Violation of a causal inequality in a spacetime with definite causal order
Processes with an indefinite causal structure may violate a causal inequality, which quantifies quantum correlations that arise from a lack of causal order. In this paper, we show that when the inequalities are analysed with a Gaussian-localised field theoretic definition of particles and labs, the causal indeterminacy of the fields themselves allows a causal inequality to be violated within the causal structure of Minkowski spacetime. We quantify the violation of the inequality and determine the optimal ordering of observers.
It is customary to think of physical processes and phenomena as built from events with definite causal relations. Recently, there has been a great interest in whether more general causal structures are possible. A main motivation is the expectation that a fundamental theory combining the indeterminacy of quantum physics and the dynamical causal structure of general relativity should include indefinite causal structures Hardy (2007); Brukner (2014). Processes with no definite causal structure have also been proposed as possible resources for a variety of tasks Chiribella et al. (2013); Chiribella (2012); Araújo et al. (2014); Feix et al. (2015); Guérin et al. (2016); Ebler et al. (2017), with an ongoing effort towards their practical realisation Procopio et al. (2015); Rubino et al. (2017a, b); Goswami et al. (2018).
The correlations between events in a definite causal structure satisfy causal inequalities Oreshkov et al. (2012); Oreshkov and Giarmatzi (2016); Branciard et al. (2016); Abbott et al. (2016), derived from the assumption that only one-way signalling is possible: if an event is the cause of an event , then cannot be the cause of . A violation of such inequalities would imply that no definite causal order between the events exists. It has been shown that it is possible to violate the causal inequalities within a framework that only assumes the local validity of quantum theory but makes no assumptions regarding a possible background causal structure Oreshkov et al. (2012). The physical interpretation of such a framework is however still uncertain.
In practice, a causal inequality could be violated trivially simply by allowing parties to exchange information across an extended period of time; any probability distribution can be obtained in this way. The interest in the subject derives from the possibility—so far only theoretically speculated—that the inequalities might be violated under stricter conditions, thus demonstrating genuinely new types of causal relations. In Ref. Oreshkov et al. (2012) these conditions were proposed to be that of closed laboratories—each event is generated through a single operation on a physical system, which cannot interact with the outside world during the operation—and of free choice—an experimenter can perform an arbitrary operation in the closed lab and the choice of operation is not caused by any other variable relevant to the system under investigation. To date no physical process has been proposed that can violate causal inequalities under such conditions.
Here, we propose a protocol in which two parties can violate a causal inequality by acting on Gaussian-localised field modes of photons in Minkowski spacetime. This is possible because the modes are extended in time, so that each intersects the future light-cone of the other. From an operational point of view, the modes provide a realisation of closed laboratories in which agents can freely choose arbitrary operations, satisfying the conditions for a genuine violation of the inequalities. An alternative perspective, where laboratories are defined in terms of the background space-time, would rather suggest that the violation is due to the failure of the closed-lab condition. We comment how this latter perspective is problematic, since any finite-energy mode is necessarily temporally extended, and a small violation of the inequalities is always possible.
Causal inequalities—We consider two parties, (Alice) and (Bob), who receive classical inputs , and generate classical outputs , , respectively. For simplicity, we restrict to binary variables and assume that the inputs are uniformly distributed, for any pair of values , .
The goal for the parties is to guess each other’s input, i.e., to maximise the probability Branciard et al. (2016)
A definite causal order between the labs imposes constraints on the probability of success: if Alice can signal to Bob, Bob cannot signal to Alice and vice versa. Even if the causal order between the labs is unknown, or decided with some probability by some external variables, the probability of success is bounded by the causal inequality Branciard et al. (2016)
This inequality (a simplified version of the original one Oreshkov et al. (2012)) must be satisfied if the operations producing the correlations are each performed between two time instants, defined with respect to a background causal structure, and the system on which Alice (Bob) performs the operation is isolated from the outside world between those two instants. In a quantum setting, the times at which operations are performed can be subject to indeterminacy. This opens the possibility of violating a causal inequality with operations that still satisfy a reasonable ‘closed laboratory’ assumption. As sketched in Ref. Oreshkov et al. (2012), a ‘closed lab’ can be defined operationally and without reference to a background causal structure in terms of the possible operations that can be performed in it. If a party is free to choose any operation that formally transforms an input Hilbert space to an output Hilbert space, and each operation can in principle be verified through tomography by external parties feeding appropriate states and performing appropriate measurements, we say—by definition—that the party acts in a closed lab. Crucially, the input and output Hilbert spaces do not have to be identified with instants in time: even when a background spacetime structure is assumed, quantum labs can be delocalised in time Oreshkov (2018).
Violation of causal inequality with field modes— We now present a scenario that, by exploiting temporally-delocalised field modes, enables the violation of the above inequality while satisfying the closed-laboratory assumption. In particular, we consider Gaussian-localised single-particle excitations of optical field modes in Minkowski spacetime,
where is a polarisation index and the mode is defined by a Gaussian superposition of plane wave modes with annihilation operators
where we use units for which , are single frequency Minkowski operators and is the Minkowski vacuum which is annihilated () by the Minkowski operators. Note that Eq. (3) is a pure state and so contains all information about the particle. This Gaussian-localised particle has a central wave number of and is peaked along the trajectory with a spatio-temporal width of . More realistically we can also require a transverse Gaussian profile for the mode that localises the particle in the transverse directions as well. However, provided we assume that all operations are carried out close to the focus of the mode then the paraxial approximation implies that the dimensional description of the mode in Eq. (4) is a good approximation to the full dimensional description.
A party (respectively, ) that can perform arbitrary operations on—and only on—the single-particle states of such a mode effectively defines a ‘closed lab’. To make this definition operationally meaningful, we assume that mode selective mirrors at the input () and output () allow only a single spatio-temporal mode, (), to enter and leave Alice’s (Bob’s) lab (see Fig.1). Modes that are orthogonal to () are completely reflected. In this way the operations in each lab are restricted to a single-spatio-temporal mode, centred around an event , . (We assume the mirrors are polarisation insensitive and so allow either polarisation mode to enter or leave.) Passive mirrors and lenses external to the labs are allowed to direct and focus fields into the labs and to direct fields away from the labs.
The closed-lab assumption requires that each party can perform arbitrary operations on the respective single-particle space. Possible operations include unitaries, projective measurements of states , and preparation of states in the same modes. More general operations could require interactions with a local ancilla, e.g., applying a controlled unitary on input and output system followed by a detection of control and input system, Fig. 1(a). Interactions with an ancilla do not violate the closed-lab assumption as long as the ancilla is not correlated with any other system outside the lab. Crucially, the assumption can be verified operationally, separately for each lab, by an external party sending selected states to the input mirror and performing measurements at the output. The verifier would then be able to tomographically reconstruct the operations, certifying that each party is indeed free to perform an arbitrary operation on the respective mode.
We now consider the specific set-up of Fig. 1(b) and assume that Alice and Bob’s modes have the same width . In general, this need not be the case, but as we are trying to maximise the violation, this is the simplest choice. Also, for simplicity we assume all operations and detections have unit efficiency.
The protocol proceeds in the following way. Alice measures the polarisation state of her incoming mode in the horizontal/vertical basis and records her guess for Bob’s bit. Three results are possible: (i) a -polarized photon is detected; (ii) a -polarized photon is detected; (iii) no photon is detected. In case (i) Alice records , in case (ii) she records , and in case (iii) she randomly chooses to record a zero or a one. Simultaneously, Alice prepares the single photon state: , choosing the polarisation to be or according to the value or of the random bit she is trying to send Bob. As the mode of the photon matches the acceptance mode of the output mirror, it escapes from Alice’s lab with no attenuation. Bob’s protocol is identical except that he measures and prepares the single photon states , matching the acceptance mode of his input and output mirrors respectively. We have defined Alice and Bob’s modes as right moving modes, i.e., localised on the trajectories . We assume Bob is to the right of Alice (see Fig. 2) and allow a passive mirror outside Bob’s station to reflect Bob’s output from right-moving to left-moving. A similar mirror outside Alice’s lab reflects left-moving modes back into right moving modes that impinge on Alice’s mode selective input mirror. In the following we will ignore the slight asymmetry of this situation and assume the effective propagation distance between the labs is simply .
Given our assumptions about the ideal operation of the components it is clear that if Alice (Bob) detects a photon in their polarisation detector they will successfully determine the bit value sent by Bob (Alice). Hence, in order to calculate the value of (Eq. 1) we need to determine the probability for Alice (Bob) to detect the photon prepared by Bob (Alice). We can calculate the transmission probability for an excitation of Alice’s mode to get through Bob’s input mirror via the absolute square of the overlap between their modes: 111 can be thought of as the Feynman propagator except our excitations are Gaussian-localised particles.
where , with the assumption that and using the usual commutation rule . The above analysis can be repeated for a photon from Bob to Alice and we obtain the similar result:
We can now specify the probability that Bob measures Alice’s bit correctly as the probability that the photon is transmitted through Bob’s mirror, after which he can definitely know the bit value, plus the probability that the photon is reflected multiplied by the probability he correctly guesses Alice’s bit, i.e. . Hence we obtain,
Similarly for Alice measuring Bob’s qubit,
The probability of success is therefore,
This is our main result—for any choice of a finite and , timings can be found for which (Eq. 2).
We now investigate the optimal that maximises this probability of success. From the perspective of perfectly localised particles this should be the case when but here there is the competing effect of delocalisation. As a result, the best-case scenario depends on the parameters. For , it is optimised by . When , the optimal is . In this regime, the average send times of Alice and Bob are no longer light-like separated, instead and become increasing more symmetric as gets smaller. When , the optimum separation in time is where and we have the symmetric case.
In the asymmetric case where ,
and we have a violation of the inequality for any . In the symmetric case, and the probability of success is,
for which when . In all cases, it is always possible for . In the limit of strong photon and lab delocalisation , the probability goes to 1, i.e. , approaching a maximal violation of the inequality. It may seem that in the limit we obtain perfect localisation, and we get back the causal inequality where . However, this is an unphysical limit. In order for our solutions to be valid we require (this ensures that the mode function doesn’t bifurcate into both right and left moving components). As a result implies and hence infinite energy.
Conclusion—Causal inequalities represent interesting constraints only if additional conditions are imposed on how the correlations are generated—with no restrictions, it is always possible to generate arbitrary correlations, without the need of quantum effects or exotic spacetime geometry. Although the inequalities are device and theory independent, the conditions on the protocols are model-dependent and have to rely on additional assumptions.
Crucial to the original formulation of Ref. Oreshkov et al. (2012) is the assumption of closed laboratories, which prevents exploiting simple multi-round protocols. We have considered a possible natural background-independent formalisation of this assumption, namely the identification of closed laboratories with field modes. We have presented a protocol where operations matched to particular field modes enable a violation of a causal inequality.
However, when analysed from the perspective of a background causal structure, the same protocol may seem to violate the closed-laboratory assumption: The two ‘laboratories’ act on delocalised modes and therefore sit in regions that are extended in time, both future and past light cone of each region have a large overlap with the other region, and information can freely travel between the two.
Nonetheless, it is questionable whether it is physically meaningful to take the existence of a background causal structure as a primitive notion. Spacetime points are sometimes a useful abstraction of physical events. In classical physics we often consider (point) particles which are perfectly localised, thus physical events such as ‘particle enters lab’ correspond to a spacetime point or a spacetime event. Such cannot be said for a quantum particle which is always delocalised. Spacetime events are therefore of limited use when we consider quantum physics. Thus, it is perhaps better to consider spacetime events/points as a useful mathematical tool than a primitive constituent of physical theory. With this view, events do not exist on their own: they make sense as relational properties between physical degrees of freedom, quantum fields in our case. It is therefore more meaningful to adopt a background-independent notion of local degrees of freedom. Furthermore, sharply-localised modes are unphysical in quantum field theory as they would be associated with infinite energy Knight (1961); Licht (1963). Thus, it would never be possible to strictly satisfy the closed lab assumption, as formulated from the background causal structure point of view. This is a manifestation of the well-known problem of localisation in QFT Hegerfeldt (1974); Vázquez et al. (2014); Schroer (2010) (tightly related with the entanglement in the quantum vacuum Unruh (1976); Summers and Werner (1985); Bombelli et al. (1986); Summers and Werner (1987); Redhead (1995); Halvorson and Clifton (2000); Reznik (2003); Calabrese and Cardy (2004); Zych et al. (2010); Ibnouhsein et al. (2014); Su and Ralph (2016)), namely the question of which quantum degrees of freedom should be associated with local spacetime regions Newton and Wigner (1949); Fleming (1965); Segal and Goodman (1965); Fleming (2000); Halvorson (2001); Piazza and Costa (2007); Costa and Piazza (2009); Cacciatori et al. (2009); Schroeren (2010). Here we have exposed yet another manifestation of this issue: The localisation problem challenges a meaningful, background-independent definition of causal relations in quantum field theory. A formulation of quantum mechanics with no background causal structure Oreshkov et al. (2012) that includes quantum fields will necessarily have to face this issue.
As the violation of a causal inequality is possible with measurements in a fixed basis, the ‘local operations’ cannot be embedded in the ‘process matrix formalism’ in which fixed-basis measurements in a bipartite scenario always lead to definite causal order Oreshkov et al. (2012); Baumann and Brukner (2016). This leaves open the question of whether, in order to be compatible with field theory, the process matrix formalism needs to be extended to allow for non-linear probabilities or whether the basic structure and the assumption of closed laboratories need to be reformulated in order to exclude such possibilities.
Acknowledgements.This research was funded in part by the Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology (Project No. CE110001027). F.C. acknowledges support through an Australian Research Council Discovery Early Career Researcher Award (DE170100712). This publication was made possible through the support of a grant from the John Templeton Foundation. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation. We acknowledge the traditional owners of the land on which the University of Queensland is situated, the Turrbal and Jagera people.
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