## Abstract

A likelihood order is defined over linear subspaces of a finite dimensional Hilbert space. The question arises as to when such an order can be represented by a quantum probability. We introduce a few behaviorally plausible axioms that provide the answer in two cases: pure state and uniform measure. The general problem is answered by using duality-like conditions.

The general problem of characterizing the partial orders that admit a quantum representation by behaviorally justified axioms remains open.

## 1. Introduction

According to the subjective approach, probabilities are merely degrees-of-belief of a rational agent. These degrees-of-belief might be indicated by the agent's willingness to bet or take other actions (de-Finetti 1974). Savage (1954) derives both probabilities and utilities from rational preferences (i.e. that satisfy some putative properties) alone. Such preferences induce, in particular, a preference order over events. That is, an agent who holds rational preferences could indicate which of two events is more likely and moreover, this likelihood order is transitive. Savage's first step is to derive a (finitely additive) probability that represents the likelihood order.

In this paper, we adopt a similar approach and apply it to the quantum framework without going beyond probabilities. While classical probability is defined over subsets (events) of a state space, quantum probability is defined over subspaces of Hilbert space. Furthermore, disjointness of the classical model is replaced by orthogonality.

Formally, let be a separable Hilbert space. A quantum probability measure *μ* over assigns a number between 0 and 1 to every closed subspace that satisfies whenever and . Gleason's theorem (1957) states that, if every quantum measure *μ* is induced by a self-adjoint non-negative operator *T* with trace 1 in the following way: for every subspace *A*, where is the orthogonal projection over *A*.

We assume the existence of a likelihood order ⪯ over subspaces of a given finite-dimensional Hilbert space. The statement ‘*A* is less likely than *B* in one's eyes’ could be understood operationally: one would prefer betting that *B* occurs than that *A* occurs (in the corresponding physical measurements).

We say that the likelihood order ⪯ can be *represented by a quantum probability* *μ* if if and only if . The goal of the line of research presented here is to find plausible properties (axioms, in the jargon of decision theory), preferably rationality-motivated, that ensure that ⪯ is representable by a quantum probability.

Throughout, it is assumed that ⪯ possesses three properties. The first is that every subspace is more likely than the zero-dimensional one. The second is that a subspace *B* is more likely than *A* if and only if is more likely than , whenever and . That is, adding or deleting a subspace which is orthogonal to both *A* and *B* would preserve the likelihood order.

The classical counterpart of the third property is a consequence of the second. However, in the quantum model it has to be explicitly assumed. It states that if *B* is more likely than *A*, then the orthogonal complement of *B* is less likely than that of *A*.

Savage (1954) also assumes these three axioms but, in order to obtain a representation by a measure, he needs an additional, less motivated, property that concerns with the richness of the state space. This one dictates that the state space could be split into mutually disjoint arbitrarily small (with respect to the likelihood order) subsets. The lack of a quantum counterpart (in the case of a finite-dimensional Hilbert space) of such an Archimedean property makes our study different from that of Savage.

Our main results refer to likelihood orders that can be represented by two types of quantum measures. The first is the most important from a physical point of view. The probabilities of this type are called *pure states* and are of the form for some unit vector . That is, the probability of a subspace *A* is the length squared of the projection of the vector *p*. By Gleason's theorem, these measures are the extreme points of the convex set of all quantum probabilities. We characterize the likelihood orders that can be represented by a pure state.

The second main result characterizes the likelihood orders that can be represented by the uniform distribution, defined by . This is the only quantum measure that obtains a discrete set of values.

We also provide a full characterizing of the partial orders that admit a quantum representation using a duality-like condition, which is less behaviourally justified. However, the general representation problem in terms of axioms that are well motivated from a behavioural viewpoint remains open.

Subjective analysis of quantum probability has been treated in the literature by several authors. Deutsch (1999) assumes that an agent assigns a value to any possible outcome of any possible measurement. Deutsch's analysis hinges on what he calls the ‘principle of substitutibility’, which is similar to the independence axiom of von Neumann & Morgenstern (1944). Barnum *et al*. (2000) claimed that Deutsch's proof relies on a tacit symmetry assumption. Deutsch's approach has been defended by Wallace (2003) who argued that the symmetry assumption is in fact a direct consequence of a ‘physicality’ assumption saying that an agent assigns the same value to two games instantiated by the same physical process. In a complementary paper, Wallace (2005) weakens Deutsch's rationality axioms and replaces the numerical assignments to games with preference order between games. Motivated by the Everett interpretation of quantum mechanics, the physical state is a primitive in the analysis of Deutsch and Wallace. In contrast, in our analysis the primitive of the model is the logic structure—the structure of subspaces of a Hilbert space, as formulated by von Neumann (1955).

Caves *et al*. (2002) assume that the agent has degrees-of-belief that determine the odds under which he is willing to take a bet. With the assumption that the agent cannot be attacked by a Dutch book and an additional assumption about ‘maximal information’, they showed that these degrees-of-belief must be given by a pure state. Pitowsky (2003) assumed that for every possible measurement the agent has a certain probability over the corresponding outcomes. From a few natural axioms, he derives the probabilistic structure over quantum mechanics. The main difference between their approaches and ours is that we do not assume that the agent has quantitative assignments of probabilities (i.e. numerical degrees-of-belief) to subspaces. Instead, the primitive of our model is a qualitative belief given by the likelihood order. This follows Savage's theory of subjective probability in classical decision theory. The approach is behavioural in nature. An agent chooses one alternative from a finite set of choices and an outside observer records the choices made. When the possible alternatives are bets on outcomes of various measurements, the choices of the decision maker reveal his opinion about which outcome of such measurements is more likely. This partial order (i.e. ‘being more likely than’) over outcomes of different measurements is the primitive of our study. A representation of the partial order by a quantum probability would mean that the agent acts as if he has quantitative degrees-of-belief that obey the rules of quantum probability.

In the model of Gyntelberg & Hansen (2004), the primitive is a preference order over *acts*, which are functions that assign to outcomes of a measurement a consequence. As in Anscombe & Aumann (1963), the set of consequence is convex. The goal of Gyntelberg & Hansen (2004) is to integrate, by means of a quantum probability and a utility function, a preference order related to the ‘small world’ to a global picture.

The paper is structured as follows. Section 2 characterizes the likelihood orders that admit a quantum probability representation in terms of continuity and a duality-like condition, called the cancellation condition. Section 3 introduces the main axioms. Sections 4 and 5 are devoted to the main results: representation by a pure state and by a uniform distribution. Section 6 provides an example of a likelihood order that satisfies the main axioms except for continuity, and cannot be represented by a quantum measure. The paper is concluded with §7 that provides some final comments and open problems.

## 2. The cancellation condition and continuity

Let be a finite dimensional Hilbert space and let ⪯ be a weak order over linear subspaces of , that is ⪯ is reflexive (i.e. for every *A*, ), transitive (i.e. for every , if and then ) and complete (i.e. for every *A, B*, or or both). We call ⪯ the *likelihood* order, and when , we say that *B* is *more likely than* *A*. Denote by the equivalence relation induced by ⪯ (i.e. if and ) and by the corresponding strict order (i.e. if and ).

### (a) The cancellation condition

Cancellation condition (e.g. Fishburn 1999) is a well-known property of a weak order in the classical framework:

*Let* ⪯ *be a weak order over subsets of a finite set* . *For* *, denote by* *the indicator function of A. Then there exists an additive probability measure μ over* *such that* *for every* *if and only if the following conditions hold.*

*For every*, ..

*For every n, if**are subsets of**such that**and**for every i, then**for every i*.

Let *μ* be a quantum probability measure over . We say that *μ* *represents* ⪯, if for every two subspaces *A, B* of .

In the quantum framework, orthogonal projections will replace the indicator functions that appear in lemma 6.1.

The likelihood order ⪯ satisfies the *cancellation condition* if, for every subspaces of , and *n* positive numbers , , if and for every , then for every .

### (b) Continuity

The cancellation condition by itself is not sufficient to ensure the existence of a representative measure (e.g. 2 below). Similar examples appear in the classical framework, when one tries to extend proposition 2.1 to an infinite . In the current framework, in order to obtain a characterization of the likelihood orders that can be represented by a quantum measure, we need the additional assumption that ⪯ is continuous w.r.t. the natural topology over subspaces.

Let *A* be a subspace and *r* be a positive number. Denote by *U* the unit ball, . By we denote the *r*-neighbourhood of *A* restricted to the unit ball, , where is the ball of radius *r* around *x*. For two subspaces *A* and *B*, we denote and . The Hausdorff metric is defined asNote that the topology induced by the Hausdorff metric is the same as the topology induced by the Euclidean distance between the corresponding projections and .

The likelihood order ⪯ is *lower semi-continuous* if, for every subspace *B*, the set of the subspaces *A* such that is open with respect to *δ*.

*Let* ⪯ *be a likelihood order. There exists a quantum probability measure that represents* ⪯ *if and only if the following conditions are satisfied.*

*For every subspace A of**;**;*⪯

*is lower semi-continuous**;*⪯

*satisfies the cancellation condition*.

Assume first that ⪯ is represented by a quantum probability *μ*. Then by Gleason's theorem there exists a non-negative operator *T* with trace 1 such that for every subspace *A* of *V*. In particular, the function is continuous and therefore, the order that it represents is lower semi-continuous. As for the cancellation condition, let be subspaces such that and , where , , are positive numbers. It follows thatSince for every *i*, it follows that , which implies .

Assume now that the likelihood order ⪯ satisfies the conditions of theorem 2.5. Consider the finite dimensional Hilbert space of Hermitian operators over with the inner product of two Hermitian operators *S* and *T* being . Denote1 and . From the cancellation condition it follows that and are disjoint.

The separation theorem (Rockefeller 1970) ensures that there is a non-zero linear functional on the space of Hermitian operators, represented in this case by some Hermitian operator *T*, such that for every (since is a subspace) and for every .

Since for , it follows that if . Let . By the definition of , . Due to lower semi-continuity, there exists some neighbourhood of *B* such that for every and therefore, for every . Since *U* contains as an interior point it follows that is strictly positive. We conclude that *T* represents ⪯.

Finally, for every *A*, since . Therefore, *T* is positive semidefinite. Moreover, since , it follows that . Define . We obtain that ⪯ is represented by , is positive semidefinite and , as desired. ▪

The cancellation condition (even in the classical framework) is difficult to justify. It is desirable to derive a probability representation of a likelihood order over linear subspaces from more plausible assumptions.

## 3. de-Finetti's and other axioms

The most natural condition is de-Finetti's. When applied to classical probability it states that if *C* is disjoint of , then *B* is preferred to *A* iff is preferred to . In the quantum framework, it takes following form.

For every linear subspaces of , if and , then iff .

In the classical framework, it easily follows from de-Finetti's axiom that if , then . In the quantum framework, we need to require it explicitly.

For every two linear subspaces *A*, *B* of , if , then .

As illustrated by the following example there might be weak orders that satisfy de-Finetti's axiom and not negation.

Let be . Any monotonic (w.r.t. set inclusion) weak order on satisfies de-Finetti's axiom but it might not satisfy negation. Let be such a weak order. As for a higher dimensional Hilbert space, let be . Let *p* be the northern pole of the unit ball, *E* be the equator, and define for any unit vector *u*.

Define ⪯ as follows: if *A* and *B* are two subspaces of different dimensions, then if the dimension of *A* is greater than that of *B*. If and , where *u* and *v* are unit vectors, then either when and or when and . Finally, if *A* and *B* are two-dimensional subspaces, then either when and and . The weak order ⪯ preserves de-Finetti's axiom but since does not preserve negation on *E*, so does not on .

We will also need the obvious assumption that any subspace is as preferred as the zero-dimensional one. Formally,

For every subspace *A* of , .

Note that monotonicity and de-Finetti's axiom together imply that if then for every pair *A*, *B* of subspaces. Thus, ⪯ is monotonic with respect to set inclusion.

In the sequel, we will say that a weak order ⪯ satisfies the *standard assumptions*, if it satisfies de-Finetti's axiom, negation and monotonicity.

Do the standard assumptions guarantee that ⪯ can be represented by a measure? The following is a counterexample.

Let be two different quantum probability measures over and define the *lexicographic order* induced by and as follows. if either or and . Then, ⪯ satisfies the standard assumptions. Furthermore, it satisfies the cancellation condition.

The lexicographic order cannot be represented by a measure since it lacks the following property:

There is a countable set of subspaces, , such that for any two subspaces *B* and *C* such that , there is that satisfies .

As indicated by Debreu (1954), separability is necessary for ⪯ in order to be represented by a real function (not necessarily a measure).

## 4. Pure states in

The most important probabilities from the physical point of view are those of the form for some unit vector . These distributions are sometimes called *pure states*. It follows from Gleason's theorem that pure states are the extreme points of the convex set of quantum probabilities. From a decision theoretic point of view, pure states correspond to situations of, where there exists a maximal measurement (i.e. a complete set of orthogonal one-dimensional subspace) whose outcome the agent can predict with certainty (see Caves *et al*. (2002) for a more detailed discussion).

It is clear that if *μ* is a pure state and ⪯ is the induced likelihood order, then the one-dimensional subspace spanned by *p* is equivalent (under ) to . In this section, we prove the inverse statement. We say that ⪯ is *non-trivial* if there exists a subspace *A* such that . The proof closely follows the proof of Gleason's theorem in Cooke *et al*. (1985).

Let ⪯ *be a weak order over subspaces of a finite dimensional real-Hilbert space that satisfies the standard assumptions and* *separability*. *Assume that there exists a one-dimensional subspace P such that* . *Let p be a unit vector in P. If* ⪯ *is non-trivial, then* ⪯ *is represented by the pure state* .

Let *E* be the orthogonal complement of . By negation, since it follows that . Let *A* be a subspace of , be and be the one-dimensional subspace of that is spanned by (the orthogonal projection of *p* over *A*). Then, since , it follows from monotonicity that . Since and , it follows from de-Finetti's axiom that . Thus, the likelihood order ⪯ is determined by its restriction to one-dimensional subspaces. Moreover, since , it is sufficient to prove that the likelihood order over one-dimensional subspaces is represented by *μ*. Slightly abusing notation, we will identify a unit vector *u* in with the one-dimensional subspace spanned by *u*. With this convention, for every unit vector *u*, .

Let *u*, *v* be two unit vectors. We need to show that iff . By looking at the three-dimensional space spanned by , with its two-dimensional subspace we can assume w.l.o.g. that . In this case, the theorem will follow directly from the following proposition. Let be the unit sphere in . We say that ⪯ is *uniform* if all the one-dimensional subspaces are equivalent. ▪

*Let* ⪯ *be a weak order over* *that satisfies the standard assumptions, and such that the restriction of* ⪯ *to one-dimensional subspaces is separable. Assume that* ⪯ *is not uniform and it attains its minimum over* *at m. Furthermore, assume that there exists a two-dimensional subspace E such that* *for every* *. Let* *such that* *. Then*, *for every pair* *,* *iff* .

The proof of the proposition is broken into five claims. As usual, we identify elements of with their corresponding one-dimensional subspaces. ▪

*Let* *be such that* . *If* *and* *are, respectively*, *the orthogonal complements of q and r in the plane* , *then* .

Let such that . Then and . By negation, . By de-Finetti's axiom, . ▪

*Let* *be orthogonal vectors such that* . *If* *, then* .

Note that is the orthogonal complement of in . Let be the orthogonal complement of *u* in . Then . By claim 4.3, . Since , it follows that . ▪

*Assume that there exists an orthogonal triple* *such that* . *Then,* ⪯ *is uniform*.

Let . Then there exists such that . By claim 4.4, . But and therefore, again by claim 4.4, . ▪

For , we denote by the angle between *p* and *q*. Thus and . Let be the *northern hemisphere* relative to *p*, and be the *equator* relative to *p*. Let . Among the great circles which pass through *q*, there is a unique one that intersects in vector *x* orthogonal to *q*. We follow Gleason (1957) and denote this circle by . Note that *q* is the northern most point in and that is tangent to the latitude circle of *q*. We will need the following lemma, that appears in Piron (1976) (see also Cooke *et al*. 1985).

*Let* *such that* *; then there exists a finite sequence* of points in *such that* .

*Under the assumption of* *theorem* 4.1*, if* *and* *, then* .

Note that iff .

Let and . Let be the orthogonal complement of in the plane of , and be the orthogonal complement of *q* in . Since , it follows from claim 4.3 that . Moreover, only if . By induction it follows from Piron's lemma that . Furthermore, only if there exists such that . We prove that in this case ⪯ is uniform, which is excluded by assumption. This will complete the proof.

Note that for every *y* such that , . Thus, all the vectors in the band below *z* are equivalent to *m*. We now show that one can find another point , such that for every and , and thus obtaining a wider band. By iterating this argument one can get wider and wider bands until one obtains a band that is wide enough to contain three orthogonal vectors. By claim 4.5, it would imply that ⪯ is uniform.

Let be a point in for which . It follows that, for every , . Thus, is entirely contained in the band defined by *p* and *z* and therefore, . ▪

*If* *and* *, then* .

We know from claim 4.6 that for such that , . Now suppose that there exist, for some *α*, , vectors such that and . Let and . Then and therefore, at least one of the sets *Q*, *R* must be uncountable. Assume w.l.o.g. that *Q* is uncountable. For every , let be the orthogonal complements of , *respectively*, in . It follows from claim 4.3 that . Notice moreover, that , where is orthogonal to . Since increases as *q* approaches along the latitude circle of , we get uncountable set of pairs such that , but with different values of *μ* for different pairs. This, together with claim 4.6 contradicts separability. ▪

From claims 4.6 and 4.7 it follows that iff and therefore, the proof of proposition 4.2 is complete.

Back to the proof of theorem 4.1. By assumption, ⪯ is non-trivial. Therefore, there exists a subspace *A* which is strictly more likely than . Suppose that *A* is spanned by the orthogonal vectors .

*At least one* *is strictly more likely than* .

Otherwise, for every . By de-Finetti's axiom . By successively adding the 's and by using de-Finetti's axiom one obtains that , in contradiction with the assumption. ▪

By claim 4.8 we can assume that there is a vector such that . Let . Since , . Furthermore, . Let be the orthogonal complement of *x* in . Since , by claim 4.3 . As ⪯ is an order, and thus, . This implies that ⪯, when restricted to , is not uniform, as assumed by proposition 4.2. This enables us to use this proposition in order to complete the proof of theorem 4.1

No sort of continuity is assumed in theorem 4.1. Nevertheless, ⪯ is represented by a measure and is therefore continuous.

## 5. The uniform measure

The only quantum probability measures over a finite dimensional Hilbert space which receives discrete values is given by the *uniform measure*, . It turns out that this is the case characterized by the property that all one-dimensional subspaces are equally likely. Formally:

*Let* ⪯ *be a weak order over subspaces of a finite dimensional Hilbert space that satisfies* *de-Finetti*'*s* *axiom*. *If all one-dimensional subspaces are equivalent, then either* ⪯ *is trivial (i.e.* *for every subspace A of* *) or* ⪯ *is represented by the uniform measure.*

Assume that every one-dimensional subspace is equivalent to some one-dimensional subspace, say, *m*. If are two-dimensional such that is one-dimensional, we can assume that and where and . Since we get, by de-Finetti's axiom, that . If , we can find a two-dimensional subspace such that and are one-dimensional. Therefore, . Thus all two-dimensional subspaces are equivalent. By a similar argument, two subspaces of the same dimension are equivalent.

Finally, if , it follows by de-Finetti's axiom that . If , then again by de-Finetti's axiom, if and then . Using the equivalence of two subspaces with the same dimension, it follows that if , then and therefore, ⪯ is represented by . ▪

The following non-trivial fact about quantum probabilities follows from Gleason's theorem.

*Let* *be a finite-dimensional Hilbert space and μ be a quantum probability over* . *Assume that there exist one-dimensional subspaces (not necessarily orthogonal)* of *such that* *and* *for every one-dimensional subspace x of* . *Then,* *μ is the uniform measure*.

We show that this proposition is a consequence of the standard assumptions, with the additional assumption that ⪯ is continuous over one-dimensional subspaces.

The likelihood order ⪯ is *continuous over one-dimensional subspaces* if for every unit vector *v* the sets and are open.

We note that if ⪯ is continuous over one-dimensional subspaces then its restriction to one-dimensional subspaces is also separable. Indeed, let be a countable dense set w.r.t. the Euclidean topology of . For every such that , let and . Since and is connected, . As *D* is dense, there exists such that . Thus, .

We state the result in . It can easily be extended to every finite-dimensional Hilbert space.

*Let* ⪯ *be a weak order over* *that satisfies the standard assumptions. Assume that* ⪯ *is continuous over one-dimensional subspaces. If* *is a basis (not necessarily orthogonal) that satisfies* , *where m is a minimum of* ⪯*, then* *for every* .

*The theorem is proved in a few steps. Denote by M a maximum of* ⪯.

*Let* *such that* *and* . *Let* *be the orthogonal complements of u*, *v in* *, respectively*. *Then,* *and* .

Since , it follows from claim 4.3 that . But and *M* is a maximum. Therefore, . By a similar argument . ▪

*If* *such that* and *, then either all one-dimensional subspaces are equivalent or* ⪯ *is represented by a pure state*.

Let . By claim 4.4, for every . By proposition 4.2, either ⪯ is trivial or ⪯ is represented by a pure state. ▪

*If* *such that* , *and* , *then* .

Let be such that and is minimal. If then *x* cannot be orthogonal to both and . Assume, therefore, w.l.o.g. that . Let be the orthogonal complement of in the plane . By claim 5.5, . Moreover, , since . This contradicts the choice of *x*. It therefore follows that *x*=*p*, meaning that . ▪

*If m and M are any minimal and maximal elements in* *and* *(i.e.* ⪯ *is not trivial), then* .

Assume the contrary. Let *a* be the orthogonal complement of *m* in . By claim 5.5, . Let *p* satisfy and . By claim 5.7, . However, since , . Therefore, it follows from claim 5.6 that ⪯ is represented by a pure state, in which case the claim holds. ▪

We now turn to the proof of theorem 5.4. Let *M* be a maximal element. If ⪯ is not trivial then from the last claim it follows that for every *i*. This is impossible since are linearly independent and the proof is complete.

We do not know whether theorem 5.4 holds true without the assumption that ⪯ is continuous over one-dimensional spaces. The proof hinges on this assumption in two ways. First, in that ⪯ attains a minimum and a maximum. Second, in claim 5.7 *x* is chosen so that among all , is minimal. While we could explicitly assume that ⪯ attains a minimum and a maximum, we could not dispose of the continuity assumption in the proof of claim 5.7.

## 6. A counterexample

In this section, we present an example of a separable (though not continuous) weak order over subspaces of that satisfies the standard assumptions but does not admit a representation via a quantum measure. We need the following two lemmas.

*Let* ⪯ *be a weak order over one-dimensional subspaces of* *such that for every two-dimensional subspace U of* *and all one-dimensional subspaces u, v of U one has* *, where* *are the orthogonal complements of u*, *v respectively*, *in U. Then*, ⪯ *can be extended to a weak order over* *that satisfies the standard assumptions.*

We define ⪯ as follows. Let *U*, *V* be two subspaces of . If , then . If , then iff . Negation is obviously satisfied. As for de-Finetti's axiom, let *u*, *v* be two different one-dimensional subspaces and *x* be the one-dimensional subspace such that . Let be the orthogonal complements of *u*, *v* in . Then, and . Since, by the assumption of the lemma , it follows by definition of ⪯ that . ▪

The second lemma states that if ⪯ is represented by a probability measure, then the order over one-dimensional subspaces of a fixed two-dimensional subspace *U* has a very specific form. As usual we identify one-dimensional subspaces with unit vectors. If is the unit circle of *U*, the lemma essentially says that either all elements of are equivalent, or there is a single maximal element that satisfies iff is closer than to *x*.

*Let μ be a probability measure over* *and* ⪯ *the corresponding weak order over subspaces. Let U be a two-dimensional subspace of* *. Then, either all one-dimensional subspaces of U are equivalent, or there exists unit vector* *such that for every pair* *of unit vectors* iff .

By Gleason's theorem, there exists a positive semidefinite operator *T* such that . Consider the operator . This is a positive semidefinite operator. Its spectral decomposition is of the formwhere are orthogonal eigenvectors in *U* with corresponding eigenvalues such that . We assume that . It follows that for every unit vector *y* in *U*,Thus, if *α*=*β* then all are equivalent. If *α*>*β*, then is a monotonic function of . ▪

Let be a weak order on one-dimensional subspaces of that satisfies the condition of lemma 6.1 but not the condition of lemma 6.2. Define ⪯ on one-dimensional subspaces of as follows: let *p* be the northern pole of the unit sphere in . Let *u* and *v* be unit vectors, then either when and or when and . By lemma 6.1 can be extended to a weak order over that satisfies the standard assumptions. However, since the condition of lemma 6.2 is not satisfied, ⪯ cannot be represented via a quantum measure.

## 7. Final comments and open problems

### (a) Representation and continuity

In Gleason's theorem (1957), continuity is not assumed and is a consequence of the existence of a frame function. When the primitive of the model is a likelihood order, matters are different. The likelihood order in example 6.3 satisfies de-Finetti's axiom, negation, monotonicity and separability and cannot be represented by a quantum measure. This order, which is not continuous, suggests that continuity must be explicitly assumed and cannot be derived from more plausible assumptions.

The question whether every continuous likelihood order which satisfies de-Finetti's axiom, negation, monotonicity and separability can be represented by a quantum measure is still open.

### (b) Partial representation

We say that *μ partially represents* ⪯ if for every two subspaces *A*, *B* of .

It turns out (we state without a proof) that if ⪯ satisfies the cancellation condition, then there exists a quantum probability measure that partially represents ⪯. Also, from the proof of theorem 4.1 it follows that, if there exists a one-dimensional subspace *p*, such that , then (without assuming separability) ⪯ admits a partial representation by a pure state.

### (c) Qualitative additivity and discrete orders

Gleason's theorem implies that the only quantum probability measure which obtains a discrete set of values is the uniform measure. The question arises whether the same is true for likelihood orders. We say that ⪯ is *discrete* if the restriction of to one-dimensional subspaces has only finitely many equivalence classes. For instance, if ⪯ is represented by the uniform probability, then its restriction to one-dimensional subspaces has only one equivalence class.

The Kochen–Specker theorem (1967) actually refers to likelihood orders whose restriction to one-dimensional subspaces have precisely two equivalence classes. In order to prove this result using likelihood orders terms only, one needs to strengthen de-Finetti's axiom and negation. The following axiom is a consequence of de-Finetti's axiom in the classical case, but not in the quantum set-up.

Let be linear subspaces of such that and . If , then . Furthermore, strict likelihood on one of the former inequalities implies strict likelihood in the latter.

Suppose that and are orthogonal and the same for the 's. Qualitative additivity states that, if the 's are at least as likely as 's, then the subspace spanned by the 's is at least as likely as that spanned by 's. That is, adding a more likely subspace to a subspace which is already more likely, cannot make the outcome less likely.

Suppose that ⪯ is defined over and there are only two equivalence classes of one-dimensional subspaces, say, green and red. If ⪯ satisfies qualitative additivity, then in any orthogonal triple there is the same number of green representatives, and moreover, a two-dimensional subspace spanned by uni-coloured vectors contains only vectors of the same colour. These are precisely the terms of the Kochen–Specker theorem (1967). It states that there exists no likelihood order that satisfies qualitative additivity and has precisely two equivalence classes of one-dimensional spaces.

This result suggests that the only discrete likelihood order that satisfies qualitative additivity is that induced by the uniform measure.

## Acknowledgments

We are grateful to David Wallace and an anonymous referee for helpful comments. We acknowledge the support of Israel Science Foundation, grant no. 762/04.

## Footnotes

↵For every set

*X*, is the convex hull of X and is the linear subspace spanned by X.- Received May 17, 2005.
- Accepted January 11, 2006.

- © 2006 The Royal Society