Revealing quantum contextuality using a single measurement device

Quantum theory is contextual since the outcomes of measurements depend on the measurement context, namely, the set of commuting observables being measured along with the desired observable Simon Kochen (1968); Liang et al. (2011). While Bell type inequalities Bell (1966); Brunner et al. (2014); Clauser et al. (1969) can reveal quantumness of composite systems, quantum contextuality can be demonstrated on a single indivisible system. The simplest such scenario involves a three dimensional quantum system and five different projective measurements Klyachko et al. (2008). While the first proof that quantum theory is contextual was provided by Kochen and Specker Kochen and Specker (1975), over time a number of simpler and more systematic ways of revealing quantum contextuality, particularly the ones based on graph theory, have become available Cabello et al. (2014); Acín et al. (2015). The graph theoretic approach has been successful in identifying new contextual scenarios Araújo et al. (2013); Amaral et al. (2014); Kurzyński and Kaszlikowski (2012); Lisoněk et al. (2014); Cabello et al. (2015); Kleinmann et al. (2012); Cabello et al. (2016); Yu and Oh (2012); Cabello (2013); Bharti et al. (2019a), simplifying formulations of contextuality monogamy Ramanathan et al. (2012), contextuality non-locality relationship Kurzyński et al. (2014), developing robust self tests Bharti et al. (2019b) and information theoretic applications of quantum contextuality Singh et al. (2017); Cabello et al. (2011); Troupe and Farinholt (2015); Saha et al. (2019).

The approach developed by Spekkens et al. Spekkens (2005, 2014); Schmid et al. (2018) provides additional insights. Since its inception, this approach has received tremendous research attention and has been influential in exhibiting advantage in various information theoretic applications like parity oblivious bit transfer Spekkens et al. (2009a); Tavakoli et al. (2021), random access codes Ambainis et al. (2019a) state discrimination Schmid and Spekkens (2018) and quantum key distribution Chaturvedi et al. (2021); Singh et al. (2017). Spekkens’ approach generalizes the notion of contextuality to positive-operator-valued measures (POVMs), as well as provides a notion of contextuality for preparations and transformations. This generalized approach has facilitated research in other aspects of foundations of quantum theory Pusey (2014); Pusey and Leifer (2015). Furthermore, this generalization provides a technique for noise-robust experimental demonstration of contextuality Mazurek et al. (2016); Kunjwal and Spekkens (2018, 2015); Xu et al. (2016); Anwer et al. (2021) and leads to information theoretic applications of quantum situations involving preparation contextuality Spekkens et al. (2009b); Chailloux et al. (2016); Ambainis et al. (2019b). Using this approach the minimum number of measurements required to reveal quantum contextuality so far is three Kunjwal and Ghosh (2014), while no physical principle prohibits a smaller number. So far a full characterization of minimum number of measurement devices required to exhibit quantum contextual correlations has been achieved for projective measurements, however, for POVMs the scenario is completely different, non-trivial and an open research problem. Recently it was shown in Ref. Schmid et al. (2020) that it is possible to talk about quantum contextuality for only a single measurement device by using the property of convexity of ontic distributions and response functions Spekkens (2014).

In this work we analyse how coarse graining of measurement outcomes can help reveal quantum contextuality for even a single measurement device. These coarse grainings and their representation in ontic models have been well studied in Ref. Spekkens (2014). We formulate a new experimentally verifiable NC inequality for coarse grained measurement outcomes which exhibits a violation for a single carefully constructed measurement device, which is a positive operator valued measure (POVM). The POVM is executed twice sequentially to reveal the contextuality of the quantum situation involved. The POVM itself is constructed by coarse graining over the projective measurements that appear in a test for revealing quantum contextuality. While we explicitly show an example using the measurements that appear in the KCBS inequality, our result readily generalises to the case of $n$-cycle contextuality inequalities.

The inequality that we present is independent of any underlying quantum state and only depends on outcomes of the measuring device. This novel feature of our inequality clearly brings out the nature of sequential measurements. While other similar inequalities Selby et al. (2021) evaluate joint probability distributions over two or more measurements and indirectly imply conclusions on sequential measurements, we take a more direct approach and evaluate conditional probabilities instead, which provides straightforward conclusions on the sequential measurements themselves. Furthermore, our result can also incorporate more scenarios based on multiple sequential measurements. The corresponding NC and quantum bounds can be calculated following the prescription we provide for a single measurement device applied sequentially.

The paper is organized as follows. In Sec. II, we briefly review the assumptions of measurement and preparation non-contextuality. In Sec. III we describe our setup of the POVM to be measured which will be used as a signature of contextuality. In Sec. IV we derive a NC inequality based on the assumptions of measurement and preparation NC. In this section we also explicitly derive the quantum bound and show a violation of the inequality. In Sec. V we provide some conclusive remarks on our result.

The definition of measurement NC as formulated by Spekkens Spekkens (2005), relies on defining a notion of equivalence among different experimental procedures. Specifically, two measurement procedures $\mathcal{M}$ and $\mathcal{M}^{\prime}$ are deemed equivalent if they yield the same statistics for every possible preparation procedure P, that is

where $p(k|\mathcal{M},\text{P})$ is the probability of obtaining the outcome $k$ in the measurement $\mathcal{M}$ for preparation P and $\left[k|\mathcal{M}\right]$ denotes the event of obtaining the outcome $k$ in a measurement $\mathcal{M}$. Consider an ontic model where an ontic state $\lambda$ can be used to predict the measurement outcomes. Based on the definition of equivalence classes for measurement procedures, and as motivated by Leibniz’s principle of indiscernibles, the definition of measurement NC is

where $\xi(k|\lambda,\mathcal{M})$ is the epistemic response function for assigning the probability of obtaining an outcome $k$ in a measurement $\mathcal{M}$ given the ontic state $\lambda$ with

Thus, the assumption of measurement NC implies that the response function for different outcomes of equivalent measurement procedures is the same if their observed statistics are the same for all preparation procedures.

In a similar fashion, one can also define a notion of preparation NC. Firstly, we define two preparation procedures P and $\text{P}^{\prime}$ to be equivalent if they yield the same statistics for all possible measurement procedures $\mathcal{M}$,

where $\left[k|P\right]$ denotes the event in which an outcome $k$ is obtained when the preparation was $P$.

Like measurement NC, the assumption of preparation NC is stated as

where $\mu(\lambda|P)$ is an epistemic distribution over the ontic variables $\lambda$ given a preparation procedure P, with

where $\Lambda$ is the space of all ontic variables $\lambda$. Thus, two preparation procedures are assigned the same epistemic state if the observed statistics for all measurements applied on them are the same.

Here we explicitly construct a non-trivial scenario on which we can apply the assumption of measurement NC with post-processing of measurement outcomes. While the scenario leads to the exhibition of measurement NC using a single measurement device, it also serves as a stepping point in the prescription NC for arbitrary sequential measurement devices.

Consider a measurement procedure able to function in two different configurations and we assume a non-contextual ontological model explaining the statistics of both the configurations.

The schematics of our setup is shown in Fig 1. The setup is comprised of $5$ projective measurements $\{\mathcal{M}_{i}\}$, with three outcomes each. We choose these projective measurements to be the ones utilized in the KCBS scenario. We also assume that the projectors involved in each measurement are of rank-one. Using these measurements we can enable the device to work in two possible configurations as detailed below:

• C1:

  In this configuration the measurements $\{\mathcal{M}_{i}\}$ are sampled from a probability distribution $\{p_{i}\}$, $i\in\{0,1,2,3,4\}$, and the device implements the projective measurement $\{\Pi_{i},\Pi_{i\oplus 1},K_{i}\}$, where $K_{i}$ is added to complete the measurement and $\oplus$ is addition modulo $5$. The projectors $\{\Pi_{i}\}$ satisfy ${\rm Tr}(\Pi_{i}\Pi_{i\oplus 1})=0$ and are the same as the ones used in the derivation of KCBS inequality. The resultant measurement is then a POVM $\mathcal{M}$ with outcomes,

  $$\mathcal{M}:\{E_{0},E_{1},E_{2},E_{3},E_{4},\mathcal{K}\},$$

  (7)

  where $E_{i}=(p_{i}+p_{i\oplus 4})\Pi_{i}$ and $\mathcal{K}=\sum_{i}p_{i}K_{i}$ with

  $$\sum_{0}^{4}E_{i}=\sum_{i=0}^{4}(p_{i}+p_{i\oplus 4})\Pi_{i}\leq\mathds{1},$$

  (8)

  which is a consequence of completeness of measurement.

• C2:

  In this configuration we choose a particular setting $i$, implementing a projective measurement on our device

  $$\mathcal{M}_{i}:\{\Pi_{i},\Pi_{i\oplus 1},K_{i}\}.$$

  (9)

  This is akin to blocking all measurement outcomes $j\neq i$. From completeness we again have

  $$\Pi_{i}+\Pi_{i\oplus 1}\leq\mathds{1}\quad\forall i.$$

  (10)

  Five such settings are possible corresponding to the projective measurements detailed above and each is labeled by a value of $i$.

We now formulate an ontological description of the aforementioned measurement device. For each measurement outcome $\Pi_{i}$ from the measurement $\mathcal{M}_{i}$, there corresponds a response function $\xi(\Pi_{i}|\lambda,\mathcal{M}_{i})$, where $\lambda$s are the ontic variables. Therefore an ontological description of $\mathcal{M}_{i}$ will have the form

This ontological model of configuration C2 induces an ontological description for the configuration C1. This is due the fact that the generalized measurement procedure $\mathcal{M}$ can be described in terms of post-processing of projective measurements defined for C2 configuration i. e. $\{\mathcal{M}_{i}|i=0,1,2,3,4\}$. Let the ontological model of the measurement device in C1 configuration be described as,

Here $\xi^{\prime}(E|\lambda,\mathcal{M})$ denotes the response function corresponding to the outcome $E\in\{E_{0},E_{1},E_{2},E_{3},E_{4},\mathcal{K}\}$ when the system is measured in C1 configuration. Since C1 implements the projective measurement $\mathcal{M}_{i}$ with probability $p_{i}$, we have for $i\in\{0,1,2,3,4\}$,

and

such that $\xi(\Pi_{i}|\lambda,\mathcal{M}_{i})$, $\xi^{\prime}(E_{i}|\lambda,\mathcal{M})$, $\xi^{\prime}(\mathcal{K}|\lambda,\mathcal{M})$, and $\xi(K_{i}|\lambda,\mathcal{M}_{i})\in\left[0,1\right]$. It is well known that the description for response functions presented above follows from the probability theory in a straightforward way and have nothing to do with assumption of measurement non-contextuality (see Ref. Spekkens (2014), for example). Similarly, the epistemic state of a system when it is prepared in a state corresponding to the outcome $E_{i}$ of measurement procedure $\mathcal{M}$ is described as

where $\mu(\lambda|\Pi_{i},\mathcal{M}_{j})$, for $j=i,i\oplus 4$, is the epistemic state of the ontological variable when the corresponding system is prepared in $\Pi_{i}$ using measurement $\mathcal{M}_{j}$ in C2 configuration. Note that, just like Eq. (13), Eq. (14) also follows from the probability theory and has nothing to do with the assumption of preparation non-contextuality.

We are now ready to propose an inequality to be tested by sequential measurements and explicitly derive its maximum non-contextual value. We then show that a quantum description of the same leads to a violation. While we focus on the measurement device set up in Fig. 1, the technique that we use to derive the NC and quantum bound can be easily modified for arbitrary measurement devices.

The scenario is as follows. The setup with configuration C1 described in Sec. III is used to perform a measurement on a state $\rho$ which is chosen such tr$(\rho\Pi_{i})\neq 0$ $\forall\,\,i$. Two cases arise: (i) if an outcome $E_{i}$ is observed, the setup with configuration C1 is again used to perform a second measurement on the resultant post-measurement state corresponding to the outcome $E_{i}$, and (ii) in case the outcome $\mathcal{K}$ is observed in the first measurement, the corresponding experimental run is discarded. For case (i) the following proposition holds true.

It should also be noted that while quantum probabilities are given by the Born rule, we make no such claim on the probabilities derived under a non-contextual assumption. Instead, we only assume that the functional constraints on the probabilities for both the scenarios must be satisfied at all times. In this way we are able to derive Eq. (25) by using the fact that NC conditional probabilities must give the same predictions as the quantum mechanics for each sequential measurement independently.

It should be noted that the relation between $\mathcal{C}_{QM}$ and $\mathcal{C}_{NC}$ is established by alluding to a particular set of quantum states $\rho$ on which the first measurement is performed. Specifically, we require that the initial state $\rho$ should not be orthogonal to any of the projectors $\Pi_{i}$. This is to ensure that $p(E_{i})\neq 0$ $\forall\,\,i$ in the first measurement so that the quantities $p(E_{j}|E_{i})$ are well defined. Therefore, apart from the aforementioned set of states, Proposition $1$ holds true for all states.

It should also be noted that while no quantum measurements have been referred to, the orthogonality relations between them have been specified as well as the functional constraints between the outcomes of the measurement device in the two configurations.

This simplified signature of measurement NC also allows for easily incorporating arbitrary $n$-cycle contextuality scenarios ($n\geq 5$) Araújo et al. (2013) like as we did for the KCBS scenario. For an arbitrary $n$-cycle scenario instead of KCBS in the measurement device described above, we have,

where in the second equality we have used the standard Born rule, while in the first equality we only assume that the probability distributions are assigned in a non-contextual manner as explained above.

As can be seen the functional relationship between the quantum and contextual bound retains the same form. However, for increasing $n$, $p_{i}=s$ decreases, which consequently broadens the gap between the contextual and non-contextual bound.

In this work we show that the notion of preparation and measurement NC can lead to several new and interesting scenarios which exhibit a superiority of quantum over classical correlations by violating a novel NC bound on an ontic model for a single measurement device. We have detailed a scenario in which the statistics of outcomes from a single measurement device is not reproducible from an underlying non-contextual ontic model. The inequality that we propose is based on conditional probabilities of outcomes of two (or more) measurements. This way we can directly make implications on the (non-)contextual nature of sequential measurements. Our inequality can be generalized for arbitrary sequential measurements for which the NC bound can be calculated easily using our prescription.

To the best of our knowledge such a scenario has been constructed for the first time. Our results pave the way for future theoretical as well as experimental work to unearth the contextuality of arbitrary sequential quantum measurements. It would be interesting to find out whether the inequality we propose is optimal in the number of measurements and/or outcomes. Another extension of our work would entail generalization of the single measurement device scenario to incorporate multiple sequential measurements of the same or different non-contextual scenarios involving $n$-outcomes which may not be cyclic.

Acknowledgements.— JS is supported by QuantERA grant SECRET, by MCINN/AEI(Project No. PCI2019-111885-2) A acknowledge the financial support from DST/ICPS/QuST/Theme-1/2019/General Project number Q-68.