Distinguishing Initial State-Vectors from Each Other in Histories Formulations and the PBR Argument

Abstract

Following the argument of Pusey et al. (in Nature Phys. 8:476, 2012), new interest has been raised on whether one can interpret state-vectors (pure states) in a statistical way (ψ-epistemic theories), or if each one of them corresponds to a different ontological entity. Each interpretation of quantum theory assumes different ontology and one could ask if the PBR argument carries over. Here we examine this question for histories formulations in general with particular attention to the co-event formulation. State-vectors appear as the initial state that enters into the quantum measure. While the PBR argument goes through up to a point, the failure to meet some of the assumptions they made does not allow one to reach their conclusion. However, the author believes that the “statistical interpretation” is still impossible for co-events even if this is not proven by the PBR argument.

Appendix

In this appendix, we will attempt to prove the conjecture made in the text, that the allowed co-events for two different state-vectors, are disjoint. We will use the example of a qubit. However, the importance of this example, is much greater. When comparing two state-vectors,¹⁹ we can do so, by restricting attention to the 2-dimensional subspace that is spanned by the two state-vectors. While it is true, that many results hold in 2-dimensions and not for higher dimensions, the comparison of two pure states, is not one of them. The reason this is the case is because if one has two state-vectors of higher dimensions |Φ ₁〉, |Φ ₂〉 $\in\mathcal{H}$, he can find the two-dimensional subspace that is spanned by those two vectors $\mathcal{H}_{\varPhi _{1},\varPhi_{2}}=\textrm{span}(\{|\varPhi_{1}\rangle,|\varPhi_{2}\rangle\})$. One can then choose |Φ ₁〉=|0〉 and define $|1\rangle\in \mathcal{H}_{\varPhi_{1},\varPhi_{2}}$ such that it is orthogonal to |0〉, i.e. 〈0|1〉=0. Then, we can express |Φ ₂〉=cosθ|0〉+sinθ|1〉 for some angle θ, and without loss of generality, we can proceed as if our initial states |Φ ₁〉,|Φ ₂〉 were 2-dimensional.

Our attempt to prove the conjecture, consist of the explicit construction of the sets of co-events corresponding to two different initial states, that are non-orthogonal. By choosing finer or coarser grained histories (i.e. by having more moments-of-time) we can construct finer or coarser sets of co-events. The direction of proof we will follow, is to exploit some property that certain coarse-graining of histories have, namely the existence of zero covers [9]. This simplifies considerably the analysis and allows us to find a suitable coarse-graining that distinguishes the states with only 3-moments-of-time, and therefore 2³ different histories for almost all cases. Unfortunately, as we will see below, for the very special case that the angle between the two states we want to distinguish is tanθ=±1/3, this strategy is not successful. To fully prove the conjecture, one needs to consider different coarse-grainings with more moments-of-time (and therefore exponentially more possible histories). With the current development of the co-event formulation, its extremely difficult to compute the possible co-events, as the moments of time increase. Moreover, the technical trick used to prove the conjecture for all the other cases (the use of that particular zero cover), cannot be used here. However, it seems very implausible, that one can distinguish between any two state-vectors in this formalism, unless their angle is tan⁻¹(±1/3). If that was the case, it would certainly be a strange property that would require further study. It is most likely that some other finer-grained description would complete the proof, but until the relevant technical methods to efficiently compute co-events for many moments-of-time appears, our claim will remain a conjecture. We now return to prove the general case.

The histories we will consider, is essentially 3-moments-of-time. We start with some initial state |Φ〉 and then measure it three times in the basis we will give below. We assume trivial evolution (the identity), but we could easily have any Hamiltonian, and then have to choose the basis measured suitably. Given a particular Hamiltonian (non-trivial this time), one can also reproduce the result we will give, considering measurement done only in the {|0〉,|1〉} basis, by suitably choosing the time t ₁,t ₂,t ₃ that the measurements take place as we will see in the end of the appendix.

The initial state will be either |Φ ₁〉=|0〉 or any other state |Φ ₂〉=cosθ|0〉+sinθ|1〉. We consider the following two orthogonal bases:²⁰

$$\begin{aligned} \begin{aligned} |\varPsi_0\rangle&= \cos\theta|0\rangle+\sin\theta|1\rangle \\ |\varPsi_1\rangle&= -\sin\theta|0\rangle+\cos\theta|1\rangle \end{aligned} \end{aligned}$$

(36)

and

$$\begin{aligned} \begin{aligned} |\varPsi_+\rangle&= \cos(\theta+\pi/4)|0\rangle+\sin(\theta+\pi /4)|1\rangle\\ |\varPsi_-\rangle&= \cos(\theta-\pi/4)|0\rangle+\sin(\theta-\pi /4)|1\rangle \end{aligned} \end{aligned}$$

(37)

The histories considered will be: They start with the initial state |Φ _i 〉, and then are measured in the {|Ψ ₊〉,|Ψ ₋〉} basis then in the {|Ψ ₀〉,|Ψ ₁〉} basis and then again in the {|Ψ ₊〉,|Ψ ₋〉}. We will label the histories depending on the outcome of each measurement in the following way (measurements are from right to left):

$$\begin{aligned} h_1 = (\varPsi_+\varPsi_0\varPsi_+),\quad\ \ h_2=(\varPsi_+\varPsi _1\varPsi _+),\quad\ \ h_3=(\varPsi_+\varPsi_0\varPsi_-), \quad\ \ h_4=(\varPsi_+\varPsi_1\varPsi _-) \\ h_5 = (\varPsi_-\varPsi_0\varPsi_+),\quad\ \ h_6=(\varPsi_-\varPsi_1\varPsi_+),\quad\ \ h_7=(\varPsi_-\varPsi_0\varPsi_-), \quad\ \ h_8=(\varPsi_-\varPsi_1\varPsi_-) \end{aligned}$$

(38)

Histories h ₁,h ₂,h ₃ and h ₄ end at final time in the |Ψ ₊〉 while h ₅,h ₆,h ₇ and h ₈ end in |Ψ ₋〉. We compute the amplitudes of histories for |Φ ₁〉=|0〉 (the subscript at the amplitudes α ₁ signifies that it correspond to initial state |Φ ₁〉):

$$\begin{aligned} \begin{aligned} {\alpha}_1(h_1)&= 1/2\cos(\theta+\pi/4),\qquad {\alpha}_1(h_2)=1/2\cos (\theta+\pi /4)\\ {\alpha}_1(h_3)&= 1/2\cos(\theta-\pi/4),\qquad {\alpha}_1(h_4)=-1/2\cos (\theta -\pi/4)\\ {\alpha}_1(h_5)&= 1/2\cos(\theta+\pi/4),\qquad {\alpha}_1(h_6)=-1/2\cos (\theta +\pi/4)\\ {\alpha}_1(h_7)&= 1/2\cos(\theta-\pi/4),\qquad {\alpha}_1(h_8)=1/2\cos (\theta-\pi/4) \end{aligned} \end{aligned}$$

(39)

The only zero quantum measure sets are the {h ₃,h ₄} and {h ₅,h ₆} for a general angle θ.

Here we should note that there are other zero quantum measure sets only in the cases where θ=0, that reduces to |0〉 which we will see below, and for tanθ=±1/3 which is the exceptional case mentioned earlier that prevents us from providing a full proof of the conjecture. For these very special cases, the fine grained description we used here is not sufficient to prove the conjecture, and further fine graining (measurements) are required.

Returning to the general θ case, we have 4 classical co-events and the set of allowed co-events are:

$$\mathcal{C}_1=\{\{h_1\},\{h_2\},\{h_7\},\{h_8\}\} $$

(40)

For |Φ ₂〉=cosθ|0〉+sinθ|1〉 the amplitudes are (note that they are independent of θ):

$$\begin{aligned} \begin{aligned} {\alpha}_2(h_1)&=\frac{1}{2\sqrt{2}} ,\qquad {\alpha}_2(h_2)=\frac{1}{2\sqrt{2}} \\ {\alpha}_2(h_3)&=\frac{1}{2\sqrt{2}} ,\qquad {\alpha}_2(h_4)=-\frac{1}{2\sqrt{2}} \\ {\alpha}_2(h_5)&=\frac{1}{2\sqrt{2}} ,\qquad {\alpha}_2(h_6)=-\frac{1}{2\sqrt{2}} \\ {\alpha}_2(h_7)&=\frac{1}{2\sqrt{2}} ,\qquad {\alpha}_2(h_8)=\frac{1}{2\sqrt{2}} \end{aligned} \end{aligned}$$

(41)

The subscript α ₂ signifies that the initial state is |Φ ₂〉. The sets with quantum measure zero are:

$$\{h_1,h_4\},\quad\{ h_2,h_4\},\quad\{h_3,h_4\},\quad\{h_5,h_6\},\quad\{h_6,h_7\},\quad\{h_6,h_8\} $$

(42)

We see that all fine grained histories are contained in one quantum measure zero set and thus there are no classical co-events. Only pairs of histories are allowed, and we have 6 potential co-events:

$$\mathcal{C}_2=\{\{h_1,h_2\},\{h_1,h_3\},\{h_2,h_3\},\{ h_5,h_7\},\{ h_5,h_8\},\{h_7,h_8\}\} $$

(43)

It is easy to see that $\mathcal{C}_{1}\cap\mathcal{C}_{2}=\emptyset$. This is general for an arbitrary θ (other than a very special case mentioned above), and thus any two state-vectors of a qubit give rise to completely disjoint set of potential co-events and thus correspond to different ontology, in the sense discussed in the main text.

We now return at the earlier remark, that all of the above can be re-expressed in terms of measurements in the {|0〉,|1〉} basis given a Hamiltonian, if we suitably choose the times that the measurements take place. If for example we this Hamiltonian

$$H=\left( \begin{array}{c@{\quad}c}1&i\\ -i&1 \end{array} \right) $$

(44)

It gives rise to the following unitary evolution

$$U(t)=\exp(-it)\left( \begin{array}{c@{\quad}c}\cos t&\sin t\\ -\sin t&\cos t \end{array} \right) $$

(45)

We choose to measure at t ₁=(θ−π/4) and at t ₂=θ and finally at t ₃=(θ+7π/4), always in the {|0〉,|1〉} basis. It is easy to calculate that the amplitudes we get for these histories for any of the two initial states, are exactly the same as the ones we calculated earlier in the appendix in Eqs. (39 , 41), with the following adjustments. (a) In the labeling we replace Ψ ₊ and Ψ ₁ with 1 while we replace Ψ ₋ and Ψ ₀ with 0 (i.e. the history h ₃ for example, that was (Ψ ₊ Ψ ₀ Ψ ₋) is now (100)) and (b) there is an overall factor of exp(−i(θ−π/4)) in the amplitudes of all histories, which however, does not affect the quantum measure. Since the quantum measure is the same, it follows that set of allowed co-events is also the same.