The Status of Determinism in Proofs of the Impossibility of a Noncontextual Model of Quantum Theory

Abstract

In order to claim that one has experimentally tested whether a noncontextual ontological model could underlie certain measurement statistics in quantum theory, it is necessary to have a notion of noncontextuality that applies to unsharp measurements, i.e., those that can only be represented by positive operator-valued measures rather than projection-valued measures. This is because any realistic measurement necessarily has some nonvanishing amount of noise and therefore never achieves the ideal of sharpness. Assuming a generalized notion of noncontextuality that applies to arbitrary experimental procedures, it is shown that the outcome of a measurement depends deterministically on the ontic state of the system being measured if and only if the measurement is sharp. Hence for every unsharp measurement, its outcome necessarily has an indeterministic dependence on the ontic state. We defend this proposal against alternatives. In particular, we demonstrate why considerations parallel to Fine’s theorem do not challenge this conclusion.

Appendices

Appendix 1: Ontological Extension: Modelling an Unsharp Measurement Using an Outcome-deterministic Response Function

The intuition at play in premiss 2 of the argument from Sect. 3 is that one can always imagine any subjective uncertainty in the outcome of a measurement on some system $s$ as being due to uncertainty about the ontic state of some other system that, together with $s,$ determines the outcome, for instance, hidden variables in the apparatus. We will see that this is indeed the case. The resulting representation of the measurement will be called an ontological extension. This is the analogue, within the ontological model, of the Naimark extension of a measurement within operational quantum theory.¹³

Suppose that for an operational theory $\left( \mathcal {P},\mathcal {M} ,p\right)$ on system $s,$ we have found a (possibly contextual) ontological model $\left( \Lambda _{s},\mu ,\xi \right)$ on $s,$ such that the operational statistics are reproduced as

$$\begin{aligned} p(k|M,P)=\sum _{\lambda _{s}\in \Lambda _{s}}\mu (\lambda _{s}|P)\xi (k|\lambda _{s},M). \end{aligned}$$

(24)

We suppose that the ontological model is outcome-indeterministic, so that $0<\xi (k|\lambda _{s},M)<1$ for some $\lambda _{s}.$

We can define a new model with outcome-deterministic response functions as follows. Introduce an ancilla system $a$ with ontic state space equal to the unit interval (hence continuous), that is, $\Lambda _{a}=[0,1]$. The ontic state space of the composite is then $\Lambda _{sa}\equiv \Lambda _{s}\times \Lambda _{a}.$ Now define

$$\begin{aligned} \mu (\lambda _{s},\lambda _{a})\equiv \mu (\lambda _{s}|P)\mu _{0}(\lambda _{a}), \end{aligned}$$

(25)

where $\mu _{0}(\lambda _{a})$ is the uniform distribution over $\left[ 0,1\right]$. Also, define

$$\begin{aligned} \omega _{0}(\lambda _{s})&= 0, \end{aligned}$$

(26)

$$\begin{aligned} \omega _{k}(\lambda _{s})&= \sum _{j=1}^{k} \xi (j|\lambda _{s},M), \end{aligned}$$

(27)

and define outcome-deterministic response functions on $\Lambda _{s} \times \Lambda _{a}$ by

$$\begin{aligned} \xi \left( k| \lambda _{s},\lambda _{a}\right) \equiv \left\{ \begin{array}{l} 1\quad \text { if } \ \omega _{k-1}(\lambda _{s})\le \lambda _{a}\le \omega _{k} (\lambda _{s})\\ 0 \quad \mathrm{otherwise} \\ \end{array}\right. \end{aligned}$$

(28)

The new model is empirically adequate because

$$\begin{aligned}&\sum _{\lambda _{s}\in \Lambda _{s}}\int _{\Lambda _{a}}\hbox {d}\lambda _{a}\; \mu (\lambda _{s},\lambda _{a})\xi (k| \lambda _{s},\lambda _{a})\end{aligned}$$

(29)

$$\begin{aligned}&\quad =\sum _{\lambda _{s}\in \Lambda _{s}}\mu (\lambda _{s}|P) \int _{\omega _{k-1}(\lambda _{s})}^{\omega _{k}(\lambda _{s})}\hbox {d}\lambda _{a} \; \mu _{0}\left( \lambda _{a}\right) \end{aligned}$$

(30)

$$\begin{aligned}&\quad =\sum _{\lambda _{s}\in \Lambda _{s}}\mu (\lambda _{s}|P) \left[ \omega _{k}(\lambda _{s})-\omega _{k-1}(\lambda _{s})\right] \end{aligned}$$

(31)

$$\begin{aligned}&\quad =\sum _{\lambda _{s}\in \Lambda _{s}}\mu (\lambda _{s}|P)\xi (k|\lambda _{s},M)\end{aligned}$$

(32)

$$\begin{aligned}&\quad =p\left( k|P,M\right) . \end{aligned}$$

(33)

A similar trick for devising an ontological model that is outcome-deterministic was described by Bell [1] and its relevance for eliminating determinism as an assumption in the proof of Bell’s theorem was emphasized by Fine [16]. It follows that one can always extend an ontological model such that it represents measurements outcome-deterministically if one so wishes. This justifies premiss $2'$ of the second argument in Sect. 3.

Appendix 2: The Multiplicity of Naimark Extensions of a POVM and its Significance for Ontological Models

We here show explicitly that a given POVM can have two distinct Naimark extensions, that is, that there are multiple distinct choices of projective measurement on $sa$ that yield the same POVM on $s.$ Specifically, we construct two Naimark extensions of the fair coin flip POVM $\{ \tfrac{1}{2} I, \tfrac{1}{2} I\}$ (discussed in Sect. 6).

The first Naimark extension is as follows. We implement a preparation of the ancilla corresponding to the mixed state

$$\begin{aligned} \rho _a=\tfrac{1}{3}\Big \vert 1\rangle \langle 1\Big \vert +\tfrac{1}{3}\Big \vert 2\rangle \langle 2\Big \vert +\tfrac{1}{3}\Big \vert 3\rangle \langle 3\Big \vert . \end{aligned}$$

The measurement on $sa$ is the binary-outcome projector-valued measure $\{ \Pi _{sa}, I-\Pi _{sa} \}$, where

$$\begin{aligned} \Pi _{sa}&\equiv \Pi _{s,1}\otimes \left| 1\right\rangle \left\langle 1\right| +\Pi _{s,2}\otimes \left| 2\right\rangle \left\langle 2\right| +\Pi _{s,3}\otimes \left| 3\right\rangle \left\langle 3\right| . \end{aligned}$$

(34)

where $\Pi _{s,i}\equiv \left| \psi _i \right\rangle \left\langle \psi _i\right|$ and $\{ \left| \psi _i \right\rangle : i \in \{1,2,3\}\}$ are a triple pure states on the system $s$ with pairwise overlaps $|\langle \psi _i | \psi _j \rangle |^2 = \tfrac{1}{4}$ for $i\ne j$, that is, in the Bloch representation, they are separated by 120 $^{\circ }$ on an equatorial plane of the Bloch sphere. To see that the effective measurement on the system is the POVM $\{\frac{1}{2} I,\frac{1}{2} I\}$, it suffices to note that

$$\begin{aligned}&\mathrm {Tr}_a\left( \Pi _{sa} \rho _a \right) =\tfrac{1}{3}\Pi _{s,1}+\tfrac{1}{3}\Pi _{s,2}+\tfrac{1}{3}\Pi _{s,3} =\tfrac{1}{2}I. \end{aligned}$$

The second Naimark extension is as follows. We implement a preparation of the ancilla corresponding to the mixed state

$$\begin{aligned} \sigma _a=\tfrac{1}{2}\left| 1\right\rangle \left\langle 1\right| +\tfrac{1}{2}\left| 2\right\rangle \left\langle 2\right| , \end{aligned}$$

(35)

and implement a measurement on the composite of system+ancilla corresponding to the three-outcome projector-valued measure

$$\begin{aligned}&\{I\otimes \left| 1\right\rangle \left\langle 1\right| ,I\otimes \left| 2\right\rangle \left\langle 2\right| ,I\otimes \left| 3\right\rangle \left\langle 3\right| \}\end{aligned}$$

(36)

$$\begin{aligned}&\equiv \{\Pi _{sa,1},\Pi _{sa,2},\Pi _{sa,3}\} \end{aligned}$$

(37)

(although technically this is a measurement on the composite, it is obviously only nontrivial on the ancilla). Again, it is straightforward to verify that the effective measurement on the system is the POVM $\{\tfrac{1}{2}I,\tfrac{1}{2}I\}$.

The pair of projector-valued measures appearing in the two Naimark extensions, $\left\{ \Pi _{sa},I-\Pi _{sa}\right\}$ and $\{\Pi _{sa,1},\Pi _{sa,2},\Pi _{sa,3}\}$, are clearly distinct. This implies that there are quantum states on $sa$ that yield different statistics for the two. For instance, the state $\tfrac{1}{2}I\otimes \left| 3\right\rangle \left\langle 3\right|$ yields 50/50 statistics for $\left\{ \Pi _{sa},I-\Pi _{sa}\right\}$ but always yields the third outcome for $\{\Pi _{sa,1},\Pi _{sa,2},\Pi _{sa,3}\}$. This implies that although these projector-valued measures are each represented by a set of outcome-deterministic response functions in an ontological model that is noncontextual in the sense of definition 1 (by virtue of Theorem 1), nonetheless the two sets of response functions must be different to account for the differing statistics.

Appendix 3: An Explicit Example of Ontological Extension

We have seen in appendix 2 that because a quantum measurement can be Naimark-extended in many ways, and because those extensions may be statistically distinguishable, it follows that the sets of response functions that represent these Naimark extensions must also be inequivalent. This section provides a second way of understanding the fact that the cost of modelling statistically-indistinguishable measurements outcome-deterministically is that on the extended system, their representations are no longer equivalent. We consider the trick described in appendix 1 for replacing a response function on the system with an outcome-deterministic response function on an extension of the system. Specifically, we show by example that a pair of measurements that are represented by the same set of response functions on the system may need to be represented by different sets of response functions on its extension.

Let $M$ be a measurement associated with the fair coin flip POVM $\{ \tfrac{1}{2} I, \tfrac{1}{2} I\}$ that was discussed in Sect. 6. Let the second measurement $M^{\prime }$ be defined as follows.

The procedure $M'$: implement $M$, and upon obtaining outcome $b\in \left\{ 0,1\right\} ,$ output $b\oplus 1$, that is, the bit-flip of $b$.

Equivalently, we can say simply that $M'$ is a measurement of $M$ with the outcomes permuted. Clearly $M^{\prime }$ is also represented by the fair coin flip POVM. Now consider how to represent $M$ and $M^{\prime }$ within a noncontextual ontological model. As argued in Sect. 6, the set of response functions must be simply $\{ \tfrac{1}{2},\tfrac{1}{2}\}$. ¹⁴

By the construction in appendix 1 , we can model both $M$ and $M'$ on a larger system, a composite of system $s$ and an ancilla $a$ with ontic state space $\Lambda _s \times \Lambda _a$, in such a way that the response functions are outcome-deterministic. Specifically, if we apply this construction to $M$, we obtain

$$\begin{aligned} \xi \left( 0| \lambda _{s},\lambda _{a},M \right)&\equiv \left\{ \begin{array} [c]{l} 1\quad \text { if }0\le \lambda _{a}\le 1/2\\ 0 \quad \mathrm{otherwise} \end{array} ,\right. \end{aligned}$$

(43)

$$\begin{aligned} \xi \left( 1| \lambda _{s},\lambda _{a},M\right)&\equiv \left\{ \begin{array} [c]{l} 1\quad \text { if }1/2\le \lambda _{a}\le 1\\ 0\quad otherwise \end{array} \ .\right. \end{aligned}$$

(44)

Meanwhile, if we remember the definition of $M^{\prime },$ it is clear that it must be represented by a set of response functions that is simply the permutation of those for $M$,

$$\begin{aligned} \xi \left( 0| \lambda _{s},\lambda _{a},M'\right)&= \xi \left( 1|\lambda _{s},\lambda _{a},M\right) ,\end{aligned}$$

(45)

$$\begin{aligned} \xi \left( 1| \lambda _{s},\lambda _{a},M'\right)&= \xi \left( 0| \lambda _{s},\lambda _{a},M\right) . \end{aligned}$$

(46)

Now we see that although we’ve managed to have both $M$ and $M^{\prime }$ represented by sets of response functions on $\Lambda _s \times \Lambda _a$ that are outcome-deterministic, the response functions are not equivalent,

$$\begin{aligned} \xi \left( 0| \lambda _{s},\lambda _{a},M'\right)&\ne \xi \left( 0|\lambda _{s},\lambda _{a},M\right) ,\end{aligned}$$

(47)

$$\begin{aligned} \xi \left( 1| \lambda _{s},\lambda _{a},M'\right)&\ne \xi \left( 1| \lambda _{s},\lambda _{a},M\right) . \end{aligned}$$

(48)