J. Math. Phys. – in print, March 1996

Correlation Properties of Quantum Measurements |

Paul BuschDepartment of Applied Mathematics, The University of Hull, Hull HU6 7RX, UK. E-mail: and Pekka J. LahtiDepartment of Physics, University of Turku, 20014 Turku, Finland. E-mail: |

Abstract

The kind of information provided by a measurement is determined in terms of the correlation established between observables of the apparatus and the measured system. Using the framework of quantum measurement theory, necessary and sufficient conditions for a measurement interaction to produce strong correlations are given and are found to be related to properties of the final object and apparatus states. These general results are illustrated with reference to the standard model of the quantum theory of measurement.

PACS number: 03.65.Bz.

I. Introduction.

Any physical measurement is carried out in order to provide information about a specified system, its state prior to or after the measurement. The procedure generally is to ascertain the values of a pointer observable, which have become correlated with some observable of the measured system. Thus the kind of information available by a given measurement depends on the statistical dependencies established by the interaction between the apparatus, or some probe system, and the object.

The minimal content of the notion of measurement in quantum mechanics is given by the probability reproducibility requirement; according to this condition, a particular measurement scheme qualifies as a measurement of a given observable if for all initial states of the object system the associated probability distributions of are reproduced in the resulting statistics of pointer readings Regarding a large ensemble of object plus apparatus systems as one individual system, this situation can be described in terms of strong correlations between the values of the frequency operators [see, e.g., ref. 1] associated with the observable and the pointer observable, respectively. In the present context we shall not be concerned with the ensembles regarded as individual objects but rather we shall analyze statistical dependencies between individual members of the ensembles as they show up in certain correlation quantities. The following three kinds of correlations appear naturally in the measurement context: correlations between an object observable and the pointer observable; correlations between the values of these observables; and correlations between the conditional final states of the object system and apparatus, respectively. Our goal is to give exhaustive characterizations of the conditions under which these correlations are established. It will be found that the final component states of the object and apparatus must possess certain properties in order that such correlations may be strong.

Our investigation builds on previous work published in ref. 2. Correcting an erroneous characterization of strong correlations used in that paper, we give here a complete account of necessary and sufficient conditions for the occurrence of strong correlations. In addition, the scope of the results is extended beyond unitary measurements and sharp observables to cover arbitrary measurements and pointer observables and general object observables. Finally, possible fields of applications are indicated on the basis of the standard model of measurement theory, which was recently used in various proposals for quantum and atomic optics QND measurements

II. Framework.

2.1. We follow here the Hilbert space formulation of quantum mechanics in which the description of a physical system is based on a complex separable Hilbert space , with the inner product , and which builds on the dual concepts of states and observables reflecting the general structure of an experiment: preparation of the system followed by a measurement.

2.2. Let denote the set of bounded linear operators on and its subset of trace class operators. A state of is given as a positive linear operator of trace one. We let denote the set of states of , and we recall that is a (-)convex subset of , the one-dimensional projection operators (genereted by the unit vectors ) being its extremal elements. The shall be called vector states.

2.3. Let be a set and a -algebra of subsets of . An observable of is represented as (and identified with) a normalised positive operator valued measure, a pov measure, , that is, an operator valued mapping on with the properties: , , and for any disjoint sequence , with the series converging in the weak operator topology of . We recall that the projection valued measures, the pv measures, are a particular case of the pov measures; furthermore a pov measure is a pv measure, that is, for all , if and only if is multiplicative, that is, for all . Observables which are represented by pv measure are called sharp observables. It is by now well-established that the extension of the notion of observables from pv measures to pov measures is a necessity in quantum mechanics.

2.4. The probability measure

(1) |

defined by an observable and a state is related to a measurement by virtue of the minimal interpretation of quantum mechanics: the number is the probability that a measuremement of the observable E performed on the system in the state T leads to a result in the set X. The intended empirical content of this statement is the following: if the same -measurement were repeated sufficiently many times under the same conditions (characterised by ), then in the long run the relative frequency of the occurrence of the measurement results in would approach the number .

III. Measurement.

A. General.

3.1. A measurement scheme for the (object) system consists of a measuring apparatus [with its Hilbert space ], a pointer observable [with its ‘space of values’ ], an initial state of the apparatus, a measurement coupling [a linear state transformation on ], and a [measurable] pointer function , with the assumption that if is an initial state of , then is the final state of the compound object-apparatus system . Taking the partial traces of over and , respectively, one gets the corresponding reduced states and of and , respectively; then the probability measure of the pointer observable in the final apparatus state is completely determined as , .

3.2. A measurement scheme defines an observable of with the space of values via the relation: for any and ,

(2) |

This is the observable measured by means of the scheme in the sense that the totality of the distributions (for all ) of the pointer outcomes in the final apparatus states determine the pov measure via (2). A measurement scheme is a measurement of a given observable if the measured observable equals .

3.3. There is an important subclass of measurement schemes for . They consist of a sharp pointer observable , a vector state preparation of , , , , with a unitary map on . Subsuming the possible pointer function in the definition of by identification of and , we denote such a scheme and call it a unitary measurement scheme (with the understanding that is a sharp observable). It is a basic result of the quantum theory of measurement that every observable of admits a unitary measurement, that is, there is a unitary measurement scheme such that Thus the relation between measurement schemes and pov measures induced by (2) defines a map from the former onto the latter. In spite of this fundamental result, it is important to consider general meaurement schemes , since in many applications the choices of a sharp pointer and a vector state preparation of the apparatus are not physically realizable. , and a unitary measurement coupling

3.4. Another basic condition for a measurement scheme to qualify as a measurement is the requirement that the measurement should lead to a definite result. We take this requirement to entail, first of all, that the pointer observable should have assumed a definite value after the measurement. One should then be able to ‘read the actual value’ of the pointer observable and deduce from this the value of the measured observable. As well known, the task of explaining the occurrence of a definite pointer value at the end of a measuring process presents one of the major open problems in quantum mechanics. We do not enter this difficult question here. (For an overview of the issues involved, the reader may wish to consult ref. 1). There are, however, some necessary conditions for the pointer observable to assume a definite value in the final apparatus state , conditions which are tractable and which call for the study of the correlation properties of a measurement. These conditions are the subject of the present paper.

B. Reading of pointer values.

3.5. The reading of measurement outcomes involves the discrimination between the elements of a finite (or, as an idealization, countable) set of alternative pointer values. In order to formulate this idea in the general context of an -measurement , we introduce the notion of a reading scale as a countable partition of the value space of the pointer observable, = , induced by a countable partition of the value space of the measured observable, = , , for . Such a reading scale will be denoted , and we let “tensyI denote its index set.

3.6. A reading scale determines discrete, coarse-grained versions of the pointer observable and the measured observable ;

The -value refers to the pointer reading which, in turn, is correlated to the value set of the measured observable . If itself is discrete there is a natural (finest) reading scale such that and = . It should be noted that we have included the pointer function in the definition of so that the two discrete observables (3) do have the same set of values.

3.7. We say that the (discrete) pointer observable has the value in the state if and only the measurement outcome probability for this value equals one: . Since , and, in general, , the pointer observable does not have a value at the end of the measurement. It may, however, occur that the state is a mixture of eigenstates of with the weights . This is indeed a necessary condition for the assertion that the pointer observable has assumed a definite value with respect to a reading scale at the end of the measurement . We go on to specify this case further.

3.8. We consider a measurement of with a fixed reading scale . Any , , defines a (unnormalised) conditioned state:

(4) |

the state of on the condition that the pointer observable has value . The (trace) norm of this state is , and the corresponding (normalised) reduced states, the final component states of and are:

(If , we put ). The conditional interpretation of the states (4) and (5) presupposes, however, that the pointer observable has value in state , that is, for all and whenever This requirement is always satisfied if the pointer observable is sharp. In general this is a condition to be imposed on the measurement scheme. We call it the pointer value-definiteness condition and note that it may be written in either of the following equivalent forms:

for all and all initial states of .

3.9. For any reading scale and any state one has

(7) |

this is to say that the final object state behaves additively with respect to the pointer conditioning: that is, the state of on the plain condition that the measurement has been performed, is the same as the state of after the measurement conditional on the fact that the pointer value is registered with respect to the reading scale Although it also holds true that for any and

(8) |

it is not the case, in general, that the final apparatus state is conditioned with respect to ; thus, in general,

(9) |

The requirement that is a mixture of the final component states is therefore another condition on the measurement; we call it the pointer mixture condition:

(10) |

for all initial states of .

3.10. The pointer value-definiteness condition (6) and the pointer mixture condition (10) imply that the final apparatus state is a mixture of the pointer eigenstates with the weights ; this means that the final apparatus state is conditioned with respect to the reading scale One may consider the assumption that in addition to this, the state admits the ignorance interpretation with respect to the decomposition (10): that is, the apparatus [in state ] is actually in one of the component states , and this is the case with the subjective probability . As well known, such an interpretation of the mixed state is extremely problematic and in most cases impossible; but if it were the case then the pointer could be claimed to have a definite value (with respect to a reading scale ) after the measurement with the subjective probability . Setting aside the difficulties with the ignorance interpretation (and thus with explaining the occurrence of definite measurement outcomes in quantum mechanics), it still is important to investigate more closely the conditions (6) and (10) and to see how these possible properties of a measurement are related to the structure of the final state of the object system.

3.11 Theorem. Let be a measurement of an observable and any reading scale. For any initial state of the object system, the condition implies the conditions and :

If is a unitary measurement , then and are equivalent conditions for any initial vector state of .

Proof: : For each , let be the support projection of , that is, the smallest projection such that . Then one gets (for ):

By the definition of one also has

Combining () and () and using the fact that yields

() |

From this one obtains , which gives . Using , one shows similarly that

Inserting this in (), multiplying each term with its adjoint and summing over , one obtains

Taking the partial trace with respect to finally yields . : This implication will be shown for a unitary measurement and for vector state preparations . In that case and . Denoting the biorthogonal decomposition of this state as , with , we obtain . Now implies that all the projections commute with . Therefore one can choose the orthonormal system such that or . Thus there is a renumbering of this system, , such that . It follows that there are corresponding renumberings and such that . Then . Since the subsets of vectors with different values of are mutually disjoint and therefore orthogonal, one concludes that holds. This completes the proof.

It can be demonstrated by means of examples that the implication need not hold if the measurement is not unitary or if the initial pointer state is not pure

C. First kind and repeatable measurements.

3.12. A measurement of an observable is of the first kind if the probability for a given result is the same both before and after the measurement, that is, for any and for all ,

(11) |

Unitary measurement schemes with a coupling , , (on ) and (on ) self-adjoint, do give rise to such measurements; we refer to Sec. 8 for an analysis of this model.

3.13. A measurement of an observable is repeatable if its repetition does not lead to a new result. One way to express the requirement is the following: for any and , if , then

(12) |

(where is defined by (3a), (4) and (5a) with ). Equivalently, is a repeatable -measurement if for any and , for which , it holds true that

(13) |

Another basic result of measurement theory is that an observable which admits a repeatable measurement is discrete

3.14. According to (13), a repeatable measurement drives the object system into an eigenstate of the measured observable . The orthogonality conditions of Theorem 3.11 are then satisfied and the final apparatus state is the mixture of the eigenstates of with the weights .

3.15. It is evident that repeatable measurements are also of the first kind. However, as will be demonstrated in Sec. 8, a first kind measurement need not be repeatable, though for sharp observables the two notions coincide

IV. Statistical dependence and correlations. A measurement of an observable brings the compound object-apparatus system into an entangled state . The possibility of transferring information from to rests on the fact that this state entails statistical dependencies between quantities pertaining to these systems. Accordingly, three types of correlations inherent in the state are of special interest for characterising the measurement: ) correlations between the measured observable and the pointer observable; ) correlations between the corresponding values of these observables; and ) correlations between the final component states of the two subsystems. For their study it is helpful to recall some basic notions and facts concerning the relation between statistical dependence and correlation.

4.1. Let be a probability measure on the real Borel space , and let and be the marginal measures of with respect to a Cartesian coordinate system: for ,

(14) |

These marginal measures correspond to the coordinate projections (random variables) and in the sense that , that is, for all , . Assume that the expectations and the variances of are well defined and finite: , , and let . The (normalised) correlation of the marginal measures and in is then defined as:

(15) |

(whenever . The Schwarz inequality entails . The marginals are uncorrelated if (that is, ), strongly correlated if (that is, ), and strongly anticorrelated if (that is, ). The strong correlation conditions can also be written in terms of the coordinate projections and :

(Here we have introduced the function . A case of special interest arises when the marginals and have the same (finite) first and second moments so that , . Then one has:

4.2. The notion of correlation can be applied to quantify the degree of mutual dependence of the marginal measures. In order to avoid dealing with unnecessary complications, we assume that and are no -valued measures; equivalently, we let . and are independent if . Otherwise, are dependent. They are completely dependent if there is a (measurable) function such that for . That is, the marginal measure suffices to determine the whole measure . The relation of complete dependence is symmetric with respect to the two marginals only if is bijective. This is the case of concern here.

4.3. It is evident that the statistical independence of implies . However, the latter condition is not sufficient to ensure their independence. (For a counter example, see, for instance ref. 9). On the other hand, eqs. (16a,b) show that strong (anti)correlation entails complete dependence, the dependence being given by the linear function . Indeed, the condition (-a.e.) implies that for all and for which . Thus, in particular, for any and , and with denoting the complement of one has = 0 = . The additivity properties of allow one then to verify that for all , , that is, and are completely dependent with . By a direct computation one can confirm that the converse implication holds true whenever the function is linear. Therefore, we have:

In both cases the constants are , , so that .

V. Strong correlations between observables.

5.1. According to the condition (2), in an -measurement the initial -outcome distribution is recovered from the final -outcome distribution. In addition to this basic requirement, a measurement may also establish complete statistical dependence between the measured observable and the pointer observable after the measurement; that is, the observables and may become strongly correlated in the final object-apparatus state . In order to avoid technical complications in the formulation of this correlation, we assume that the value space of is the real Borel space, . Then for any state the map

(19) |

extends to a probability measure on The marginal distributions are

Denoting the correlation of and in as , we say that the measurement of produces strong observable-(anti)correlation in state if this number equals . According to (18), this occurs exactly when the probability measures (20a,b) are completely dependent, with the function . In order to analyze the statistical dependence of and we shall make use of the concept of a state transformer (also known as an instrument) associated with a measurement.

5.2. Consider a measurement of . Any defines a nonnormalised state

(21) |

the (trace) norm of which is . Taking the partial trace of over one gets the (nonnormalised) reduced state of ,

(22) |

For any and , , and is a (contractive) state transformation. The mapping has the measure property for any disjoint sequence and for all . Moreover, for any . We call the state transformer induced by the measurement . It describes the object system’s state changes under the measurement, and it uniquely defines the measured observable via the relation . We note also that , and, in particular, .

5.3. The probability measure (19) can be written as

(23) |

and the second marginal is . The strong (anti-)correlation then amounts to

(24) |

A special case of complete dependence arises with being the identity function:

(25) |

This relation is easily seen to coincide with (12) Thus, if valid for all states , (25) expresses the repeatability of the measurement, and we may conclude that any repeatable measurement leads to strong observable-correlations. The repeatability condition (25) is not necessary for the strong observable-correlation (24).

5.4. Condition (25) implies, in particular, the equality of the marginal measures of Eqs. (20a,b): for all ,

(26) |

This is just the first-kind property of the measurement. It may occur that these marginal measures coincide irrespectively of whether (25) holds or not; in that case conditions (17a,b) give the relevant characterisations of strong (anti)correlations.

5.5 Theorem. Let be a measurement of an observable , and let be any reading scale. Then implies , where:

If the reading scale is finite, then and are equivalent.

Proof: The eigenstate condition is equivalent with the repeatability condition (with respect to ). Therefore, if holds, then also is true. It remains to show that implies whenever is finite. According to (18a), the statement is equivalent to the complete dependence, , with a bijective linear mapping , , between those values for which (and hence ). Case 1. Let be such that for all . Then correlates, via , all values with values . Since is onto and monotonically increasing, . But the complete dependence condition, with , is nothing but Eq. (25) (with respect to ), which is equivalent to . Case 2. Let be any state such that holds exactly for all , a proper nonempty subset of . Take any for which exactly for all , the complement of . Then the reasoning of Case 1 applies to for all . Inserting in this equation the relation , which holds for , it follows that for . But this relation holds trivially also for since in that case . This completes the proof. . Hence,

VI. Strong correlations between values.

6.1. The observable measured by the scheme with the reading scale is discrete. One may therefore ask to what degree the values of this observable and the pointer observable become correlated in the measurement. To answer this question requires studying the correlation of the -th values of these observables in the final object-apparatus state, that is, the correlation of quantities and in the state :

(27) |

The respective quantities are easily determined:

Strong correlation is then equivalent to

(29) |

whenever the right-hand side is nonzero.

6.2. Assume that the final component state is a 1-eigenstate of (whenever ); then one obtains = = for all . It follows that and thus . On the other hand, the relation together with implies . Therefore the correlation equals 1 whenever .

Another interesting implication of the eigenstate condition and the ensuing equality is the fact that the state is a 1-eigenstate of . With these observations we have established the following result.

6.3 Theorem. Let be a measurement of an observable and let be any reading scale. Then for any state of , implies and :

This result entails that a repeatable measurement is a strong value-correlation measurement. Moreover, a necessary condition for to be a repeatable measurement is that the final component state of is a 1-eigenstate of the pointer observable, that is, must fulfil the pointer value-definiteness condition. We recall that this last property and in addition the pointer mixture property arise already as consequences of the mutual orthogonality of the component states of (Theorem 3.11). The notion of a correlation between values suggests that the observables in question do have definite values; yet it turns out that strong value-correlation does not require pointer value-definiteness, nor repeatability. Even the combination of and does not require the property to hold, as can be demonstrated by simple examples

6.4 Theorem. Let be a measurement of a sharp observable and any reading scale. For any initial state of , is equivalent to :

Proof: In view of Theorem 6.3 we only need to show that implies . Hence let hold for each . Condition implies . Similarly the relation implies . From Eqs. (28) we obtain , , and therefore

This implies . On the other hand,

Using , one concludes that . But the last equation is equivalent to . This completes the proof.

VII. Strong correlations between final component states.

7.1. In the two preceding sections it was demonstrated in which way strong observable and value correlations serve as characterisations of repeatable measurements. The corresponding eigenstate condition entails, in particular, that the final component states of the object associated with different outcomes are mutually orthogonal, . In some cases this orthogonality can be characterised in terms of strong correlations between the final component states of and .

Consider a measurement scheme of an observable with respect to a reading scale . We say that , with , is a strong state-(anti)correlation measurement of if for each initial state of it correlates strongly the final component states and of the object and the apparatus. This calls for the study of the correlation of the probability measure defined by the self-adjoint operators and and the final object-apparatus state .

7.2 Theorem. Let be a measurement of an observable and any reading scale. For any initial state of the object system for which the component states and are vector states, is equivalent to :

Proof: The equivalence is shown to hold under the assumptions and . These two relations imply that is a vector state of the product form, that is,

() |

If holds, then by Theorem 3.11, fulfils the pointer value-definiteness condition . Thus for both implications one can make use of the fact that . Then () implies

With this one computes:

: is equivalent to , one has , and . Thus , that is, .: Let . Using the inequalities , , one concludes that . Since , one also has , and therefore . But from the definition of one has , so that the equality of these numbers implies whenever , that is . This completes the proof.

7.3. One may also ask whether the requirement of strong correlation between the final and states and imposes any constraint on the measurement scheme under consideration. That this cannot be expected in general can be seen in the case of a unitary measurement . Note first that the reduced states of have the same spectra, including multiplicities. The spectral decompositions can be given in terms of orthonormal systems , defined by the biorthogonal decomposition (), and a straightforward calculation shows that

(30) |

Hence these states are always strongly correlated.

VIII. Examples.

8.1. A particularly interesting class of measurements arises if the coupling is generated by a unitary map of the form

(31) |

where and are self-adjoint operators in and , respectively, and is a coupling constant. The operator is usually taken to represent the (sharp) observable one aims to measure. In order to specify the full measurement scheme and thus the actually measured observable, one neeeds to choose the pointer observable and fix the initial preparation of the apparatus; the measured observable is then given by eq. (2). Using the spectral decomposition of , , and denoting

(32) |

the final apparatus state, for , assumes the form

(33) |

Since it is of interest to compare the measured observable with we assume from the outset that the value space of is . In view of the coupling constant () it is also convenient to introduce a pointer function . The observable measured by the scheme takes then the following form: for any ,

(34) |

The structure of the operators show that in general the measured observable is not the sharp observable , but a smeared version of it One may ask which choices of and would possibly yield <