aa r X i v : . [ qu a n t - ph ] A p r A unique quasi-probability for projective yes-no measurements
Lars M. Johansen
Department of Technology, Buskerud University College, N-3601 Kongsberg, Norway ∗ (Dated: October 29, 2018)From an analysis of projective measurements, it is shown that the Wigner rule is the uniqueoperational quasi-probability for the post-measurement state. A unique pre-measurement quasi-probability is derived from a principle of invariance of measurement disturbance under orthogonalprojector complementation. Physical arguments for this principle are given. The informationallycomplete complex extension of the quasi-probability is also derived. Nonclassicality of this quasi-probability is due to measurement disturbance. The same quasi-probability follows from weakmeasurements. PACS numbers: 03.65.Ta, 03.67.-a,03.65.Wj
In its most general form, classical physics deals withprobability distributions over a Boolean logic of events.Quantum mechanics is also a probabilistic theory. Everystatement the theory makes about observations is proba-bilistic. However, it is not a classical probabilistic theory[1, 2]. Successive measurements give probabilities depen-dent on measurement order. This is due to measurementdisturbance. Measurement disturbance in itself is not anonclassical phenomenon. But in quantum mechanics,measurement disturbance is of an entirely different na-ture than in classical physics. It is not due to an imperfec-tion of the instrument, but due to the invasive nature ofquantum measurements. Classically, if one has sufficientinformation about how a measurement disturbs a system,it may be possible to compensate or subtract this distur-bance. But is compensation of measurement disturbancepossible also in quantum mechanics? Obviously, it re-quires a closer characterization of the quantum mechani-cal measurement disturbance. However, the prospects ofsuch compensation or subtraction might seem dim [3].Wigner found a classical-like representation of quan-tum mechanics in terms of a quasi-probability over phasespace [4]. The essential nonclassical aspect of the Wignerdistribution is that it may take negative values. TheWigner distribution, together with a number of otherquasi-probability distributions, have become useful toolsfor distinguishing quantum and classical effects. In finitedimensional Hilbert space, generalizations of the Wignerdistribution have also been proposed (see [5] and refer-ences therein). The Wigner distribution may be derivedfrom a set of classical conditions. A basic condition isthat unitary time evolution should correspond to classicalpropagation in phase space [2]. Whereas unitary evolu-tion is associated with a closed system that is not subjectto observation, measurements lead to non-unitary evolu-tion. In this Letter, we derive a unique extension of theclassical joint probability concept by analyzing projectivemeasurements. We shall proceed by imposing conditionsmotivated by physical arguments. This leads, in the end,to a complex quasi-probability that was originally dis-cussed by Dirac [6]. If negative “probabilities” is what it takes to makequantum mechanics look like a classical theory [7], itwould be of interest to find an operational explanation ofsuch a phenomenon. The Wigner distribution itself doesnot seem to give an answer. Although the Wigner dis-tribution may be reconstructed using a variety of meth-ods [8, 9], a direct physical interpretation of negativityof the Wigner distribution in terms of measurements ofthe phase space observables themselves has not been ob-tained. With the quasi-probability derived in this Let-ter, we find a direct physical interpretation of nonclassical(i.e. negative and imaginary) quasi-probabilities in termsof measurement disturbance.The discovery of weak values due to weak measure-ments [10] came as a surprise to many. The notion thata measurement somehow could give values outside theeigenvalue spectrum seemed heretic. However, within thequasi-probability formalism derived here, both projectiveand weak measurements appear as consistent manifesta-tions of the same underlying “reality”. In fact, in theformalism derived here weak values appear as a naturalconsequence of projective measurements.We will consider projective measurements [11] in aHilbert space of arbitrary, finite dimensions. Projectivemeasurements are ideal, in the sense that they are repeat-able while disturbing the system as little as possible. Anyideal measurement, quantum or classical, may be decom-posed into elementary yes-no measurements. In quantummechanics, such measurements are represented by pro-jectors, the quantum generalization of classical events.A projector α = α has eigenvalues 1 and 0, correspond-ing to outcomes yes or no, true or false. The probabilityof a yes in the projective measurement of this projectoron a system prepared in a state ρ is Tr ρα . The jointprobability for positive outcomes in successive projectivemeasurements of the two projectors α and β is given bythe Wigner rule Tr ραβα [12]. A measurement in the op-posite order gives the probability Tr ρβαβ . If the twoprojectors commute, both of these joint probabilities re-duce to the order independent joint probability Tr ραβ .If the two projectors do not commute, the expressionTr ραβ is in general complex. This does not make muchsense as a joint probability [6]. Or does it?We seek an operational quasi-probability F ( ρ, α, β ).Operational here means that it should be connected asclosely as possible to the classical definition of a jointprobability in terms of successive measurements. Theorder of the arguments refer to an initial preparation ρ ,a first measurement α and a second measurement β . Wealso introduce the orthogonal complements ˜ α = 1 − α and˜ β = 1 − β . In correspondence with classical probabilitytheory, the first condition on the quasi-probability is: Condition 1. F ( ρ, α, β ) + F ( ρ, α, ˜ β ) = Tr ρα, (1a) F ( ρ, α, β ) + F ( ρ, ˜ α, β ) = Tr ρβ. (1b)This, of course, implies that the distribution is normal-ized to unity.After a projective measurement of a projector α , theensemble corresponding to the eigenvalue 1 is representedby the state αρα/ Tr ρα [11]. The complete initial ensem-ble is represented after the measurement by the state [11]Λ α ( ρ ) = αρα + ˜ αρ ˜ α. (2)We shall refer to such measurements as nonselective,and we shall refer to the state transformation Λ α as theL¨uders map.We first find the post-measurement quasi-probability F [Λ α ( ρ ) , α, β ]. A state σ is undisturbed by a measure-ment Λ α if Λ α ( σ ) = σ . The state Λ α ( ρ ) is undisturbedby a measurement Λ α since Λ α [Λ α ( ρ )] = Λ α ( ρ ). A sub-sequent measurement of β can be made without it beingdisturbed by the preceding α measurement. The jointprobability obtained in a successive measurement of α and β on Λ α ( ρ ) will not be influenced by measurementdisturbance. We may therefore identify this joint prob-ability with F [Λ α ( ρ ) , α, β ]. The Wigner rule applied tothe state Λ α ( ρ ) is F [Λ α ( ρ ) , α, β ] = TrΛ α ( ρ ) αβα . Thissimplifies to: Condition 2. F [Λ α ( ρ ) , α, β ] = Tr ραβα. (3)It is easily verified that this quasi-probability satisfiesthe marginality conditions (1). This is due to the lack ofmeasurement disturbance. Of course, Eq. (3) refers to aparticular measurement order. As such, this is not a jointprobability in the full classical sense. It is reasonable toassume that a pre-measurement quasi-probability shouldnot depend on the order in which subsequent measure-ments are performed, so that F ( ρ, α, β ) = F ( ρ, β, α ) . (4)We may note that the distribution Tr ρα Tr ρβ satisfiesboth this symmetry relation and the marginality condi-tions (1). As such it is a joint probability which is even nonnegative. However, it does not fulfill the condition(3).We shall not impose the order symmetry (4) in thederivation of the quasi-probability F ( ρ, α, β ). However,we shall use it to argue for another symmetry condition.To this end, we introduce the “disturbance” or the changeof the quasi-probability F due to the L¨uders map Λ α ,∆ F ( ρ, α, β ) = F ( ρ, α, β ) − F [Λ α ( ρ ) , α, β ] . (5)We apply the symmetry condition (4) to the state Λ α ( ρ ), F [Λ α ( ρ ) , α, β ] = F [Λ α ( ρ ) , β, α ] . (6)By using Eqs. (3) and (5) this implies thatTr ραβα = TrΛ α ( ρ ) βαβ + ∆ F [Λ α ( ρ ) , β, α ] . (7)Likewise, by exchanging β with its orthogonal comple-ment ˜ β in the equation above, we find thatTr ρα ˜ βα = TrΛ α ( ρ ) ˜ βα ˜ β + ∆ F [Λ α ( ρ ) , ˜ β, α ] . (8)On comparing Eqs. (7) and (8), we find that∆ F [Λ α ( ρ ) , ˜ β, α ] = ∆ F [Λ α ( ρ ) , β, α ] . (9)Thus, we see that the change of the quasi-probability isinvariant under the exchange of the projector that is mea-sured first (the one causing “disturbance” to the other)and its orthogonal complement.We now assume that the symmetry (9) applies to anystate ρ . In reference to an opposite measurement orderof that in Eq. (9), the third condition on the quasi-probability is: Condition 3. ∆ F ( ρ, ˜ α, β ) = ∆ F ( ρ, α, β ) . (10)Next, we calculate the marginal of the change ∆ F over α . By using Eqs. (1b) and (5) we find that∆ F ( ρ, α, β ) + ∆ F ( ρ, ˜ α, β ) = 12 [Tr ρβ − TrΛ α ( ρ ) β ] . (11)By inserting (10) into (11) we find that∆ F ( ρ, α, β ) = 12 [Tr ρβ − TrΛ α ( ρ ) β ] . (12)Therefore we have [13] F ( ρ, α, β ) = Tr ραβα + 12 [Tr ρβ − TrΛ α ( ρ ) β ] . (13)This is the resulting pre-measurement quasi-probability.We see that it differs from the post-measurement jointprobability only if the probability for β is disturbed bythe preceding measurement of α . The only possibility forthe pre-measurement quasi-probability to become nega-tive is that the measurement disturbance is sufficientlylarge.By inserting (2), Eq. (13) may also be written in theform F ( ρ, α, β ) = 12 Tr [ ρ ( αβ + βα )] . (14)Here we can see that the quasi-probability indeed satisfiesthe condition (4) of order symmetry. So, order symmetry(4) follows from the assumption of disturbance symme-try (10). This quasi-probability has been discussed byvarious authors [14, 15].It follows from the analysis in Ref. [16] that the distri-bution (14) is bounded between − / ρ and the twoprojectors α and β correspond to socalled trine states[17].One of the main purposes with a quasi-probability dis-tribution is to provide an alternative representation ofquantum states. It has been shown that the distribution(14) defined over classical phase space determines thestate uniquely [18]. However, the informational contentof this distribution is not complete in general. For exam-ple, in two-dimensional Hilbert space, the most generaldensity matrix contains three real parameters. It may beshown that at least for a large class of observables, thedistribution (14) contains fewer parameters.In order to complete the information contained in (14),we may consider a complex extension of the distribution.A possible complex extension of (14) is rather obvious,since it is the real part of the complex expression Tr ραβ .This is in fact a quasi-probability in its own right, givingcorrect marginal probabilities. It was first explored inphase space by Kirkwood [19], and generalized to arbi-trary observables by Dirac [6]. However, Dirac could notfind the physical interpretation of this expression, and sohe did not pursue it further.Here, we shall derive the complex extension of (14) bysome further conditions. Thus, we seek a complex quasi-probability G ( ρ, α, β ) = F ( ρ, α, β ) + i I ( ρ, α, β ) . (15)where the real part F ( ρ, α, β ) is given by (14) and theimaginary part I ( ρ, α, β ) is to be determined.The argument leading to Eq. (3) still stands. Thisimplies Condition 4. I [Λ α ( ρ ) , α, β ] = 0 . (16)The complex quasi-probability must fulfill the samemarginality conditions as the real distribution. Other-wise, it would lead to complex expectation values of her-mitian observables. We must therefore impose the fol-lowing: Condition 5. I ( ρ, α, β ) + I ( ρ, α, ˜ β ) = 0 , (17a) I ( ρ, α, β ) + I ( ρ, ˜ α, β ) = 0 . (17b) Classically, the pre-measurement state and the post-measurement state is the same. That is why joint prob-abilities may be defined directly in terms of successivemeasurements. Quantum mechanically, a number of dif-ferent pre-measurement states will give rise to the samepost-measurement state Λ α ( ρ ). We will now first find anequivalence class of pre-measurement states that givesrise to the same post-measurement state. To this end,we introduce the unitary operator e iφα . Since e iφα γ = γ if α and γ are orthogonal projectors, and e iφα α = e iφ α ,this operator will be denoted as a selective phase rota-tion operator. A selective phase rotation of the initialstate ρ gives the state ρ φα = e iφα ρe − iφα . We notice thatthe post-measurement state (2) may be written in theform [20] Λ α ( ρ ) = ( ρ + ρ πα ) /
2. This is a state consist-ing of a classical mixture of the original state ρ and thesame state selectively phase rotated an angle π . Thus,the phase related to the projector α has been completelyrandomized. This is an example of complete decoher-ence, but only with respect to a single projector (notethat the complement ˜ α has also decohered).Since the L¨uders map Λ α entails phase randomizationdue to the action of the selective phase rotation operator e iφα , a selective phase rotation of the pre-measurementstate ρ does not alter the post-measurement state Λ α ( ρ ).Hence, it can be shown that Λ α ( ρ φα ) = Λ α ( ρ ). Fur-thermore, any state ρ φα gives rise to the same jointprobability in a successive measurement of α and β ,Tr ρ φα αβα = Tr ραβα . Therefore, it is not possible to dis-tinguish the states ρ φα by measuring α and β successively.We shall refer to them as “classically equivalent” with re-spect to the successive measurement of α and β . Theyalso give rise to the same post-measurement distribution F (cid:2) Λ α ( ρ φα ) , α, β (cid:3) = Tr ραβα . However, they give rise todifferent changes of the quasi-probability ∆ F ( ρ φα , α, β ).We have already seen that the imaginary term I ( ρ, α, β ) should be a “ disturbance term” in the senseof Eq. (16). The simplest possibility is to assume thatit takes the same form as the real disturbance term∆ F ( σ, α, β ) for some state σ . This would imply thatEqs. (16) and (17a) are automatically satisfied. How-ever, we must have σ = ρ . Otherwise, there would be aconflict between conditions (11) and (17b). Our assump-tion will be that I ( ρ, α, β ) should take the same form asthe real disturbance term (12), but for one of the classi-cally equivalent states ρ φα , Condition 6. I ( ρ, α, β ) = 12 (cid:2) Tr ρ φα β − TrΛ α ( ρ ) β (cid:3) . (18)By combining Eqs. (17b) and (18), we find that φ = π/ G ( ρ, α, β ) = Tr ραβ + 12 [Tr ραβ − TrΛ α ( ρ ) β ]+ i h Tr ρ π α β − TrΛ α ( ρ ) β i . (19)On inserting the expression (2) for Λ α ( ρ ) we find thatEq. (19) simplifies to G ( ρ, α, β ) = Tr ραβ. (20)This is recognized as the complex quasi-probability dis-cussed by Dirac [6]. Note that the sign in Eq. (18) wasarbitrarily chosen. If an opposite sign had been assigned,we would have found the quasi-probability Tr ρβα . Wehave found no way of distinguishing between these twoalternatives. We also note that (20) does not fulfill theorder symmetry (4), but the condition G ( ρ, β, α ) = G ( ρ, α, β ) ∗ . (21)Exchanging the measurement order is equivalent to com-plex conjugation.The distribution (20) is informationally complete, i.e.,it determines the density matrix uniquely, provided thatthe two projectors belongs to two different projection val-ued measures where every projector is trace 1 and the twoprojection valued measures have no common elements[13]. This means that any pair of nondegenerate observ-ables describe a quantum state completely provided onlythat they cannot both have a well-defined value for anystate whatsoever. This is a beautiful illustration of theprinciple of complementarity.The quasi-probability (20) is closely related to weakmeasurements [21]. The real part may be observed di-rectly in terms the correlations between pointer positionsin successive measurements of the two projectors, pro-vided that the interaction with the first pointer is suffi-ciently weak [22]. This scheme requires no post-selection.This is a different manifestation of the uniqueness of thisdistribution.The distribution (20) may find applications e.g. inthe theory of quantum information, communication andcomputing, where the goal is to construct procedures thatoutperform any classical counterpart. Due to its infor-mational completeness, it may become a useful tool ininformation retrieval from successive measurements. Itcan also be mentioned that (20) has been discussed inconnection with linearly positive histories [16, 23, 24, 25].The derivation of the real quasi-probability (14) relieson the assumption (10) of disturbance symmetry. Thedistribution (14) satisfies order symmetry (4). Thus,order symmetry (4) follows from disturbance symmetry(10). We demonstrated that for particular class of states,disturbance symmetry (10) could be derived from order symmetry (4). It is still an open question whether dis-turbance symmetry (10) may be derived from order sym-metry (4) for arbitrary states.In conclusion, we have derived a unique quasi-probability for arbitrary projectors from the analysis ofsuccessive projective measurements. Nonclassical prop-erties of the quasi-probability is closely related to mea-surement disturbance. It unites the concept of projectiveand weak measurements in a common formalism.The author acknowledges many stimulating talks withPier A. Mello. 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