The argument for QM – version 2 58 

In the preceding section, I assumed a very conservative view of what consti- tutes the theory of QM and how its models should be interpreted. In this section, I’ll run the argument again, this time taking a more catholic view. I will again ignore time dependence for simplicity, but to be as inclusive as possible, I’ll consider the space of state representations in QM to consist of full set of density operators on the relevant Hilbert space. This means that both ‘pure’ states equivalent to rays in the Hilbert space and ‘mixed’ states— convex combinations of pure states—will count as potential models of the theory. The theory in this view consists of the set of density operators on a Hilbert space and a dynamical equation known as the von Neumann Equa- tion:

i~½_= [½;H^]

This equation specifies how density operators representing physical states evolve in time or, in the time-independent case, stipulates what states are compatible with a given Hamiltonian.

In this construal of QM, models of the theory are just single density opera- tors. In the case of a single-particle system, each density operator can be in- terpreted as assigning a full set of properties to a particle in the weak sense of bearing properties that I introduced in Section 3.5. That is, we can drop the eigenvector-eigenvalue link and interpret each density operator as assigning to a particle a set of probability distributions over all the values the particle

ε

59 might be measured to have for each observable. Equivalently, the density operator assigns to each particle a set of expectation values with respect to all observables, and we treat these as the vague properties of the particle. In this approach, even mixed states—which assign no properties to particles under the eigenvector-eigenvalue link—can still be interpreted in terms of a particle with a determinate set of properties.

Of course, we can no longer use the Fubini-Study metric to assign distances between particle interpretations, since that metric refers to rays in the Hilbert space. Instead, we need a metric on density operators. To once again make the argument easy to visualize, I will take the distance between any two den- sity operators ½1 and ½2 to be twice the standard ‘trace distance’48:

dTR(½i; ½j) = Trk½i¡½jk

The operation indicated by Trk ¢ k is the ‘trace-norm’.49 _{Once again, there is a} one-one correspondence between models of the theory (density operators) and particle specifications (unique assignments of properties corresponding to sets of expectation values). So once again, we can define a metric on the space of specifications by way of the metric on state representations. In par- ticular, the distance between two specifications is just the trace distance be- tween the two density operators mapped to those specifications in the process of interpretation.

If we consider the case of two-dimensional particle states (e.g. just the spin- degree of freedom of a spin-½ particle), then this choice of metric again makes the argument easy to visualize. In the two-dimensional case, we can represent density operators as vectors in a three-dimensional Euclidean space. In particular, for each density operator ½ (represented as a 2£2 ma- trix in the basis of eignestates of spin in the ^z-direction) there exists a unique vector ~r such that k~rk ·1 and

½= I^+~r¢~¾ 2 ,

where _I^ _{is the identity operator and ~¾ is a vector containing the three Pauli ma-} trices (¾_x = (0 1

1 0); ¾y = (0i ¡0i); ¾z = (10 ¡01)). Geometrically, the space of

vectors ~r is just the set of points of the Bloch sphere, including those of the interior. That is, for every point inside or on the surface of a Euclidean sphere of radius 1, there is corresponding vector ~r and thus a corresponding density

operator ½. The points on the surface of the sphere (those we considered in the

48 _{See e.g. (Nielsen and Chuang 2000), Section 9.2.1.} 49 _{See (Omnès 1994, 245).}

60 previous example) correspond to pure states, while those interior to the sphere correspond to the mixed states. If we map density operators onto points of the Bloch sphere in this way, the trace distance turns out to be equivalent to the Euclidean distance k~ri¡~rjk.

To visualize the incompatibility result in this framework, choose a threshold dis- tance ²¿1. Now choose any two points ~r1 and ~r2 in the sphere such that

k~r1¡~r2k>2². If MA(iii) holds for QM, we must be able to find a two-particle

model that—when interpreted—yields two copies of a particle specification cor- responding to the point ~r¤ _{and such that k~r}¤_¡~r

1k ·² and k~r¤¡~r2k ·².

Geometrically, this means we need to find two points that lie within a ball of radius ² centered at the point ~r¤ _{but that are nonetheless more than 2² apart}

(see Figure 6.2). This is clearly impossible. Thus, even in this liberal reading, QM cannot satisfy MA(iii) if PI is imposed.

Figure 6.2 The argument visualized for pure and mixed states in the Bloch Sphere. For QM to be compatible with MA, it must be possible to locate two specifi- cations r1 and r2 that are within a distance ε of the specification r* (and thus within the

shaded sphere) but that are nonetheless a distance of more than 2ε apart. This is clearly impossible since it requires one member of the pair to be both within and without the shaded sphere.

ε

r1

r2

61