No Fine theorem for macrorealism: Limitations of the Leggett-Garg inequality
Tests of local realism and macrorealism have historically been discussed in very similar terms: Leggett-Garg inequalities follow Bell inequalities as necessary conditions for classical behavior. Here, we compare the probability polytopes spanned by all measurable probability distributions for both scenarios and show that their structure differs strongly between spatially and temporally separated measurements. We arrive at the conclusion that, in contrast to tests of local realism where Bell inequalities form a necessary and sufficient set of conditions, no set of inequalities can ever be necessary and sufficient for a macrorealistic description. Fine’s famous proof that Bell inequalities are necessary and sufficient for the existence of a local realistic model, therefore cannot be transferred to macrorealism. A recently proposed condition, no-signaling in time, fulfills this criterion, and we show why it is better suited for future experimental tests and theoretical studies of macrorealism. Our work thereby identifies a major difference between the mathematical structures of local realism and macrorealism.
The violation of classical world views, such as local realism Bell (1964) and macrorealism Leggett and Garg (1985); Leggett (2002), is one of the most interesting properties of quantum mechanics. Experiments performed over the past decades have shown violations of local realism in various systems Freedman and Clauser (1972); Aspect et al. (1982); Weihs et al. (1998), while violations of macrorealism are on the horizon Palacios-Laloy et al. (2010); Goggin et al. (2011); Xu et al. (2011); Dressel et al. (2011); Fedrizzi et al. (2011); Waldherr et al. (2011); Athalye et al. (2011); Souza et al. (2011); Zhou et al. (2015); Knee et al. (2012); Suzuki et al. (2012); George et al. (2013); Katiyar et al. (2013); Emary et al. (2014); Asadian et al. (2014); Robens et al. (2015); White et al. (); Knee et al. (). The latter endeavors pave the way towards the experimental realization of Schrödinger’s famous thought experiment Schrödinger (1935). In the future, they might offer insight into important foundational questions, such as the quantum measurement problem Leggett (2005), and allow experimental tests of (possibly gravitational) extensions of quantum mechanics Romero-Isart (2011).
Historically, the discussion of tests of macrorealism (MR) follows the discussion of tests of local realism (LR) closely: Leggett-Garg inequalities (LGIs) Leggett and Garg (1985) are formulated similarly to Bell inequalities Bell (1964); Clauser et al. (1969); Clauser and Horne (1974), and some concepts, e.g. quantum contextuality Kochen and Specker (1967), are connected to both fields Avis et al. (2010); Kleinmann et al. (2012); Araújo et al. (2013); Kujala et al. (2015); Dzhafarov et al. (2015). However, recently, a discrepancy between LR and MR has been identified: Whereas Fine’s theorem states that Bell inequalities are both necessary and sufficient for LR Fine (1982), a combination of arrow of time (AoT) and no-signaling in time (NSIT) Kofler and Brukner (2013) equalities are necessary and sufficient for the existence of a macrorealistic description Clemente and Kofler (2015). A previous study Clemente and Kofler (2015) also demonstrated that LGIs involving temporal correlation functions of pairs of measurements are not sufficient for macrorealism, but did not rule out a potential sufficiency of other sets of LGIs, e.g. of the CH type Clauser and Horne (1974); Mal and Majumdar (), leaving open the possibility of a Fine theorem for macrorealism. Moreover, cases have been identified where LGIs hide violations of macrorealism Avis et al. (2010) that are detected by a simple NSIT condition Kofler and Brukner (2013). The latter fails for totally mixed initial states, where a more involved NSIT condition is required Clemente and Kofler (2015). These fundamental differences between tests of local realism and macrorealism seem connected to the peculiar definition of macrorealism Maroney and Timpson (); Bacciagaluppi (2015).
In this paper, we analyze the reasons for and the consequences of this difference. We show that the probability space spanned by quantum mechanics (QM) is of a higher dimension in an MR test than in an LR test, and we analyze the resulting structure of the probability polytope. We conclude that inequalities—excluding the pathological case of inequalities pairwise merging into equalities—are not suited to be sufficient conditions for MR, and form only weak necessary conditions. Fine’s theorem Fine (1982), therefore cannot be transferred to macrorealism (unless one uses potentially negative quasi-probabiltities Halliwell ()). Our study thus identifies a striking difference between the mathematical structures of LR and MR. While current experimental tests of macrorealism overwhelmingly use Leggett-Garg inequalities, this difference explains why NSIT is better suited as a witness of non-classicality, i.e. why it is violated for a much larger range of parameters Kofler and Brukner (2013); Clemente and Kofler (2015).
Let us start by reviewing the structure of the LR polytope (LR), as described in refs. Pironio (2005, 2014); Brunner et al. (2014). Consider an LR test between parties . Each party can perform a measurement in one of settings . Each setting has the same number of possible outcomes , and, to allow for all possible types of correlations, it may measure a distinct property of the system. We can define probability distributions for obtaining outcomes , given the measurement settings . If a party chooses not to perform a measurement, the corresponding “setting” is labeled , and there is only one “outcome” labeled (e.g. when only the first party performs a measurement). We leave out final zeros, e.g. . Note that this convention differs from the literature for LR tests, where the case of no measurement is often left out Pironio (2005); Brunner et al. (2014), but simplifies the comparison between LR and MR tests. Each experiment is then completely described by probability distributions; it can be seen as a point in a probability space .
We now require normalization of the probabilities. There are linearly independent normalization conditions, as each probability only appears once:
Because of the special case of no measurements (), here (and in the following equations) we have abbreviated the notation of the summation: The possible values of , in fact, depend on . The normalization conditions reduce the dimension of the probability space to
Furthermore, the positivity conditions
restrict the reachable space to a subspace with the same dimension, but they are delimited by flat hyperplanes. The resulting subspace is called the probability polytope .
In an LR test with space-like separated parties, special relativity prohibits signaling from every party to any other,
These no-signaling (NS) conditions restrict the probability polytope to a NS polytope (NS) of lower dimension. Taking their linear dependence, both amongst each other and with the normalization conditions, into account, we arrive at dimension Pironio (2005)
Since quantum mechanics obeys NS, and due to Tsirelson bounds Cirel’son (1980), the space of probability distributions from spatially separated experiments implementable in quantum mechanics, , is located strictly within the NS polytope. Furthermore, the space of local realistic probability distributions, LR, is a strict subspace of . It is delimited by Bell inequalities (e.g. the CH/CHSH inequalities for ) and positivity conditions, and therefore forms a polytope within Fine (1982); Pironio (2005). In summary, we have , with . The structure of the NS, and LR spaces is sketched on the left of fig. 1.
In a test of MR, temporal correlations take the role of an LR test’s spatial correlations. Instead of spatially separated measurements on systems by different observers, a single observer performs sequential (macroscopically distinct) measurements on one and the same system. Again, each measurement is either skipped (“0”) or performed in one of 111In contrast to LR tests, where is required to observe quantum violations, allows for violations of MR, and is in fact the most considered case in the literature. settings, with possible outcomes each. With this one-to-one correspondence, the resulting probability polytope in the space is identical to the one in the Bell scenario. However, without further physical assumptions, no-signaling in temporally separated experiments is only a requirement in one direction: While past measurements can affect the future, causality demands that future measurements cannot affect the past. This assumption is captured by the arrow of time (AoT) conditions:
Counting the number of equalities in eq. 6 shows that their number is
where the first factor in the sum counts the setting and outcome combinations for times , excluding the choice of all , and the second factor the number of settings at time . All listed conditions are linearly independent due to their hierarchical construction, see fig. 2. However, a number of the normalization conditions for the marginal distributions, already subtracted in eq. 2, are not linearly independent from AoT, and thus become obsolete. Their number is obtained by counting the different settings in eq. 6:
The remaining normalization conditions are the ones for probability distributions with just one measurement and for the “0-distribution”; there are such distributions. Taking eq. 2, subtracting eq. 7 and adding eq. 8, we conclude that the AoT conditions restrict the probability polytope to an AoT polytope (AoT) of dimension
By simple extension of the proof in ref. Clemente and Kofler (2015), the set of all no-signaling in time (NSIT) conditions,
is, together with AoT, necessary and sufficient for macrorealism. To get from AoT to the macrorealism polytope, MR, we therefore require a linearly independent subset of these conditions. However, since the AoT conditions from eq. 6 plus the NSIT conditions from eq. 10 are equivalent to the NS conditions from eq. 4, we arrive at MR with the same dimension as the LR polytope:
We are left with the question of how the space of probability distributions realizable from temporally separated experiments in quantum mechanics, , relates to AoT. Fritz has shown in ref. Fritz (2010) that for , if we allow for positive-operator valued measurements (POVMs). Let us now generalize his proof to arbitrary . We do so by constructing a quantum experiment that produces all possible probability distributions which are allowed by AoT.
|LR test||MR test|
|Number of unnormalized distributions|
Consider a quantum system of dimension , with states enumerated as . As with the probability distributions, final zeros may be omitted. The initial state of the system is . Now, POVMs are performed on the system. The measurements are chosen such that depending on their setting and outcome they take the system to the corresponding state: Performing a measurement on a system in state with setting and obtaining outcome should leave the system in state . This is accomplished by choosing Kraus operators for the -th measurement in basis for outcome as
For , the first sum in eq. 12 reduces to the single term , while the second sum remains unchanged. The second sum in eq. 12 is necessary for the completeness relation . The above definitions also work for , where , and . The conditional probabilities in eq. 12 can be obtained from the probabilities using the assumption of AoT:
This construction gives a recipe to obtain any point in the AoT probability space in a quantum experiment. We have therefore shown that for any choice of .
Note that the probability distributions constructed above can also be achieved by a purely classical stochastic model, albeit with invasive measurements. Such an experiment would therefore not convince a macrorealist to give up their world view. For that to happen, an experiment needs to properly address the clumsiness loophole Leggett and Garg (1985); Wilde and Mizel (2012); Moreira et al. (2015). The relevant methods previously established for the LGI can also be applied to NSIT-based experiments Knee et al. ().
Since AoT is a polytope, with POVMs is also a polytope, and no non-trivial Tsirelson-like bounds exist. If, on the other hand, we only allowed projective measurements, we would have with non-trivial Tsirelson-like bounds, as shown in ref. Fritz (2010). In this case, would not be a polytope. It is easy to see that QM with projectors is unable to reproduce some probability distributions: , fulfills AoT but cannot be constructed in projective quantum mechanics, since the initial state must be an eigenstate of the first measurement. Here we consider the general case of POVMs.
In summary, we have
with , and dimensions
Finally, let us compare the characteristics of quantum mechanics in LR and MR tests. Trivially, QM fulfills NS between spatially separated measurements, and AoT between temporally separated measurements 222To show that QM fulfills NS, we consider a setup with only two parties, 1 and 2, performing measurements with POVM elements and , respectively, on a two-particle state . We then calculate . To show that QM fulfills AoT, we consider a setup where are measured at time on state , and are measured at time . We then have , where is the state after measurement of at time 1 with outcome , evolved to time 2. The proofs for more parties or more measurement times follow straightforwardly. . While and LR have the same dimension and are separated by Bell inequalities, and MR span subspaces with different dimensions. Inequalities can never reduce the dimension of the probability space, since they act as a hyperplane separating the fulfilling from the violating volume of probability distributions. We conclude that no combination of (Leggett-Garg) inequalities can be sufficient for macrorealism.
The observation that inequalities cannot be sufficient for macrorealism, and the differences in the structure of the probability space shown above, present fundamental discrepancies between LR and MR. Fine’s observation Fine (1982) that Bell inequalities are necessary and sufficient for LR can therefore not be transferred to the case of LGIs and MR. More precisely, Fine’s proof uses the implicit assumption of NS, which is obeyed by all reasonable physical theories, including QM. However, the temporal analogue to NS is the conjunction of AoT and NSIT, where AoT is obeyed by all reasonable physical theories, while NSIT is violated in QM. Therefore,
where “BIs” and “LGIs” denote the sets of all Bell and Leggett-Garg inequalities, respectively.
Moreover, since MR is a polytope with smaller dimension than , LGIs can only touch MR (i.e. be tight) at one facet, i.e. a positivity constraint, as sketched in fig. 1 on the right. A comparable Bell inequality, sketched in fig. 1 on the left as BI’, clearly illustrates the limitations resulting from this requirement. In an experimental test of MR, using a LGI therefore needlessly restricts the parameter space where violations can be found. The favorable experimental feasibility of NSIT is demonstrated by the theoretical analyses of refs. Kofler and Brukner (2013); Clemente and Kofler (2015), as well as the recent experiment of ref. Knee et al. (). Note also the mathematical simplicity of the NSIT conditions when compared to the LGI. We conclude that for further theoretical studies and future experiments it might be advantageous to eschew the LGIs and rather use NSIT.
Acknowledgements.We acknowledge support from the EU Integrated Project SIQS.
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