# Imprecise probability for non-commuting observables

###### Abstract

It is known that non-commuting observables in quantum mechanics do not have joint probability. This statement refers to the precise (additive) probability model. I show that the joint distribution of any non-commuting pair of variables can be quantified via upper and lower probabilities, i.e. the joint probability is described by an interval instead of a number (imprecise probability). I propose transparent axioms from which the upper and lower probability operators follow. They depend only on the non-commuting observables and revert to the usual expression for the commuting case.

###### pacs:

03.65.-w, 03.67.-aNon-commuting observables in quantum mechanics do not have a joint
probability wigner_no_go ; hudson_soto ; gudder ; deMuynck ; busch [see
section 1.1 of the Supplementary Material for a reminder]. This is
the departure point of quantum mechanics from classical probabilistic
theories malley ; it lies in the core of all quantum oddities.
There are various quasi-probabilities (e.g., Wigner function) which
have features of joint probability for (loosely defined) semiclassical
states hillery ; ferrie ; ballentine . Quasi-probabilities do have
two problems: (i) they (must) get negative for a class of
quantum states, thereby preventing any probabilistic
interpretation for them ^{1}^{1}1Negative probabilities were not
found to admit a direct physical meaning mueck (what can be
less possible, then the impossible?). In certain cases what seemed
to be a negative probability was later on found to be a local value
of a physical quantity, i.e. physically meaningful, but not a
probability mueck . Mathematical meaning of negative
probability is discussed in Refs. khren ; gabor .. (ii)
Even if the quasi-probability is positive on a certain state, it is
not unique, i.e. there can be other (equally legitimate)
quasi-probability that is positive (and has other expected features of
probability) on this state, e.g. in quantum optics there is Wigner
function, P-function, Terletsky-Margenau-Hill function etc. Despite of the drawbacks, quasi-probabilities do have many
applications hillery ; ferrie ; gardiner ; blako ; armen , since they
still possess certain features of joint probability, e.g. they
reproduce the marginals
hudson_soto ; deMuynck ; hillery ; ferrie ; armen . One can relax this
requirement ^{2}^{2}2Employing instead the unbiasedness: the averages
of the non-commuting quantities are reproduced correctly
arthurs ., as done for joint measurements of non-commuting
variables deMuynck ; busch ; de ; arthurs ; abn ; yuko . Such measurements
have to be approximate, since they operate on an arbitrary initial
state deMuynck ; busch . They produce positive probabilities for
the measurement results, but it is not clear to which extent these
probabilities are intrinsic uffink , i.e. to which extent they
characterize the system itself, and not approximate measurements
employed. Alternatively, one can consider two consecutive measurements
of the non-commuting observables bub ; hughes . These two-time
probabilities do not (generally) qualify for the joint probability of
the non-commuting observables; see section 1.2 of the Supplementary
Material.

It is assumed that the sought joint probability is linear over the
state (density matrix). If this condition is skipped, there are
positive probabilities that correctly reproduce marginals for
non-commuting observables cohen_zap ; yuko , e.g. simply the
product of two marginals ballentine . However, they do not
reduce to the usual form of the joint quantum probability for commuting observables ^{3}^{3}3Given two projectors and
and state , this product is , while the correct form for is .; hence their physical meaning is unclear
ballentine .

The statement on the non-existence of joint probability concern the usual precise and additive probability. This is not the only model of uncertainty. It was recognized since early days of probability theory shafer that the probability need not be precise: instead of being a definite number, it can be a definite interval good ; kuz ; walley ; fine_freq ; see bumbarash for an elementary introduction.

Instead of a precise probability for an event , the measure of uncertainty is now an interval , where are called lower and upper probabilities, respectively. Qualitatively, () is a measure of a sure evidence in favor (against) of . The event is surely more probable than , if . The usual probability is recovered for . Two different pairs and can hold simultaneously (i.e they are consistent), provided that and for all . In particular, every imprecise probability is consistent with , .

It is not assumed that for all there is a true (precise, but unknown) probability that lies in . This assumption is frequently (but not always fine_1994 ) made in applications kuz ; walley , and it did motivate the generalized Kolmogorovian axiomatics of imprecise probability fine_freq ; see section 2.1 of the Supplementary Material. Imprecise joint probabilities in quantum mechanics are to be regarded as fundamental entities, not reducible to a lack of knowledge. They do need an independent axiomatic ground.

My purpose here is to propose a transparent set of conditions (axioms) that lead to quantum lower and upper joint probabilities. They depend only on the involved non-commuting observables (and on the quantum state).

Previous work. In 1967 Prugovecki tried to describe the joint probability of two non-commuting observables in a way that resembles imprecise probabilities prug . But his expression was not correct (it still can be negative) ballentine ; see also khren in this context. In 1991 Suppes and Zanotti proposed a local upper probability model for the standard setup of Bell inequalities (two entangled spins) suppes_zanotti ; see also barros ; hartmann . The formulation was given in the classical event space of hidden variables, and it is not unique even for the particular case considered. It violates classical observability conditions for the imprecise probability fine_1994 ; fine_freq ; suppes_zanotti . In particular, no lower probability exists in this scheme. Despite of such drawbacks, the pertinent message of suppes_zanotti is that one should attempt at quantum applications of the upper probabilities that go beyond its classical axioms. More recently, Galvan attempted to empoy (classical) imprecise probabilities for describing quantum dynamics in configuration space galvan . For a general discussion on quantum versus classical probabilities see khren_book .

Notations. All operators (matrices) live in a finite-dimensional Hilbert space . For two hermitean operators and , (larger or equal) means that all eigenvalues of are non-negative, i.e. for any . The direct sum of two operators refers to the block-diagonal matrix: of is the subspace of vectors , where . For orthogonal (sub)spaces and , the space is formed by all vectors , where and . is the unity operator of . and are the unity and zero matrices, respectively. . The range

Axioms for quantum imprecise probability. Existing axioms for imprecise probability are formulated on a classical event space with usual notions of con- and disjunction and complemention good ; kuz ; walley ; fine_freq ; good ; see section 2 of the Supplementary Material for a reminder. For quantum probability it is natural to start from a Hilbert space and introduce upper and lower probabilities as operators. The axioms below require only the most basic feature of upper and lower probability and demand its consistency with the quantum joint probability whenever the latter is well-defined.

The usual quantum probability can be defined over (hermitean) projectors jauch ; jauch_book . A projector generalizes the classical notion of characteristic function. Each uniquely relates to its eigenspace . refers to a set of hermitean operators :

(1) |

is a projector to an eigenspace of or to a direct sum of such eigenspaces, i.e. refers to an eigenvalue of or to a union of several eigenvalues. The quantum (precise and additive) probability to observe is , where the density matrix defines the quantum state deMuynck ; busch ; jauch ; jauch_book .

Let be another projector which refers to the set of observables. Generally, . Given the density matrix , we seek upper and lower joint probabilities of and (i.e. of the corresponding eigenvalues of and ):

(2) |

where and are hermitean operators. Their dependence on and can be expressed via Taylor series. We impose the following conditions (axioms):

(3) |

(4) |

(5) |

(6) |

(7) |

Eq. (2) implies that and depend on and only through and . This non-contextuality feature holds also for the ordinary (one-variable) quantum probability bell ; smerd . Provided that the operators and are found, and can be found in the usual way of quantum averages.

Conditions (3) stem from that are demanded for all density matrices . Eq. (4) is the symmetry condition necessary for the joint probability. Eq. (5) is reversion to the commuting case. In particular, (5) ensures and . Since means that is anywhere, the latter equality is the reproduction of the marginal probability (which cannot be recovered by summation, since the probability model is not additive).

For the joint probability is . This expression is well-defined (i.e. positive, symmetric and additive) also for or (but not necessarily ). If , one obtains by measuring ( is not disturbed) and then . Alternatively, one can obtain it by measuring the average of an hermitean observable . Thus (5, 6) demands that and are consistent with the joint probability , whenever the latter is well-defined.

Finally, (7) means that () can be measured simultaneously and precisely with or with (on any quantum state), a natural condition for the joint probability (operators).

If there are several candidates satisfying (3–7) we shall naturally select the ones providing the largest lower probability and the smallest upper probability.

CS-representation will be our main tool. Given the projectors and , Hilbert space can be represented as a direct sum dix ; halmos ; hardegree [see also section 3 of the Supplementary Material]

(8) |

where the sub-space of dimension is formed by common eigenvectors of and having eigenvalue (for ) and (for ). Depending on and every sub-space can be absent; all of them can be present only for . Now is the intersection of the ranges of and . has even dimension halmos ; hardegree , this is the only sub-space in (8) that is not formed by common eigenvectors of and . There exists a unitary transformation

(9) |

so that and get the following block-diagonal form related to (8) halmos :

(10) | |||

(11) |

where and are invertible square matrices of the same size holding

(12) |

Now and are sub-spaces of . One has and , where is the operator analogue of the angle between two spaces. are absent, if and do not have any common eigenvector. This, in particular, happens in .

The main result. Note that if (3–7) holds for and , they hold as well for and , because for . Section 4 of the Supplementary Material shows how to get and from (3–7) and (10, 11):

(13) | |||

(14) |

Let be the projector onto intersection of and . We now return from (13, 14, 9) to original projectors and [see section 4 of the Supplementary Material] and obtain the main formulas:

(15) | |||

(16) |

For , , and we revert to . Note that .

Physical meaning of and . When looking for a joint probability defined over two projectors and one wonders whether it is just not some (operator) mean of and . For ordinary numbers and there are 3 means: arithmetic , geometric and harmonic . Now (15) is precisely the operator harmonic mean of and anderson_duffin1

(17) |

where is the inverse of if it exists, otherwise it is the pseudo-inverse; see section 5 of the Supplementary Material for various representations of and . More familiar formula is

(18) |

The intersection projector appears in jauch ; jauch_book ; jauch_piron ; busch ; bell . It was stressed that cannot be a joint probability for non-commutative and de . Its meaning is clear by now: it is the lower probability for and . is non-zero only if (or ), since two different rays cross only at zero.

Let us now turn to . The transition probability between 2 pure states is determined by the squared cosine of the angle between them: . Eq. (13) shows that depends on , where is the operator angle between and . Note from (10, 11) that the eigenvalues of , which hold are the eigenvalues of , and—as seen from (13)—they are also (doubly-degenerate) eigenvalues of . Thus we have a physical interpretation not only for (transition probability), but also for eigenvalues of ( and have the same eigenvalues).

Eqs. (13, 14, 9) imply that the upper and lower probability operators can be measured simultaneously on any state [cf. (7)]:

(19) |

The operator quantifies the uncertainty for joint probability, the physical meaning of this characteristics of non-commutativity is new.

Section 7 of the Supplementary Material calculates the upper and lower probabilities for several examples.

Note that the conditional (upper and lower) probabilities are straighforward to define, e.g. [cf. (2)]: .

The distance between two probability intervals and can be calculated via the Haussdorff metric alefeld

(20) |

which nullifies if and only if and , and which reduces to the ordinary distance for usual (precise) probabilities. Now

(21) |

means that the pair of projectors is surely more probable (on ) than ; see section 7 of the Supplementary Material for examples. Note from (15, 16) that if

(22) |

holds for , then it also hods for (and vice versa). Though in a weaker sense than (21), (22) means that and together is more probable than neither of them together (which is the pair ). Eqs. (21, 22) are examples of comparative (modal) probability statements; see ochs in this context.

Further features of and are uncovered when looking at a monotonic change of their arguments; see section 6 of the Supplementary Material. Section 7 discusses concrete examples.

Summary. My main message is that while joint precise probability for non-commuting observables does not exist, there are well-defined expressions for upper and lower imprecise probabilities. They can have immediate applications as shown in section 7 of the Supplementary Material for simple examples. Not less important are the open question suggested by this research, e.g. what is the most convenient way of defining averages with respect to quantum imprecise probability, or are there even more general axioms that involve the density matrix non-lineary and reduce to the linear situation when (5, 6) (effective commutativity) holds.

I thank K.V. Hovhannisyan for discussions.

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## I Supplementary Material

Imprecise probability for non-commuting observables by Armen E. Allahverdyan

This Supplementary Material consists of seven sections. All of them can be read independently from each other.

Sections 1, 2 and 3 recall, respectively, the no-go statements for the joint quantum probability, generalized axiomatics for the imprecise probability and the CS-representation. This material is not new, but is presented in a focused form, adapted from several different sources.

Section 4 contains the derivation of the main result, while sections 5 and 6 demonstrate various feature of quantum imprecise probability.

Section 7 illustrates it with simple physical examples.

## Ii 0. Notations

We first of all recall the employed notations. All operators (matrices) live in a finite-dimensional Hilbert space . For two hermitean operators and , means that all eigenvalues of are non-negative, i.e. for any . The direct sum of two operators refers to the following block-diagonal matrix:

is the range of (set of vectors , where ). is the unity operator of . is the subspace of vectors with .

and are the unity and zero matrices, respectively.

In the direct sum of two sub-spaces, it is always understood that and are orthogonal. The vector sum of (not necessarily orthogonal) sub-spaces and will be denoted as . This space is formed by all vectors , where and .

## Iii 1. Non-existence of (precise) joint probability for non-commuting observables

### iii.1 1.1 The basic argument

Given two sets of non-commuting hermitean projectors:

(23) | |||

(24) |

we are looking for non-negative operators such that for an arbitrary density matrix

(25) |

These relations imply

(26) |

Now the second (third) relation in (26) implies (). Hence .

Thus, if (e.g. when and are one-dimensional), then , which means that the sought joint probability does not exist.

### iii.2 1.2 Two-time probability (as a candidate for the joint probability)

Given (23, 24), we can carry out two successive measurements. First (second) we measure a quantity, whose eigen-projections are (). This results to the following joint probability for the measurement results [ is the density matrix]

(28) |

Likewise, if we first measure and then , we obtain a quantity that generally differs from (28):

(29) |

If we attempt to consider (29) [or (28)] as a joint additive probability for and , we note that (29) [and likewise (28)] reproduces correctly only one marginal:

(30) |

One can attempt to interpret the mean of (28, (29)

(31) |

as a non-additive probability. This object is linear over , symmetric (with respect to interchanging and ), non-negative, and reduces to the additive joint probability for . The relation can be interpreted as consistency with the correct marginals (once is regarded as a non-additive probability, there is no point in insisting that the marginals are obtained in the additive way).

However, the additive joint probability is well-defined also for (or for ). If holds, is not consistent with , i.e. depending on , and both

(32) |

are possible.

To summarize, the two-time measurement results do not qualify as the additive joint probability, first because they are not unique (two different expressions (28) and (29) are possible), and second because they do not reproduce the correct marginals. If we take the mean of two expressions (28) and (29) and attempt to interpret it as a non-additive probability, it is not compatible with the joint probability, whenever the latter is well-defined.

## Iv 2. Axioms for classical imprecise probability

### iv.1 2.1 Generalized Kolmogorov’s axioms

Given the full set of events , and defined over sub-sets of (including the empty set ) satisfy kuz ; walley ; fine_freq :

(33) | |||

(34) | |||

(35) | |||

(36) | |||

(37) |

where includes all elements of that are not in , and where means intersection of two sets; holds for elementary events.

Here are some direct implications of (33–37).

(39) | |||||

Eq. (39) follows directly from (36). Eq. (39) follows from (36, 35). Next relation:

(40) |

which, in particular, implies

(41) |

To derive (40), note that (36, 35) imply or , which is the first inequality in (40). The second inequality is derived via (37, 35).

The following inequality generalizes the known relation of the additive probability theory

(42) |

To prove (42), we denote , which means . Now

(43) | |||||

(44) |

where in (43) [resp. in (44)] we applied the first [resp. the second] inequality in (40).

Note that the (non-negative) difference between the upper and lower probabilities also holds the super-additivity feature (cd. (37))

(45) |

### iv.2 2.2 Joint probability

### iv.3 2.3 Dominated upper and lower probability

## V 3. Derivation of the CS-representation

### v.1 3.1 The main theorem

Let and are two subspaces of Hilbert space that hold ( is the orthogonal complement of )

(54) |

The simplest example realizing (54) is when and are one-dimensional subspaces of a two-dimensional .

Let and be projectors onto and respectively. Now is the projector of , and let be the projector . Employing the known formulas (see e.g. hardegree )

(55) |

we get from (54)

(56) |

which means that should be even for
(54) to hold ^{4}^{4}4Eq. (56) can be derived by
noting that implies .
Indeed, if , where , then
. Hence . Let us mention for
completeness that . Indeed, let
us assume that , and . Then . The last relation means that either , or
, which contradicts to (54)..

Here is the statement of the CS-representation halmos : after a unitary transformation and can be presented as

(57) |

where all blocks in (57) have the same dimension .

Next, let us show that

(59) |

Since , means . Indeed, we have , which together with [see (54)] leads to . , we need to show that for any

Eq. (59) implies that there is the well-defined polar decomposition [ is hermitean, while is unitary]

(60) |

We transform as

(61) |

where . We shall now employ the fact that the last matrix in (61) is a projector:

(62) |

The first and second relations in (62) show that . Then the third relations produces . Since (due to ), we conclude that . The rest is obvious.

### v.2 3.2 Joint commutant for two projectors

### v.3 3.3 General form of the CS representation

The above derivation of (57) assumed conditions (