aa r X i v : . [ qu a n t - ph ] A ug The Garrison–Wong quantum phase operator revisited
The Garrison–Wong quantum phase operator revisited
Jan van Neerven Delft University of Technology, Faculty EEMCS/DIAM, P.O. Box 5031, 2600 GA Delft,The Netherlands a) (Dated: 21 August 2020) We revisit the quantum phase operator Φ introduced by Garrison and Wong. Denoting by N the number operator,we provide a detailed proof of the Heisenberg commutation relation Φ N − N Φ = iI on the natural maximal domain D ( Φ N ) ∩ D ( N Φ ) as well as the failure of the Weyl commutation relations, and discuss some further interesting proper-ties of this pair. I. INTRODUCTION A Heisenberg pair is an ordered pair ( A , B ) of (possiblyunbounded) self-adjoint operators, acting on the same Hilbertspace H , such that for all h ∈ D ( AB ) ∩ D ( BA ) we have ABh − BAh = ih . Here, D ( AB ) = { h ∈ D ( B ) : Bh ∈ D ( A ) } and similarly theother way around. A Weyl pair is an ordered pair ( A , B ) of (possibly unbounded) self-adjoint operators, acting on thesame Hilbert space H , such that for all t , s ∈ R we have theoperator identity e isA e itB = e − ist e itB e isA . Here, ( e isA ) s ∈ R and ( e itB ) t ∈ R are the strongly continuousone-parameter groups generated by iA and iB in the senseof Stone’s theorem. By straightforward differentiation (seeRef. ) every Weyl pair is seen to be a Heisenberg pair, but theconverse is false unless A + B is essentially self-adjoint (thisis the Rellich–Dixmier theorem, see Ref. (Theorem 4.6.1)for a precise statement).The standard textbook example (see Ref. (Section 12.2),Ref. (Section 2.11)) of a Heisenberg pair that is not a Weylpair is the pair ( A , B ) on H = L ( T ) (where T is the unit circlein the complex plane) given by A f ( θ ) = θ f ( θ ) , B f ( θ ) = i f ′ ( θ ) . This is a variation of the standard position-momentum pair.The aim of this short note is to revisit, from a mathemati-cian’s point of view, some well known facts about the number-phase pair ( N , Φ ) on the Hilbert space H ( D ) (where D is theopen unit disc in the complex plane; the relevant definitionsare given below). The interest of this pair derives from it be-ing another example of a Heisenberg pair that is not a Weylpair. This fact is well known and contained in Ref. , exceptfor some details concerning domains which we provide here.We also point out some further interesting features of this pair,providing along the way a rigorous justification of some ob-servations in Ref. . As such this note does not contain newresults, but we hope that it could be of some use to the moremathematically inclined reader interested in the subject. a) Electronic mail: [email protected]
II. QUANTUM PHASE AND NUMBER
The problem of defining quantum phase operators has beenconsidered by many authors and has been re-viewed in several places . It has recently found applica-tion in the context of quantum computing and quantum er-ror correcting codes . The proposal by Garrison andWong is particularly attractive from a mathematical per-spective. On the Hilbert space H : = H ( D ) whose elements consist of the holomorphic functions f ( z ) = ∑ n ∈ N c n z n on the unit disc D for which k f k : = ∑ n ∈ N | c n | is finite, they consider the bounded self-adjoint Toeplitz oper-ator Φ with symbol arg ( z ) , defined for functions f , g ∈ H bythe relation ( Φ f | g ) = π Z π − π θ f ( e i θ ) g ( e i θ ) d θ . (1)Here we identify the function f ( z ) = ∑ n ∈ N c n z n with theFourier series f ( e i θ ) = ∑ n ∈ N c n e in θ and similarly for g . Thechoice of Φ as the quantum phase operator has been criticallyevaluated on physical grounds by several authors .The number operator is the unbounded self-adjoint opera-tor N in H given by N f ( z ) = z f ′ ( z ) on its maximal domain D ( N ) = n f = ∑ n ∈ N c n e n ∈ H : ∑ n ∈ N n | c n | < ∞ o . The spectrum of N is given by σ ( N ) = N : = { , , , . . . } and Ne n = ne n , n ∈ N , where the functions e n ( z ) : = z n , n ∈ N , form an orthonormal basis of eigenvectors in H .he Garrison–Wong quantum phase operator revisited 2In what follows we write [ Φ , N ] : = Φ N − N Φ for the com-mutator of Φ and N , which we view as an operator defined onits maximal domain D ([ Φ , N ]) : = D ( Φ N ) ∩ D ( N Φ ) = D ( N ) ∩ D ( N Φ ) , where D ( Φ N ) : = { f ∈ D ( N ) : N f ∈ D ( Φ ) = H } = D ( N ) , D ( N Φ ) : = { f ∈ D ( Φ ) = H : Φ f ∈ D ( N ) } . III. MAIN RESULT
It was shown by Garrison and Wong that the Heisenbergcommutation relation [ Φ , N ] f : = ( Φ N − N Φ ) f = i f holds for all functions f in a suitable subspace Y , introducedin Lemma 5, which is dense in H and contained in the domainof the commutator [ Φ , N ] . This fact, which we take for grantedfor the moment, self-improves as follows. Proposition 1
For all f ∈ D ([ Φ , N ]) one has [ Φ , N ] f = i f . Proof.
Let us denote by A and B the operator [ Φ , N ] with do-mains D ( A ) = Y and D ( B ) = D ([ Φ , N ]) . Then both A and B are densely defined and we have A ⊆ B . By Lemma 4, A issimply the restriction of the bounded operator iI to Y . Thisoperator is closable and since Y is dense its closure equals A = iI with domain D ( A ) = H .The self-adjointness of Φ and N immediately implies that ( i [ Φ , N ] f | g ) = i (( N f | Φ g ) − ( Φ f | Ng )) = ( f | i [ Φ , N ] g ) for all f , g ∈ D ([ Φ , N ]) . This means that iB is symmetric. Inparticular, iB (and hence B ) is closable, a closed extensionbeing given by its adjoints.It now follows that iI = A ⊆ B and therefore we must have D ( B ) = H . As a result, B = A = iI , and the asserted resultfollows. (cid:3) In the terminology introduces earlier, the proposition saysthat ( Φ , N ) is a Heisenberg pair. That it is not a Weyl paircan be seen by checking against the conditions of the Rellich–Dixmier theorem (as in Ref. ) or by noting that the Stone–von Neumann uniqueness theorem (see Ref. (Chapter 14))implies that both operators in a Weyl pair must be unbounded. Remark 2
We could generalise the definition of a Heisenbergpair by insisting only that the commutation relation
ABh − BAh = ih hold for all h ∈ Y , where Y is some given densesubspace of H contained in D ( AB ) ∩ D ( BA ) . The above proofcan be repeated verbatim to show that this definition, whichis the one used in Ref. , is equivalent to the one given in theIntroduction. Remark 3
The following observation serves to justify our ap-proach of interpreting the commutator [ Φ , N ] in terms of itsmaximal domain: Let iN be the generator of a bounded C -group on a Banach space X. There does not exist a boundedlinear operator T on X with the following two properties: (i) for all x ∈ D ( N ) one has T x ∈ D ( N ) ; (ii) the identity T Nx − NT x = ix holds for all x ∈ D ( N ) . Indeed, this is an immediate consequence of the second partof Ref. (Theorem 3) to A = B = iN and C = I . In our settingwhere N is the number operator, the arguments in the preced-ing remark imply that the operator Φ fails property (i) for thefunction x = , the constant-one function.Let us now give a detailed derivation of the Garrison–Wongresult, filling in some domain issues along the way. We splitthe result into two lemmas, Lemmas 4 and 5. The startingpoint is the following explicit representation for Φ , which fol-lows readily from (1): ( Φ e m | e n ) = π Z π − π θ e i ( m − n ) θ d θ = − i ( − ) m − n m − n δ m = n . Since Ne n = ne n , this gives ( Ne m | Φ e n ) − ( Φ e m | Ne n )= − i (cid:16) ( − ) m − n mm − n − ( − ) m − n nm − n (cid:17) δ m = n = − i ( − ) m − n δ m = n . It follows that if f , g ∈ D ( N ) are finite sums of the form f = ∑ ℓ j = c j e j and g = ∑ ℓ j = d j e j , then ( N f | Φ g ) − ( Φ f | Ng )= − i ℓ ∑ j , k = ( − ) j − k c j d k δ j = k = i ( f | g ) − i ℓ ∑ j = c j d j − i ℓ ∑ j , k = ( − ) j + k c j d k δ j = k = i ( f | g ) − i (cid:0) ℓ ∑ j = ( − ) j c j (cid:17)(cid:16) ℓ ∑ k = ( − ) k d k (cid:17) . For arbitrary f = ∑ j ∈ N c j e j and g = ∑ j ∈ N d j e j in D ( N ) (withconvergence of the sums in H ) we consider the truncations f ℓ = ∑ ℓ j = c j e j and g ℓ = ∑ ℓ j = d j e j , which satisfy f ℓ , g ℓ ∈ D ( N ) and f ℓ → f and g ℓ → g in the graph norm of D ( N ) . In combi-nation with the boundedness of Φ this gives ( N f | Φ g ) − ( Φ f | Ng )= lim ℓ → ∞ (( N ℓ f | Φ g ℓ ) − ( Φ f ℓ | Ng ℓ ))= i ( f | g ) − i (cid:16) ∑ j ∈ N ( − ) j c j (cid:17)(cid:16) ∑ k ∈ N ( − ) k d k (cid:17) , where the limits in the last step exist by the absolute summa-bility ∑ j ∈ N | c j | (cid:16) ∑ j ∈ N ( j + ) (cid:17) / (cid:16) ∑ j ∈ N ( j + ) | c j | (cid:17) / using the Cauchy–Schwarz inequality. Both terms in the right-hand side product are finite, the second because we are assum-ing that f ∈ D ( N ) . In particular, if f = ∑ j ∈ N c j e j belongs tohe Garrison–Wong quantum phase operator revisited 3 D ( N ) , then the series defining f converges absolutely on D ,and therefore such functions extend continuously to D .The following lemma gives a necessary and sufficient con-dition for functions f ∈ D ( N ) to satisfy the Heisenberg com-mutation relation. It provides some details for Ref. (Eq.(4.8)) as well as a converse to it. Lemma 4
For a function f = ∑ j ∈ N c j e j in D ( N ) the followingassertions are equivalent: (1) f ∈ D ([ Φ , N ]) and [ Φ , N ] f = i f ; (2) ∑ j ∈ N ( − ) j c j = ; (3) f ( − ) = .Proof. The equivalence (2) ⇔ (3) is clear by the preceding ob-servations.(1) ⇒ (2): In the converse direction, if f ∈ D ( N ) belongs to D ([ Φ , N ]) and [ Φ , N ] f = i f , then the above computation gives i ( f | f ) = ([ Φ , N ] f | f ) = i ( f | f ) − i (cid:12)(cid:12)(cid:12) ∑ j ∈ N ( − ) j c j (cid:12)(cid:12)(cid:12) and therefore ∑ j ∈ N ( − ) j c j = . (2) ⇒ (1): Let f , g ∈ D ( N ) and suppose that f = ∑ j ∈ N c j e j with ∑ j ∈ N ( − ) j c j = . The above computation then gives ( N f | Φ g ) − ( Φ f | Ng ) = i ( f | g ) and therefore | ( Φ f | Ng ) | ( k Φ N f k + k f k ) k g k . This bound shows that Φ f ∈ D ( N ⋆ ) = D ( N ) , which subse-quently gives f ∈ D ([ Φ , N ]) and ([ Φ , N ] f | g ) = ( Φ N f | g ) − ( N Φ f | g ) = i ( f | g ) . This being true for all g in the dense subspace D ( N ) , it followsthat [ Φ , N ] f = i f . (cid:3) These results imply the following curious cancellation re-sult:
If a sequence of complex scalars ( c n ) n ∈ N satisfies (i) ∑ n ∈ N n | c n | < ∞ ; (ii) ∑ n ∈ N n (cid:12)(cid:12)(cid:12) ∑ m ∈ N m = n ( − ) m − n m − n c m (cid:12)(cid:12)(cid:12) < ∞ ,then ∑ n ∈ N ( − ) n c n = . To see this, note that by (2), for functions f = ∑ n ∈ N c n e n wehave Φ f ( z ) = ∑ m ∈ N c m Φ e m ( z ) = − i ∑ n ∈ N (cid:16) ∑ m ∈ N n = m ( − ) m − n m − n c n (cid:17) z n after changing the order of summation. Thus (i) and (ii) saythat f ∈ D ( N ) and f ∈ D ( N Φ ) , respectively, so together theysay that f ∈ D ([ Φ , N ]) . For such functions, Proposition 1asserts that the Heisenberg commutation relation holds, andtherefore the stated conclusion holds by virtue of Lemma 4.The next lemma from Ref. implies that D ([ Φ , N ]) is densein H . The simple proof is included for the sake of complete-ness. Lemma 5
The subspace Y of H consisting of all functions f ∈ D ( N ) satisfying the equivalent conditions of Lemma 4 is densein H.Proof. By the lemma 4, for all integers k > f k : = e + ∑ k − j = k e j + belongs to Y . Moreover we have k e − f k k = / k . As a result, e belongs to the closure Y of Y in H . Again by the lemma, for all n ∈ N we have e n + e n + ∈ Y .This implies that e n ∈ Y for all n ∈ N , and therefore Y is densein H . (cid:3) It follows from Lemma 5, D ([ Φ , N ]) is dense in H . In thelight of this, the following negative result is perhaps somewhatsurprising. Proposition 6
The domain D ([ Φ , N ]) is not dense in D ( N ) with respect to the graph norm of the latter.Proof. We begin by observing that Φ e m ( z ) = − i ∑ n ∈ N n = m ( − ) m − n m − n z n . (2)By taking m =
0, for e = this gives Φ ( z ) = i ∞ ∑ n = ( − ) n n z n = − i log ( + z ) . (3)For all g ∈ D ([ Φ , N ]) we have, using that N ⋆ = N = ( | [ Φ , N ] g ) = ( | Φ Ng ) = ( Φ | Ng ) . By the definition of adjoint operators, we have ∈ D ([ Φ , N ] ⋆ ) if and only if there exists a constant C such that for all g ∈ D [ Φ , N ]) we can estimate | ( | [ Φ , N ] g ) | C || g || . If that is thecase, we also obtain that | ( Φ | Ng ) | C || g || , g ∈ D [ Φ , N ]) . (4)Suppose now, for a contradiction, that D ([ Φ , N ]) is dense in D ( N ) with respect to the graph norm. Then, by density, (4)implies the stronger statement | ( Φ | Ng ) | C || g || , g ∈ D ( N ) . But this is equivalent to asserting that Φ ∈ D ( N ⋆ ) = D ( N ) .But in that case Φ ( z ) = − i log ( + z ) (cf. (3)) would extendcontinuously to D , which is not the case.Since D ([ Φ , N ]) is dense in H the adjoint operator [ Φ , N ] ⋆ is well defined, and the preceding argument proves that ifhe Garrison–Wong quantum phase operator revisited 4 D ([ Φ , N ]) is dense in D ( N ) , then D ([ Φ , N ] ⋆ ) . But thenwe arrive at the contradiction D ([ Φ , N ] ⋆ ) = D ([ Φ , N ] ⋆ ) = D (( iI ) ⋆ ) = H , using Ref. (Theorem 1.8) to justify the first equality. (cid:3) Let V be the contraction on H defined by the left shift Ve : = , Ve n : = e n − , n > . Identifying the functions e n with elements of L ( T ) , the two-sided left shift on L ( T ) is a unitary extension of V and istherefore given by a unique projection-valued measure P on T . Compressing P to H produces a positive operator-valuedmeasure (POVM) Q on T such that V k = Z T λ k d Q ( λ ) , k = , , . . . , and this property uniquely characterises Q as a POVM. Forthe details the reader is referred to Ref. ; see also .As implicitly observed on page 87 of Ref. , the Garrison–Wong operator Φ can be characterised in terms of Q as fol-lows. Proposition 7 Φ = Z T arg ( λ ) d Q ( λ ) . The rigorous interpretation of the integral on the right-handside is as follows. Denoting by B b ( T ) the Banach space ofall bounded Borel measurable functions on T , one uses theboundedness of the Borel calculus of the projection-valuedmeasure P associated with V to obtain that there exists aunique linear mapping Ψ : B b ( T ) → L ( H ) , the space ofbounded operators on H , satisfying Ψ ( B ) = Q B , B ⊆ T Borel , and k Ψ ( f ) k k f k ∞ , f ∈ B b ( T ) . It further satisfies Ψ ( f ) ⋆ = Ψ ( f ) , f ∈ B b ( T ) . We now define Z T f d Q : = Ψ ( f ) , f ∈ B b ( T ) . For all f , f ∈ A ( T ) , the uniform closure of the trigonometricpolynomials in C ( T ) , we have Ψ ( f ) Ψ ( f ) = Ψ ( f f ) , but this property does not extend to general functions f , f ∈ B b ( T ) . Proof of Proposition 7.
We equipartition T = ( − π , π ] into k subintervals of length 2 π / k by setting I j : = ( − π + π ( j − ) / k , − π + π j / k ] for j = , . . . , k . Then, by the continuity of Ψ : f R T f d Q , Z T arg ( λ ) d Q ( λ ) = lim k → ∞ Z T arg k ( λ ) d Q ( λ )= lim k → ∞ k ∑ j = π jk Q ( I j ) , where arg k ( λ ) : = ∑ Nj = π jk I j ( λ ) . To compute Q ( I j ) we usethat Q is the compression to H of the projection-valued mea-sure P associated with the two-sided shift U on L ( T ) . Thelatter is given by P ( I n ) f = I n f for f ∈ L ( T ) . Accordingly,if we denote the inclusion mapping H L ( T ) by J , then forall f ∈ H we have Q ( I j ) f = J ⋆ P ( I j ) J f = J ⋆ I j f . It follows that Z T arg ( λ ) d Q ( λ ) = lim k → ∞ k ∑ j = π jk J ⋆ I j , identifying I j with the multiplication operator f I j f from H to L ( T ) . On the other hand, by the definition of the opera-tor Φ , ( Φ f | g ) = Z T arg ( λ ) f ( λ ) g ( λ ) d λ , f , g ∈ H , we have Φ f = J ⋆ ( arg ( · ) f ( · ))= lim k → ∞ J ⋆ ( arg k ( · ) f ( · )) = lim k → ∞ k ∑ j = π jk J ⋆ ( I j f ) . This completes the proof. (cid:3)
Remark 8
The arguments used in the proof imply that Q B = B has measure 0. It follows that the inte-gral R T f d Q is well defined for functions f ∈ L ∞ ( T ) . Withessentially the same proof as above one shows that for any φ ∈ L ∞ ( T ) the bounded Toeplitz operator T φ on H with sym-bol φ is given by T φ = Z T f d Q . As observed in Ref. , the POVM Q obeys the following“covariance property”. For the reader’s convenience we in-clude the simple proof. Proposition 9
For all t ∈ R and Borel sets B ⊆ T we havee itN Q ( B ) e − itN = Q ( e it B ) , where ( e itN ) t > is the unitary C -group on H generated by iNand e it B = { e it λ : λ ∈ B } is the rotation of B over angle t. he Garrison–Wong quantum phase operator revisited 5 Proof.
The properties of the projection-valued measure P usedin the proof of Proposition 7 imply that for the trigonometricfunctions e k , k ∈ N , we have ( Q ( B ) e − itN e n | e m ) = ( P ( B ) Je − itN e n | Je m ) = e − int ( B e n | e m ) while at the same time ( e − itN Q ( e it B ) e n | e m )= ( P ( e it B ) Je n | Je itN e m ) = e − itm ( e it B e n | e m )= e − itm Z e it B λ n − m d λ = e − itm Z B ( e − it µ ) n − m d µ = e − itn Z e it B µ n − m d µ = e − int ( B e n | e m ) . Since the span of the trigonometric functions is dense in H ,this completes the proof. (cid:3) This contrasts with the failure of the
Weyl commutation re-lations e itN e is Φ e − itN = e − ist e is Φ , s , t ∈ R . (5)This failure is usually demonstrated by showing that (5) wouldimply the identity a = e − i Φ N / , where a is the annihilationoperator associated with N (so that a ⋆ a = N ); this identity issubsequently shown to be impossible if at the same time Φ isto be self-adjoint (see Ref. ).Here, by elementary methods, we will give a direct proofof the more precise result that the Weyl relation (5) fails forevery fixed s = Proposition 10
Let T be an arbitrary bounded operator onH. If s ∈ R is such that for all t ∈ R one hase itN e isT e − itN = e − ist e isT , (6) then s = . The same conclusion holds if we assume that T isa (possibly unbounded) self-adjoint operator on H.Proof. Suppose that s ∈ R is such that (6) holds for all t ∈ R .Choose n , m ∈ N so that ( e isT e n | e m ) =
0. Applying (6) to e n and taking inner products with e m , we obtain e it ( m − n ) ( e isT e n | e m ) = e − ist ( e isT e n | e m ) . This can hold for all t ∈ R only if e it ( m − n ) = e − ist for all t ∈ R ,forcing s = n − m ∈ Z .Suppose next that s = k ∈ Z is such that (6) holds for all t ∈ R . If k >
1, the above argument shows that we must have ( e isT e n | e m ) = n − m = k , which implies that e isT e n is amultiple of e n − k if n > k and e isT e n = n k −
1. Givena fixed n ∈ N , it follows that e imsT e n = m ∈ N . But this is impossible as it would lead to thecontradiction e n = e − imsT e imsT e n = . If k −
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