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

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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.

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Notes

  1. A deeper discussion, could involve questioning the whole concept of wavefunction and Hilbert space as starting point of quantum theory, but this is beyond the scope of this paper.

  2. One can view the De Broglie-Bohm theory as a histories formulation, however in this paper we use the term for formulations based on the Feynman path integral and the decoherence functional, which is not the case for De Broglie-Bohm.

  3. This is achieved by restricting the path integral to particular sets of histories and weighting differently each of those.

  4. Note that measurements are not understood as actual measurements with an external experimenter, but rather by introducing different projections at different times and mathematically fine graining histories. It is always possible however, to view the histories space differently, in terms of the finest possible description and its coarse-grainings, that does not require the concept of projection operator as a prerequisite.

  5. Here 〈Ψ 1|Ψ 2〉≠0.

  6. Note that the above can be generalised for n copies of the system.

  7. \(|{-}\rangle=1/\sqrt{2}(|0\rangle-|1\rangle)\).

  8. Note, that following Hartle and Sorkin, we are adopting the path integral view of histories that takes the stance that there exist a unique (preferred) fine grained description, i.e. paths in the generalised configuration space. Other points of view, such as Isham’s, are compatible with the one we take, at least in most ordinary cases.

  9. Initially this requirement was weaker, namely that D(A,A)≥0 and was called positivity. However, it is believed that this stronger requirement is more physical and we will adopt this convention here.

  10. The sum in the expression should be replaced with an integral if we consider continuous histories.

  11. Other attempts have been made, such as using some special precluded sets, at the original papers introducing the quantum measure [10, 11] by Sorkin, or extending the concept of probability to maintain a single-history-realised view, more recently in [12] by Gell-Mann and Hartle.

  12. This is not necessarily true if one considers the weaker notion of positive decoherence functionals.

  13. Strictly speaking, from the real part of the off-diagonal terms. The role the complex part plays will be briefly discussed later.

  14. Here we have presented the analysis for the specific example of the qubit in those two states. As in the original paper [1] it can be extended for arbitrary states.

  15. This particular example, is weakly decoherent, but one can construct more complicated examples that even this is not true.

  16. In his work on extending the probabilities [28], Hartle has also point out this strange property for his extended probabilities. According to his work, however, when restricted to “settleable” questions the probabilities of composite systems, behave classically.

  17. Of course one would need to show that the real decoherence functionals give rise to all the properties one would expect from experiments.

  18. However, the examples with purely real decoherence functional, involve higher cardinality histories space Ω, since no such example exist in the 2 histories space mentioned above.

  19. For simplicity, of a finite dimensional Hilbert space. For infinite dimensions more care is needed.

  20. Note that |Φ 2〉=|Ψ 0〉.

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Acknowledgements

The author is very grateful to Rafael Sorkin, for bringing the problem to his attention, many discussions and reading and commenting an earlier draft. He acknowledges the COST Action MP1006 “Fundamental Problems in Quantum Physics” and also the Perimeter Institute for Theoretical Physics, Waterloo, Canada, for hospitality while carrying out part of this work.

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Correspondence to Petros Wallden.

Appendix

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,Footnote 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:Footnote 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=(θπ/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(θπ/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.

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Wallden, P. Distinguishing Initial State-Vectors from Each Other in Histories Formulations and the PBR Argument. Found Phys 43, 1502–1525 (2013). https://doi.org/10.1007/s10701-013-9759-8

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