Exact Quantum Correlations of Conjugate Variables From Joint Quadrature Measurements
aa r X i v : . [ qu a n t - ph ] M a y Exact Quantum Correlations of Conjugate Variables From Joint QuadratureMeasurements
S. M. Roy HBCSE,Tata Institute of Fundamental Research, Mumbai (Dated: July 23, 2018)We demonstrate that for two canonically conjugate operators ˆ q, ˆ p ,the global correlation h ˆ q ˆ p +ˆ p ˆ q i − h ˆ q ih ˆ p i , and the local correlations h ˆ q i ( p ) − h ˆ q i and h ˆ p i ( q ) − h ˆ p i can be measured exactly by VonNeumann-Arthurs-Kelly joint quadrature measurements . These correlations provide a sensitiveexperimental test of quantum phase space probabilities quite distinct from the probability densitiesof q, p . E.g. for EPR states, and entangled generalized coherent states, phase space probabilitieswhich reproduce the correct position and momentum probability densities have to be modified toreproduce these correlations as well. PACS numbers: 03.65.Ud, 03.67.Ac,03.65.Yz,42.50.-pKeywords: Position-momentum correlations, Von Neumann-Arthurs-Kelly joint quadrature measurements,EPR states, generalized coherent states.
Von Neumann-Arthurs-Kelly Joint Measure-ments of Conjugate Variables in Quantum Me-chanics . Correlations between conjugate observables,being rather different from Bell type correlations [1]among commuting observables, is a largely unexploredarea with possible fundamental importance. We presenta method for exact measurement of local and global cor-relations between conjugate observables in quantum me-chanics. We use the Von Neumann-Arthurs-Kelly JointMeasurements, realizable as heterodyne measurementsin quantum optics. As a first application these corre-lations are used to experimentally test proposed phasespace probabilities and to constrain costruction of suchprobabilities so that they reproduce not only the quan-tum probability densities of conjugate observables butalso their correlation.Von Neumann [2] not only proposed a model Hamil-tonian for accurate measurement of a single observable ˆ q but also noted that joint approximate measurements ofcanonically conjugate observables ˆ q, ˆ p were possible withaccuracy limited by the uncertainty principle . Arthursand Kelly [3] generalized the Von Neumann Hamilto-nian to realize such joint measurements and deduce ameasurement uncertainty relation which was soon provedwith full generality by Arthurs and Goodman [4]. Braun-stein, Caves and Milburn [5] analyzed the optimum ini-tial conditions and deduced that the optimum Arthurs-Kelly joint distribution for ˆ q, ˆ p is just the Husimi Q func-tion [6]. Stenholm [7] proposed a realizable quantumoptics Hamiltonian for such measurements. The vari-ous facets of the uncertainty relations so revealed havebeen reviewed by Busch, Heinonen and Lahti [8]. Themost satisfying thing is that Heterodyne measurementsin quantum optics [9] can practically realize joint mea-surements of any pair of rotated conjugate quadratures,( a exp ( − iθ ) / √ h.c., − ia exp ( − iθ ) / √ h.c. ) whereˆ q, ˆ p are given in terms of the photon annihilation andcreation operators a, a † ,ˆ q = ( a + a † ) / √ , ˆ p = ( a − a † ) / ( i √ . Arthurs-Kelly Results . Their idea is that the sys-tem (with position and momentum operators ˆ q, ˆ p ) inter-acts with an apparatus which has two commuting ob-servables x , x and approximate values of system posi-tion and momentum are extracted from accurate jointobservation of x , x . The Von Neumann-Arthurs-Kellyinteraction during the time interval ( t , t + T ) is, H = K (ˆ q ˆ p + ˆ p ˆ p ) , (1)where ˆ p , ˆ p are canonical conjugates of x , x respec-tively, the coupling K is large, and T is small, with KT = 1. During interaction time, H is so strong thatthe free Hamiltonians of the system and apparatus areneglected. Arthurs and Kelly start with the system-apparatus initial state, ψ ( q, x , x , t ) = φ ( q ) χ ( x ) χ ( x ) (2)where, φ ( q ) is the system state and the apparatus stateis given by, χ ( x ) = π − / b − / exp( − x / (2 b )) , (3) χ ( x ) = π − / (2 b ) / exp( − b x ) , (4)and b/ √ x in the initial apparatusstate. They solve the Schr¨ o dinger equation exactly andobtain the final joint probability density of the apparatusvariables to be just the Husimi function [6], P ( x , x ) = |h φ b,x ,x | φ i| / (2 π ) , (5)where φ b,x ,x ( q ) = (2 πb ) − / exp( iqx − ( x − q ) / (4 b )) (6)is a minimum uncertainty system state centred at q = x , p = x . Note that for any value of b , h x i = h ˆ q i , and h x i = h ˆ p i , but the dispersions in x , x are larger thanthose for the corresponding system variables q, p ,(∆ x ) = (∆ q ) + b , (∆ x ) = (∆ p ) + 14 b ; (7)they obey the “measurement or noise ” uncertainty rela-tion , (units ~ = 1 ), ∆ x ∆ x ≥ b . Here the minimum uncertainty istwice the usual “ preparation uncertainty”. Arthurs andGoodman [4] gave a beautiful proof of this fundamentaluncertainty relation, independent of any particular choiceof the Hamiltonian. Further,for the x distribution toapproximate q distribution closely, we need b ≪ ∆ q ; forthe x distribution to approximate p distribution closely,we need b ≫ (∆ p ) − . P ( x ) ≡ Z P ( x , x ) dx = (2 π ) − / b − Z dq | φ ( q ) | exp ( − ( x − q ) / (2 b )) → b → | φ ( x ) | , (9) P ( x ) ≡ Z P ( x , x ) dx = (2 π ) / b Z dp | ˜ φ ( p ) | exp ( − ( x − p ) (2 b )) → b →∞ | φ ( x ) | , (10)where ˜ φ ( p ) denotes the Fourier transform of φ ( q ). Exact Measurement of Quantum CorrelationsBetween Conjugate Variables.
The above equationsshow that the exact position and momentum probabilitydensities of the system are recovered by the Arthurs-Kellymeasurement in the limits b → b → ∞ respec-tively,i.e. in two experiments with very different initialapparatus states. It is a pleasant surprise that the jointmeasurement can nevertheless give local and global cor-relations between ˆ q ,and ˆ p exactly. We define, h ˆ p i ( q ) ≡ h Λ( q )ˆ p + ˆ p Λ( q ) i h Λ( q ) i ; h ˆ q i ( p ) ≡ h Λ( p )ˆ q + ˆ q Λ( p ) i h Λ( p ) i , (11)where h A i denotes the quantum expectation value ofa self-adjoint operator A , and the projection operatorsΛ( q ) , Λ( p ) are defined by,Λ( q ) = | q ih q | , Λ( p ) = | p ih p | . (12)For a pure state | φ i we have the explicit expressions, h ˆ p i ( q ) = Re ( φ ∗ ( q )( − i ) ∂φ ( q ) /∂q ) | φ ( q ) | , (13) h ˆ q i ( p ) = Re ( ˜ φ ∗ ( p )( i ) ∂ ˜ φ ( p ) /∂p ) | ˜ φ ( p ) | . (14)We shall see that the local correlations h ˆ p i ( q ) − h ˆ p i ,and h ˆ q i ( p ) − h ˆ q i can be measured exactly for arbitrary q and p respectively for appropriate values of b . The globalcorrelation h ˆ q ˆ p + ˆ p ˆ q i− h ˆ q ih ˆ p i is in fact exactly measurablefor any value of b . For the Arthurs-Kelly measurement we define as for aclassical distribution, h x i A − K ( x ) ≡ Z x P ( x , x ) dx /P ( x ) , (15) h x i A − K ( x ) ≡ Z x P ( x , x ) dx /P ( x ) , (16) h x x i A − K ≡ Z x x P ( x , x ) dx dx . (17)Substituting the value of P ( x , x ), and doing the inte-gral over x we obtain, Z x P ( x , x ) dx = ( b √ π ) − Z dqdq ′ φ ( q ) φ ∗ ( q ′ )exp ( − ( x − q ) + ( x − q ′ ) b ) i ∂δ ( q − q ′ ) ∂q = Re Z dqb √ π exp ( − ( x − q ) b ) φ ∗ ( q )( − i ) ∂φ ( q ) ∂q (18)where δ ( q − q ′ ) is the Dirac delta function. Similarly, weobtain, Z x P ( x , x ) dx = b p /πRe Z dp exp ( − b ( x − p ) ) ˜ φ ∗ ( p ) i ∂ ˜ φ ( p ) ∂p . (19)Taking the limits of b going to 0 and ∞ yield respectively, h x i A − K ( x ) → b → h ˆ p i ( q = x ) , (20) h x i A − K ( x ) → b →∞ h ˆ q i ( p = x ) . (21)Thus we have proved that the quantum position proba-bility density and the local correlation h ˆ p i ( q ) − h ˆ p i canbe measured exactly with the initial condition b → h ˆ q i ( p ) − h ˆ q i can be measured exactly withthe very different initial condition b → ∞ . A similarcalculation shows that for any value of b , h x x i A − K = h ˆ q ˆ p + ˆ p ˆ q i , (22)the global correlation is exactly measured in the Arthurs-Kelly (A-K) measurement. Thus, the A-K measurementswith b → b → ∞ equip us with exact probabilitydensities of position and momentum as well as their exactlocal and global correlations. Experimental test of phase space probabilitiesby correlation measurements . We demonstrate thatexact measurement of the correlations is a valuable toolto discriminate between various phase space probabil-ity densities which may give exactly the same posi-tion and momentum probability densities. The tremen-dous progress initiated by research on Bell inequalitiesand quantum contextuality [1], and their extension tophase space [12] teaches us that in 2 N dimensional phasespace, a positive density can have a maximum of N + 1marginals reproducing quantum probability densities forarbitrary states. (E.g. for N = 2, probability densitiesof ( q , q ) , ( p , q ) , ( p , p ) can be reproduced.) Of coursewe know that all marginals of Wigner’s quasi-probabilitydistribution [13] agree with the corresponding quantumprobablities for the state | φ i .But we shall only cosiderpositive densities as candidates for a probability inter-pretation. De Broglie and Bohm [10] proposed a posi-tive phase space density which reproduces the quantumposition probability density but fails to agree with thequantum momentum probability density [11]. The mostgeneral positive densities with two marginals reproduc-ing quantum position and momentum probabilities [14],and with N + 1 marginals reproducing the correspondingquantum probabilites are also known [12]. Roy and Singh[15] built a new causal quantum mechanics symmetric in q, p in which the phase space density obeys positivityand the marginal conditions on momentum and positionprobabilities . For example for N = 1, the two densities ρ ǫ ( q, p ) = | φ ( q ) | | ˜ φ ( p ) | δ (cid:18)Z p −∞ dp ′ | ˜ φ ( p ′ ) | − Z ǫq −∞ dq ′ | φ ( ǫq ′ , t ) | (cid:19) , (23)where ǫ = ± Z ρ ǫ ( q, p ) dp = | φ ( q ) | ; Z ρ ǫ ( q, p ) dq = | ˜ φ ( p ) | . (24)To demonstrate the discriminatory power of the quantumcorrelation measurements we shall use them in severalconcrete examples to test these two phase space densities( for ǫ = ± | φ ( q ) | | ˜ φ ( p ) | , all of which reproduce quantum q, p probability densities.(i) Free particle spreading wave packets for non-relativistic particle of mass m . At the time t of theA-K measurement, let˜ φ ( p ) = ( πα ) − / exp [ − ( p − β ) α − it p m ] , (∆ p ) = α , (∆ q ) = 1 + ( αt /m ) α . (25)The Roy-Singh q, p symmetric causal quantum mechanicsgives, for ǫ = ± h p i ( q ) ± − h ˆ p i = ± ∆ p ∆ q ( q − βt /m ) , (26)whereas the Arthurs-Kelly correlation is, h x i ( x ) A − K − h ˆ p i = p (∆ q ∆ p ) − / q ) + b ( x − βt /m ) , (27) and it’s limit b → ǫ = − ǫ = 1 Roy-Singh causal density (seefigure) approaches unity for b/ ∆ q ≪ q ∆ p ≫ / D q D p = D q D p = D q D p = D q D p = b D q0.800.850.900.951.00Corr . AKCorr . RS FIG. 1: For the free particle expanding Gaussian wave packet,the ratio of the correlation h x i ( x ) −h x i in the Arthurs-Kellymeasurement to h p i ( q ) − h p i in the ǫ = 1 Roy-Singh causalphase space density is plotted for various values of b/ ∆ q andof ∆ q ∆ p .The causal correlation agrees with the quantum cor-relation (i.e. the b → ǫ = 1 and ǫ = − relations at given p , the Roy-Singh causal quantum me-chanics gives, for ǫ = ± h q i ( p ) ± − h ˆ q i = ± ∆ q ∆ p ( p − β ) , (28)which agrees only for ǫ = 1 and only for large ∆ q ∆ p withthe quantum correlation which is the b → ∞ limit of theArthurs-Kelly correlation , h x i ( x ) A − K − h ˆ q i = p (∆ q ∆ p ) − / p ) + (4 b ) − ( x − β ) . (29)For the global correlation, the Roy-Singh causal quantummechanics with ǫ = ± gives, h qp i ± − h q ih p i = ± q ∆ p, (30)of which only the ǫ = 1 correlation agrees with the quan-tum correlation, h ˆ q ˆ p + ˆ p ˆ q i − h ˆ q ih ˆ p i = p (2∆ q ∆ p ) − , (31)provided that 2∆ q ∆ p ≫ n -th excited states of the oscillator of frequency ω , φ n,α ( q, t ) = h q − ¯ q | n i exp ( − iωt ( n + 12 ) + i ¯ p ( q − ¯ p , (32)where α = A exp ( − i ( ωt + θ )), ¯ q ≡ Reα, ¯ p ≡ Imα , and
A, θ are real constants. Here the Roy-Singh causal quan-tum mechanics with ǫ = ± h p i ( q ) ± − h ˆ p i = ± ( q − ¯ q ) , h q i ( p ) ± − h ˆ q i = ± ( p − ¯ p ) , (33) h qp i ± − h q ih p i = ± (2 n + 1) . (34)In contrast quantum mechanics gives zero for the abovethree correlations and thus agrees with the correlationlessphase space density. Phase space probabilities reproducing quantumposition and momentum probabilities and corre-lations exactly . Surprisingly, in both the examplesconsidered above, convex combinations of the Roy-Singhphase space densities with ǫ = ± ρ ( q, p ) C = λ + ρ + ( q, p ) + λ − ρ − ( q, p ) , (35)0 ≤ λ ± ≤ , λ + + λ − = 1 . (36)where the state dependent constants λ ± are chosen to re-produce the quantum global correlation h ˆ q ˆ p + ˆ p ˆ q i− h ˆ q ih ˆ p i yield local correlations also equal to the correspondingquantum local correlations. Explicitly, in cases (i) and(ii) of Gaussian packets and generalized coherent states,( i ) λ ± = 12 ± p − (2∆ q ∆ p ) − , ( ii ) λ ± = 1 / . (37) EPR states . A normalizable version of the originalEPR state [17] | q − q = q i| p + p = P i of two particlesis, φ ( q − q , p + p ) = φ ( q − q ) ˜ φ ( p + p ) , (38)where the individual Gaussian wave functions, φ ( q − q ) = ( πα ) − / exp ( − ( q − q − q ) α ) , (39)˜ φ ( p + p ) = ( πα ) − / exp ( − ( p + p − P ) α ) (40)are sharply peaked at q − q − q = 0 and p + p − P =0 respectively in the limits α → , α →
0. We nowconstruct the phase space density, ρ C ( q − q , ( p − p )2 ) ρ C ( ( q + q )2 , ( p + p )) , (41) with the two factors ρ C and ρ C made to fit respec-tively the ( q − q , ( p − p ) /
2) and ( q + q ) / , ( p + p ))correlations in the Gaussian states φ , φ using convexcombinations of the Roy-Singh phase space densities de-scribed above. This phase space density reproduces ex-actly, the above quantum correlations as well as quantumjoint probability densities of the four commuting pairs ofvariables q − q , ( q + q ) / q − q , p + p ; ( q + q ) / , p − p ; p + p , ( p − p ) / Entangled generalized coherent states .For twomodes of light with the same frequency an exact solutionof the Schr¨ o dinger equation at time t is the entangledgeneralized coherent state , φ m,α (( q + q ) / √ , t ) φ n,β (( q − q ) / √ , t ) , (42)where m, n are integers, α, β complex constants and thefactors φ m,α , φ n,β are generalized coherent states definedbefore . A phase space probability reproducing the rel-evant quantum correlations and probabilities exactly is, ρ mC ( ( q + q ) √ , ( p + p ) √ ρ nC ( ( q − q ) √ , ( p − p ) √ ρ mC , ρ nC are arithmetic means of the ǫ = ± Roy-Singh phase space densities for the states φ m,α , φ n,β re-spectively. Future directions . The central point is the exactmeasurability of local and global correlations betweenconjugate observables. Actual joint quadrature measure-ments to test their correlations will be very interesting.The Arthurs-Kelly joint measurements and hence thepossibilities of exact measurements of quantum correla-tions between conjugate variables can be generalized to2 N -dimensional phase space. An interesting question istriggered by the success in exact reproduction of cho-sen quantum correlations and probabilities in the specialstates (including entangled states) considered. Can weconstruct phase space probabilities reproducing quantumposition and momentum probabilities and their correla-tions exactly for every quantum state ?I wish to thank Arunabha S. Roy for the crucial fig-ure on experimental test of correlations , Virendra Singhfor many intensive discussions and the Indian NationalScience Academy for financial support. [1] J.S. Bell, Physics
1, 195 (1964); A.M. Gleason,
J. Math.& Mech.
6, 885 (1957); S. Kochen and E.P. Specker,
J.Math. & Mech.
17, 59 (1967).[2] J. Von Neumann,
Math. Foundations of Quantum Me-chanics , Princeton University Press (1955).[3] E. Arthurs and J. L. Kelly,Jr.,
Bell System Tech. J.
Phys. Rev. Lett.
Phys.Rev.
A43,1153 (1991).[6] K. Husimi,
Proc. Phys. Math. Soc.Japan , 22,264 (1940).[7] S. Stenholm,
Ann. Phys.
Phys. Reports
IEEE Trans. Inf. Theory ,24,657 (1978);25,179 (1979); 26,78 (1980).[10] L. de Broglie, “Nonlinear Wave Mechanics, A Causal In-terpretation”, (Elsevier 1960); D. Bohm,
Phys. Rev.
Phys. Rev.
96, 208 (1954);P.R. Holland,
The Quantum Theory of Mo-tion , Cambridge University Press (1993) and
Foundationsof Physics
28, 881 (1998).[11] T. Takabayasi,
Prog. Theor. Phys.
8, 143 (1952).[12] G. Auberson, G. Mahoux, S. M. Roy and V. Singh,
Phys.Lett.
A300, 327 (2002);
Journ. Math. Phys.
44, 2729-2747(2003), and 45,4832-4854 (2004); A. Martin and S.M.Roy,
Phys. Lett.
B350, 66 (1995) ; S. M. Roy,
Int. J.Mod. Phys.
Phys. Rev.
40, 749 (1932).[14] L. Cohen and Y.I. Zaparovanny, J. Math. Phys. 21, 794(1980); L. Cohen, ibid 25, 2402 (1984).[15] S.M. Roy and V. Singh
Mod. Phys. Lett.
A10, 709 (1995);S.M. Roy and V. Singh,
Phys. Lett.
A255, 201 (1999).[16] S. M. Roy and V. Singh,
Phys. Rev.
D25,3413 (1982); I.R. Senitzky,
Phys. Rev.