ∈ N with R ⊆ S, we have that f −1(R) ⊆ and only if J satisfies the following two conditions for f −1(S), so that J ′(R) ≤ J ′(S), which again, follows from all Q ∈B(R2) and all |ψi ∈ H. the fact that J is a gPVM. R (1) If there exist R1, R2 ∈ B( ) with R1 × R2 ⊆ Q and As such, for given diagonalizable PVMs A and B, and A(R1) ∧ B(R2)|ψi = |ψi, then J(Q)|ψi = |ψi. Borel measurable function f : R2 → R,

(2) If for every R1, R2 ∈B(R) with R1 ×R2 ⊆ Q we have −1 f(A, B) := JAB ◦ f A(R1) ∧ B(R2)|ψi = 0, then J(Q)|ψi = 0. is a gPVM. Proof. First we will show that condition 1 above im- R2 In the sequel, we will find the property of f(A, B) in plies that J ≥ JAB, so consider any Q ∈ B( ), and the following lemma useful. any R1, R2 ∈ B(R) such that R1 × R2 ⊆ Q. Now if A(R1) ∧ B(R2)|ψi = |ψi, by assumption we must have Lemma 7. Let A, B be diagonalizable PVMs, and let 2 J(Q)|ψi = |ψi, i.e. J(Q) ≥ A(R1) ∧ B(R2). Since this is f : R → R be Borel measurable. Then, for all R ∈B(R), true for any such R1 × R2 ⊆ Q, taking the join gives that 2 f(A, B) R ∩ f(σ(A), σ(B)) = f(A, B)(R). (78) J(Q) ≥ JAB(Q). Since this is true for any Q ∈B(R ), it follows that J ≥ J . AB Proof. To reduce notational clutter, let Next we show that J ≥ J implies condition 1 AB S := f(σ(A), σ(B)). By Lemma 5, along with the above. Given Q ∈ B(R2) and |ψi ∈ H, assume that fact that f −1 ◦ f(X) ⊇ X for any set X, we have that there exist R1, R2 ∈B(R) with R1 × R2 ⊆ Q such that −1 −1 A(R1) ∧ B(R2)|ψi = |ψi. Then we have f(A, B)(E ∩ S)= JAB f (E) ∩ f (S) ≥ J f −1(E) ∩ σ(A) × σ(B) J(Q) ≥ JAB (Q) ≥ A(R1) ∧ B(R2), (75) AB  −1 = JAB ◦ f (E)= f(A, B)(E), (79) so that J(Q)|ψi = |ψi. Finally, we show that condition 2 above is equiv- and so equality holds (since f(A, B) is a gPVM, and alent to J ≤ JAB. Note that J ≤ JAB if and property 2 of Lemma 1 gives the other inequality). ⊥ ⊥ 2 only if J(Q) ≥ JAB(Q) for all Q ∈B(R ). Now assume condition 2 and consider |ψi ∈ H such that We now prove that we can construct PVMs out of these gPVMs. We first make the following definition. JAB(Q)|ψi = 0. By definition of the join, we must have R A(R1) ∧ B(R2)|ψi = 0 for all R1, R2 ∈B( ) such that Definition 6. Let M be a Boolean σ-algebra, and let R1 × R2 ⊆ Q. Then condition 2 gives that J(Q)|ψi = 0, E⊆M be a subset of M which is totally ordered under which shows that J ≤ JAB. inclusion and which furthermore generates M as a σ- 2 On the other hand, given Q ∈ B(R ), assume that algebra. Then E will be called a generating chain for M. ⊥ ⊥ J(Q) ≥ JAB(Q) , as well as that the hypothesis of con- If M = B(R), we will simply refer to E as a generating dition 2 holds for a given |ψi ∈ H. We can easily see that chain. JAB(Q)|ψi = 0, so that J(Q)|ψi = 0. As such, condition 2 holds. Theorem 7. Let (Ω, M) be a measurable space, let E be a generating chain for M, and let J be a gPVM on (Ω, M) such that there exists a finite set S ∈ M which E. Functional Calculus satisfies J(Q ∩ S) = J(Q) for all Q ∈ M. Then J|E uniquely extends to a PVM on (Ω, M)[26]. We now prove a generalization of the result that our Proof. The result will follow from Sikorski’s extension f(A, B) (previously defined in section IV) is in fact a theorem — first, let α : E → {−1, 1} be any set map, gPVM. and for any E ∈M define 1 · E := E and −1 · E := Ec, while for any P ∈ L define 1·P := P and −1·P := P ⊥. Theorem 6. Let (Ω, M) and (Ω′, N ) be measurable H We will show that for any countable E ⊆E, such that spaces, let f : Ω → Ω′ be a measurable function, and 0 ′ −1 let J be a gPVM on (Ω, M). Then J := J ◦ f is a α(E) · E = ∅, (80) gPVM on (Ω′, N ). E\∈E0 14 we have that Theorem 8. Let A, B be diagonalizable PVMs, and let f : R2 → R be continuous. Then α(E) · J(E)=0. (81) E := J ◦ f −1 (−∞, λ] (85) E^∈E0 λ AB −1 −1  First, let E+ := α ({1})∩E0 and E− := α ({−1}) ∩E0. is a spectral family of projectors on H. Since E is a generating chain for M, we have that E = E as well as E = E for some Proof. First note that Eλ is clearly a projection operator E∈E+ + E∈E− − for all λ, and that λ 7→ Eλ is obviously a monotone map E+, E− ∈M. Then T S (i.e. λ1 ≤ λ2 implies Eλ1 ≤ Eλ2 ). Next we show that c limλ→−∞ Eλ = 0 and limλ→∞ Eλ = I. Since f is contin- α(E) · E = E ∩ E uous and A and B are diagonalizable, f(σ(A), σ(B)) is a E\∈E0 E\∈E+ E\∈E− compact subset of R, so let m denote the minimum and c = E+ ∩ E− M the maximum of this set. By Lemma 5 we immedi- = ∅, (82) ately see that so that E+ ⊆ E−. Now, for any Q ∈ M, we have (by Em−1 = A(R1) ∧ B(R2) −1 assumption) that J(Q ∩ S)= J(Q). Using this, we have R1×R2⊆f _((−∞,m−1]) R1,R2∈B(R) R1⊆σ(A),R2⊆σ(B) α(E) · J(E)= J(E) ∧ J(E)⊥ = ∅ = 0 (86) E^∈E0 E^∈E+ E^∈E− ⊥ _ = J(E ∩ S) ∧ J(E ∩ S) and E^∈E+ E^∈E− ⊥ E = A(R ) ∧ B(R ) = J(E+ ∩ S) ∧ J(E− ∩ S) M 1 2 −1 ⊥ R1×R2⊆f_((−∞,M]) = J(E+) ∧ J(E−) , (83) R1,R2∈B(R) R1⊆σ(A),R2⊆σ(B) where the second to last equality holds because J is ≥ A(σ(A)) ∧ B(σ(B)) = I, (87) monotonic and S is a finite set, so that {E ∩S : E ∈E0} is a finite set as well. Now, since E+ ⊆ E−, we must have and since Eλ is an increasing function of λ, this demon- ⊥ that J(E+) ≤ J(E−), and hence that J(E+) ⊥ J(E−) , strates the desired property. which gives (combining with equation (83)) The content of the above lemma is that for A, B di- α(E) · J(E)=0. (84) agonalizable (but not necessarily with finite spectra), for the particular generating chain E^∈E0 ⋆ R The result then follows directly from (one version of) E := {(−∞, λ]: λ ∈ }, (88) the Sikorski extension theorem (theorem 34.1 in [27]). we can construct a PVM which agrees with f(A, B) on Uniqueness follows trivially from the fact that any two E⋆. In what follows, for A, B diagonalizable PVMs, we such extensions must agree on a generating chain, and will call a generating chain E appropriate if there is a hence must be equal. PVM fE (A, B) which agrees with f(A, B) on E. It is easy to see that any such PVM must be unique. Corollary 2. Let E be a generating chain, let A, B be di- We now show that the assumption that the generating agonalizable PVMs with finite spectra, and let f : R2 → set is a chain is necessary in order to construct PVMs. R be Borel measurable. Then f(A, B)|E uniquely extends to a PVM. Lemma 8. Let H be a two-dimensional Hilbert space, and let A = σx and B = σy (Pauli matrices), and as- Proof. This follows directly from the above theorem and sume that E σ-generates B(R), but that E is not a chain Lemma 7, since f(A, B) is a gPVM and we can take under ⊆. Then there exists a continuous (and hence S = f σ(A), σ(B) . 2 −1 Borel measurable) f : R → R such that JAB ◦ f |E  does not extend to a PVM with its spectra contained in Given two diagonalizable PVMs A and B with finite f(σ(A), σ(B)). spectra, a generating chain E, and a Borel measurable function f : R2 → R, we will denote the PVM agree- Proof. Since E is not a chain, there exists some c ing with f(A, B)|E (provided by corollary 2 above) as E1, E2 ∈ E such that there is some α ∈ E1 ∩ E2 c fE (A, B). Note that fE (A, B)(Q) = 0 for all Q ⊆ and some β ∈ E2 ∩ E1. Note that σ(A) × σ(B) = f(σ(A), σ(B))c. We also have the following result which {(1, 1), (1, −1), (−1, 1), (−1, −1)}. By the Tietze exten- goes beyond the finite spectra case. sion theorem, there exists a continuous f : R2 → R such 15 that f(1, 1) = f(−1, −1) = α and f(−1, 1) = f(1, −1) = (1) σ(fE (A, B)) ⊆ f(σ(A), σ(B)); β. † † † We now prove the lemma by contradiction, so assume (2) fE (UAU ,UBU ) = UfE (A, B)U for any unitary −1 operator U on H; that there is a PVM F extending JAB ◦ f which is such that σ(F ) ⊆ f(σ(A), σ(B)) = {α, β}. Then we (3) If A|ψi = a|ψi and B|ψi = b|ψi for some |ψi ∈ H, must have that then fE (A, B)|ψi = f(a,b)|ψi. F ({α, β})= I. (89) Proof. We begin with property 1 above. Assume that λ ∈ σ(f (A, B)), which is true if and only if the cor- c c E However, since β ⊆ E2, we must have E2 ⊆ {β} , responding spectral family is not constant for every in- −1 −1 c and hence f (E2) ⊆ f ({β}) , and in particular terval surrounding λ. In this case we must have, for −1 f (E2) ∩ σ(A) × σ(B) ⊆{(1, 1), (−1, −1)}. From this N n n R every n ∈ , that there exists some R1 , R2 ∈ B( ) ∩ we immediately see that n n −1 1  σ(A) × σ(B) with R1 × R2 ⊆ f ((−∞, λ + n ]) but n n −1 1 N −1 −1 R1 × R2 6⊆ f ((−∞, λ − n ]). Hence, for each n ∈ JAB ◦ f (E2)= JAB f (E2) ∩ σ(A) × σ(B)  there exists some (an,bn) ∈ σ(A)×σ(B) with f(an,bn) ∈   1 1 ≤ JAB {(1, 1), (−1,−1)}  (λ − n , λ + n ], so that limn→∞ f(an,bn)= λ. Of course, since σ(A) × σ(B) is compact, (a ,b ) has a convergent =0. (90) n n  subsequence, converging to some (a,b) ∈ σ(A) × σ(B), −1 and since f is continuous, f(a,b)= λ. A similar argument shows that JAB ◦ f (E1) = 0. But, since F is a PVM extending J ◦ f −1| , we also have Property 2 above follows directly from Lemma 9, since AB E −1 ⋆ f(A, B)= JAB ◦ f , and so for any Q ∈E , we have F ({α, β})= F ({α}) ∨ F ({β}) ≤ F (E1) ∨ F (E2) † † −1 f(UAU ,UBU )(Q)= J † † ◦ f (Q) −1 −1 UAU ,UBU = JAB ◦ f (E1) ∨ JAB ◦ f (E2) −1 = J † † f (Q) =0, (91) UAU ,UBU −1 † = U JAB f (Q) U  which is the desired contradiction. † = Uf (A, B )(Q)U . (94) We next prove our results about the properties of an The result then follows from the fact that f (A, B)(Q)= observable f (A, B) from section IV A. We begin with a E E f(A, B)(Q) for all Q ∈E, that E generates B(R), and that result which is useful in demonstrating these properties. conjugation by U induces a σ-homomorphism on LH. Lemma 9. Let A, B be diagonalizable PVMs, let U For property 3, given the hypothesis stated above, we be a unitary operator on H, and let Q ∈ B(R). Then must show that fE (A, B)({f(a,b)})|ψi = |ψi. Since † UJAB(Q)U = JUAU †,UBU † (Q). fE (A, B) is a PVM, this amounts to showing that f(A, B)((−∞,f(a,b)])|ψi = |ψi, while for any n ∈ N, Proof. As is well-known (see [28]), for any P,Q ∈ L , we H that f(A, B)((−∞,f(a,b) − 1 ])|ψi = 0. For the first have that U(P ∧ Q)U † = (UPU †) ∧ (UQU †), and also n statement, we have for any collection {Pj}j∈J ⊆ LH, where J is any set, we −1 have f(A, B)({f(a,b)})= JAB ◦ f ({f(a,b)}) † † (UPj U )= U Pj U . (92) = A(R1) ∧ B(R2) −1 j∈J  j∈J  R1×R2⊆_f (f(a,b)) _ _ R1,R2∈M Using this, we compute. ≥ A({a}) ∧ B({b}), (95) † † UJAB(Q)U = U A(R1) ∧ B(R2) U since {a}, {b}∈B(R). Now, by assumption we have both

R1×R2⊆Q A({a})|ψi = |ψi and B({b})|ψi = |ψi, and hence we also  _  R1,R2∈M have A({a}) ∧ B({b})|ψi = |ψi. Then, using equation † = U A(R1) ∧ B(R2) U (95) and the fact that f(A, B) is a gPVM, we have that

R1×R2⊆Q R1,R_2∈M  f(A, B)((−∞,f(a,b)]) ≥ f(A, B)({f(a,b)}) † † ≥ A({a}) ∧ B({b}). (96) = UA(R1)U ∧ UB(R2)U R1×R2⊆Q As such, it follows that f(A, B)((−∞,f(a,b)])|ψi = |ψi. R1,R_2∈M  Now, consider any n ∈ N. Since we = JUAU †,UBU † (Q). (93) 1 have that (−∞,f(a,b) − n ] ∩{f(a,b)} = ∅, and f(A, B) is a gPVM, we have that 1 f(A, B)((−∞,f(a,b) − n ]) ⊥ f(A, B)({f(a,b)}), Theorem 9. Let A, B be diagonalizable PVMs, and let and since f(A, B)({f(a,b)})|ψi = |ψi by above argu- ⋆ R2 R 1 E = E (equation (88)). Furthermore let f : → be ment, we must have f(A, B)((−∞,f(a,b) − n ])|ψi = 0 continuous. Then fE (A, B) satisfies for any n ∈ N. 16

Analogs of the above results also hold in a slightly dif- Of course, since E is a chain, if we have E, Eˆ ∈ E with ferent context. λ ∈ E and λ∈ / Eˆ, then clearly E 6⊆ Eˆ, and so therefore Eˆ ⊆ E. For this reason we have that Theorem 10. Let A, B be diagonalizable PVMs with finite spectra, and let E be a generating chain. Further- 2 (E ∩ S) ⊇ (E ∩ S), (105) more let f : R → R be Borel measurable. Then fE (A, B) E∈E E∈E satisfies λ\∈E λ/[∈E

(1) σ(fE (A, B)) ⊆ f(σ(A), σ(B)); so that there must be some η satisfying † † † (2) fE (UAU ,UBU ) = UfE (A, B)U for any unitary operator U on H; η ∈ (E ∩ S) but η∈ / (E ∩ S)= E ∩ S, E∈E E∈E E∈E (3) If A|ψi = a|ψi and B|ψi = b|ψi for some |ψi ∈ H, λ\∈E λ/[∈E  λ/[∈E  (106) then fE (A, B)|ψi = f(a,b)|ψi. and hence, since η ∈ S, Proof. To reduce clutter, we define S := f(σ(A), σ(B)) ′ and E := {E ∩ S : E ∈ E}, and we note that since A η∈ / E, i.e. η ∈ Ec. (107) ′ and B have finite spectra, both S and E are finite sets. E∈E E∈E First, we consider property 1, so assume that λ/[∈E λ\∈Ec λ ∈ σ fE (A, B) , so that If λ∈ / S, then clearly η 6= λ, but this means that, for  fE (A, B)({λ}) 6=0. (97) all E ∈ E, that η ∈ E if and only if λ ∈ E. However, a simple inductive argument then yields that E could not Then then generate B(R), and hence we must have that η = λ, so that λ ∈ S. fE (A, B)({λ}) ≤ fE (A, B)(E) (98) The proof of property 2 is similar to the analogous for any E ∈E such that λ ∈ E, and also statement in Theorem 9, so we omit it. For property 3, we need to show that c ⊥ fE (A, B)({λ}) ≤ fE (A, B)(E )= f(A, B)(E) (99) fE (A, B)({f(a,b)})|ψi = |ψi. Since E generates B(R), it is straightforward to show that E′ generates S c for any E ∈E such that λ ∈ E , since fE (A, B) is a PVM as an algebra, and so we have that (which agrees with f(A, B) on elements of E). This then yields {f(a,b)} = E ∩ Ec . (108) ′ ′ ⊥ E∈E E∈E f(A, B)(E) ∧ f(A, B)(E) 6=0. (100)  f(a,b\)∈E   f(a,b\)∈/E  E∈E E∈E λ^∈E λ/^∈E Hence, since fE (A, B) is a PVM, using considerations However, by Lemma 7, we have that f(A, B)(E ∩ S) = similar to those in the proof of property 1, we have that f(A, B)(E). This means that fE (A, B)({f(a,b)})= ⊥ f(A, B)(E) ∧ f(A, B)(E) 6=0. (101) f(A, B)(E) ∧ f(A, B)(E)⊥. (109) ′ ′ E∈E E∈E ′ ′ λ^∈E ^ E∈E E∈E λ/∈E f(a,b^)∈E f(a,b^)∈/E ′ Then, since E is a finite set, by property 4 in Lemma 1, ′ −1 we have Now for any E ∈E , we have f(A, B)(E)= JAB ◦f (E) by definition, and also that f(a,b) ∈ E if and only if ⊥ −1 f(A, B) E ∧ f(A, B) E 6=0, (102) (a,b) ∈ f (E). Hence by Lemma 6, we have that ′ ′ f(A, B)(E)|ψi = |ψi for each E with f(a,b) ∈ E, and E∈E E∈E ′  λ\∈E  h  λ/[∈E i also that f(A, B)(E)|ψi = 0 for each E ∈ E with f(a,b) ∈/ E, which is to say that f(A, B)(E)⊥|ψi = |ψi which means that for such E. From these considerations, along with equa- tion (109), we then deduce that E 6= E, (103) E∈E′ E∈E′ λ\∈E λ/[∈E fE (A, B)({f(a,b)})|ψi = |ψi. (110) or, equivalently

(E ∩ S) 6= (E ∩ S). (104) We next consider the the properties which are used to E∈E E∈E λ\∈E λ/[∈E establish the equality in expression (18) in section IV A. 17

Lemma 10. Let A, B be diagonalizable PVMs, let E be since (g ◦ g−1)(S)= S, we have an appropriate generating chain, and let f : R2 → R −1 −1 and g : R → R be Borel measurable, with g satisfying A(g1 (S1)) ∧ B(g2 (S2)) −1 −1 g (E) ∈E for all E ∈E. Then S1×S2_⊆f (Q) S1×S2⊆g1(σp(A))×g2(σp(B))

(g ◦ f)E (A, B)= g(fE (A, B)). (111) ≤ A(R1) ∧ B(R2) −1 g1(R1)×g2(R2)⊆f (Q) _ R2 Proof. We know that for any Q ∈E we have R1×R2∈B( ) = h(A, B)(Q). (117) −1 [(g ◦ f)E (A, B)](Q) = [JAB ◦ (g ◦ f) ](Q) −1 −1 This gives that = [JAB ◦ f ](g (Q)) −1 A(g−1(S )) ∧ B(g−1(S )) = h(A, B)(Q) = fE (A, B) ◦ g (Q) 1 1 2 2 −1 S1×S2⊆f (Q) = g(fE (A, B))(Q). (112) _ S1×S2⊆g1(σp(A))×g2(σp(B)) (118) for all Q ∈E. We also have that

−1 −1 2 A(g1 (S1)) ∧ B(g2 (S2)) Lemma 11. Let f : R → R and g1,g2 : R → R be Borel −1 measurable functions, let E be an appropriate generating S1×S2⊆f (Q) S ×S ⊆g (σ _(A))×g (σ (B)) chain, and let A, B be diagonalizable PVMs. Then for 1 2 1 p 2 p 2 h : R → R defined by (for all x, y ∈ R) h(x, y) := = g1(A)(S1) ∧ g2(B)(S2) f(g (x),g (y)) (i.e. h = f ◦ (g × g )) we have that −1 1 2 1 2 S1×S2_⊆f (Q) S1×S2⊆σp(g1(A))×σp(g2(B))

hE (A, B)= fE (g1(A),g2(B)). (113) = f(g1(A),g2(B))(Q), (119) and since the above holds for all Q ∈E, and E generates Proof. Since A, B have pure point spectra, we have (for B(R), we have any Q ∈E) hE (A, B)= fE (g1(A),g2(B)). (120) h(A, B)(Q)= A(R1) ∧ B(R2) −1 R1×R2⊆h (Q) _ R2 R1×R2∈B( ) Finally, after a useful lemma, we demonstrate that the spectral order (defined in equation (22)) is respected by = A(R1) ∧ B(R2) ⋆ −1 addition (defined relative to the generating chain E ). R1×R2_⊆h (Q) R1×R2⊆σp(A)×σp(B) Lemma. . 12. Let A, B be diagonalizable PVMs, and let ⋆ = A(R1) ∧ B(R2). (114) + and × be as defined in section IV A, where E = E . −1 Then, for any λ ∈ R, we have g1(R1)×g2_(R2)⊆f (Q) R1×R2⊆σp(A)×σp(B) . (A + B) (−∞, λ] = A (−∞,a] ∧ B (−∞,b] . This follows from a+b=λ  _  (121) −1 If, furthermore, A and B have no non-negative eigenval- h (Q)= {(a,b): f(g1(a),g2(b)) ∈ Q}, (115) ues, we also have −1 . so that (a,b) ∈ h (Q) if and only if (g1(a),g2(b)) ∈ −1 −1 (A × B) (−∞, λ] = A (−∞,a] ∧ B (−∞,b] . f (Q), and hence R1 × R2 ⊆ h (Q) if and only if −1 ab=λ g1(R1) × g2(R2) ⊆ f (Q).  a,b_≥0   (122) Now, for R1 × R2 ⊆ σp(A) × σp(B) such −1 that g1(R1) × g2(R2) ⊆ f (Q), we know that Proof. We first define Qλ := (−∞, λ] for all λ ∈ R. Then g1(R1) ⊆ g1(σp(A)) and g2(R2) ⊆ g2(σp(B)). Hence, we need to prove that −1 using that R ⊆ (g ◦ g)(R) we have . (A + B)(Qλ)= A(Qa) ∧ B(Qb). (123) −1 −1 a+b=λ h(A, B)(Q) ≤ A(g1 (S1)) ∧ B(g2 (S2)). (116) _ −1 . S1×S2_⊆f (Q) Recalling that A + B is defined by S1×S2⊆g1(σp(A))×g2(σp(B)) . (A + B)(Qλ) = A(R1) ∧ B(R2), (124) Also, for any S1 × S2 ⊆ g1(σp(A)) × g2(σp(B)), we −1 −1 −1 R R1×R2⊆_+ (Qλ) clearly have g1 (S1),g2 (S2) ∈ B( ). Additionally, R1,R2∈B(R) 18

. R2 R where ‘+’ is thought of as a map from → . We also fined as in equation (22)). Also, let +. be defined. relative have to the generating chain E⋆. Then A + C ⊑ B + C.

−1 R + (Qλ)= {(a,b) ∈ R : a + b ≤ λ}. (125) Proof. We again let Qλ := (−∞, λ] for all λ ∈ . Now, by Lemma 12, we have that −1 Since, for any a + b = λ, we have Qa × Qb ⊆ + (Qλ), . as well as Qa,Qb ∈B(R), we immediately conclude that (A + C)(Qλ)= A(Qa) ∧ C(Qc), (127) a+c=λ . _ (A + B)(Qλ) ≥ A(Qa) ∧ B(Qb). (126) as well that a+b=λ _ . (B + C)(Qλ)= B(Qb) ∧ C(Qc), (128) To establish the opposite inequality, consider any b+c=λ −1 R1, R2 ∈B(R) such that R1 × R2 ⊆ + (Qλ) for a given _ λ ∈ R. Defining α = sup R1 and β = λ − α, we have and we note that the joins in these expressions run over that α + β = λ. Moreover, we clearly have R1 ⊆ Qλ, and the same sets since a + c = λ = b + c implies a = b. Now, also R2 ⊆ Qβ, since for any b ∈ R2, we have α + b ≤ λ. since A ⊑ B, we have that A(Qµ) ≤ B(Qµ) for all µ ∈ R, But. from this, we see that every term in the join defining from which it follows that (A + B)(Qλ) (equation (124)) is less than some element in the join occurring in our desired expression (i.e. equa- A(Qµ) ∧ C(Qc) ≤ B(Qµ) ∧ C(Qc) (129) tion (123)), establishing the other inequality. for all µ ∈ R. As such, we have that A similar argument (with some subtleties. concerning negative numbers) yields the result for ×. . . (A + C)(Qλ) ≤ (B + C)(Qλ) (130) . . Lemma 13. Let A, B, and C be diagonalizable PVMs, for all λ ∈ R, or equivalently, A + C ⊑ B + C. with A ⊑ B (where ⊑ denotes the spectral order, as de-

[1] E. Arthurs and J. L. Kelly, B.S.T.J. Briefs 44, 725 (1965) ily not satisfying this requirement still corresponds to a [2] C. Y. She and H. Heffner, Phys. Rev. 152, 1103 (Dec unique self-adjoint operator, but each self-adjoint opera- 1966) tor corresponds to multiple spectral families when right- [3] J. Park and H. Margenau, Int. J. Theor. Phys. 1, 211 continuity is not imposed. Of course, given any spectral (1968) family {Eλ}λ∈R, one can form a right-continuous spec- [4] H. Yuen, Phys. Lett. A 91, 101 (1982), ISSN 0375-9601 tral family which corresponds to the same self-adjoint [5] P. Busch, Int. J. Theor. Phys. 24, 63 (1985), ISSN 0020- operator, namely {Eˆλ}λ∈R, where Eˆλ := lim Eλ′ . Fi- λ′→λ+ 7748 nally, although we restrict our discussion to the finite- 62 [6] M. G. Raymer, Am. J. Phys. , 986 (1994) dimensional case in the initial sections of this paper, 26 [7] P. Busch and P. Lahti, Found. Phys. , 875 (1996), ISSN properties (i)-(iii) above define a spectral family on any 0015-9018 separable complex Hilbert space H, and a similar 1-1 [8] P. Busch, T. Heinonen, and P. Lahti, Physics Reports correspondence between (now possibly unbounded) self- 452 , 155 (2007) adjoint operators on H and (right continuous) spectral 89 [9] P. Busch, P. Lahti, and R. Werner, Phys. Rev. A , families on H also holds in this case. 012129 (Jan 2014) [17] S. Holland, in The Logico-Algebraic Approach to Quan- [10] However, some of our results are applicable to a wider tum Mechanics, Vol. 1 (Springer Netherlands, 1975) class of observables. [18] In this case, one must replace 2P with an appropriate [11] S. Gudder, Journal of Mathematics and Mechanics 18, collection MP of subsets of P — see Theorem 3 in the 325 (1968) appendix for details in a more general context. 15 [12] V. S. Varadarajan, Comm. Pure Appl. Math. , 189 [19] Nielsen and Chuang, Quantum Computation and Quan- (1962), ISSN 1097-0312 tum Information (Cambridge University Press, 2000) A B A, B [13] If the spectrum of either or is not finite (and [ ] 6= [20] H. Robertson, Phys. Rev. 34, 163 (1929) f R2 R 0), we require that : → be continuous. [21] We actually require one technical condition on the par- [14] E. Kreyszig, Introductory Functional Analysis with Ap- titions PA and PB in order for equation (12) to hold — plications (John Wiley & Sons, 1978) see Theorem 4 in the appendix. [15] All observables on a finite-dimensional Hilbert space have [22] Just as the generating chain {(−∞,λ]}λ∈R for B(R) can a finite number of eigenvalues, and hence are bounded be used to define a spectral family {Eλ}λ∈R from which with pure point spectra. a PVM on (R, B(R)) can be constructed, we can think 2 [16] Actually, one additional requirement is needed to ob- of the generating chain E for B(R ) as giving rise to a tain this 1-1 correspondence, namely right continuity “generalized spectral family” {J(X)}X∈E from which a ′ + Eλ′ Eλ λ R 2 2 (limλ →λ = for all ∈ ). A spectral fam- PVM on (R , B(R )) can be constructed. 19

[23] This extra orthogonality condition is not necessary when [26] For any map f : X → Y , and any Z ⊆ X, we use the no- i = 1 (and thus λi−1 does not exist). tation f|Z to mean the map f restricted to the subset Z. [24] M. P. Olson, Proc. Amer. Math. Soc. 28, 537 (1971) [27] R. Sikorski, Boolean Algebras (Springer-Verlag, 1969) [25] P. R. Halmos, Measure Theory (Springer, 1978) [28] G. Kalmbach, Orthodmodular Lattices (Academic Press, Inc., 1983)