Eric Cavalcanti

Griffith University · Centre for Quantum Dynamics

Quantum mechanics is an outstandingly successful description of nature, underpinning fields from biology through chemistry to physics. At its heart is the quantum wavefunction, the central tool for describing quantum systems. Yet it is still unclear what the wavefunction actually is: does it merely represent our limited knowledge of a system, or is...

According to a recent no-go theorem (M. Pusey, J. Barrett and T. Rudolph, Nature Physics 8, 475 (2012)), models in which quantum states correspond to probability distributions over the values of some underlying physical variables must have the following feature: the distributions corresponding to distinct quantum states do not overlap. This is sign...

We formally link the concept of steering (a concept created by Schrödinger but only recently formalized by Wiseman, Jones and Doherty Phys. Rev. Lett. 98 140402 (2007)]) and the criteria for demonstrations of Einstein-Podolsky-Rosen (EPR) paradox introduced by Reid Phys. Rev. A 40 913 (1989)]. We develop a general theory of experimental EPR-steerin...

This Colloquium examines the field of the Einstein, Podolsky, and Rosen (EPR) gedanken experiment, from the original paper of Einstein, Podolsky, and Rosen, through to modern theoretical proposals of how to realize both the continuous-variable and discrete versions of the EPR paradox. The relationship with entanglement and Bell’s theorem are analyz...

We derive a new class of correlation Bell-type inequalities. The inequalities are valid for any number of outcomes of two observables per each of n parties, including continuous and unbounded observables. We show that there are no first-moment correlation Bell inequalities for that scenario, but such inequalities can be found if one considers at le...

The Choi–Jamiołkowski isomorphism is an essential component in every quantum information theorist’s toolkit: it allows to identify linear maps between two quantum systems with linear operators on the composite system. Here, we analyse this widely used gadget from a new perspective. Namely, we explicitly distinguish between its kinematical and dynam...

The Wigner’s friend thought experiment has gained a resurgence of interest in recent years thanks to no-go theorems that extend it to Bell-like scenarios. One of these, by us and co-workers, showcased the contradiction that arises between quantum theory and a set of assumptions, weaker than those in Bell’s theorem, which we named ‘Local Friendlines...

Bell nonlocality refers to correlations between two distant, entangled particles that challenge classical notions of local causality. Beyond its foundational significance, nonlocality is crucial for device-independent technologies like quantum key distribution and randomness generation. Nonlocality quickly deteriorates in the presence of noise, and...

Lawrence et al. have presented an argument purporting to show that “relative facts do not exist” and, consequently, “Relational Quantum Mechanics is incompatible with quantum mechanics”. The argument is based on a GHZ-like contradiction between constraints satisfied by measurement outcomes in an extended Wigner’s friend scenario. Here we present a...

A recent paper by two of us and co-workers \cite{CQD20}, based on an extended Wigner's friend scenario, demonstrated that certain empirical correlations predicted by quantum theory (QT) violate inequalities derived from a set of metaphysical assumptions we called "Local Friendliness" (LF). These assumptions are strictly weaker than those used for d...

The famous "Wigner's friend" paradox highlights the difficulty of modelling the evolution of quantum systems under measurement in situations where observers themselves are considered to be subject to the laws of quantum mechanics. In recent years, variations of the original Wigner's friend paradox have been recognized as fruitful arenas for probing...

The Wigner's friend thought experiment has gained a resurgence of interest in recent years thanks to no-go theorems that extend it to Bell-like scenarios. One of these, by us and co-workers, showcased the contradiction that arises between quantum theory and a set of assumptions, weaker than those in Bell's theorem, which we named "local friendlines...

Lawrence et al. have presented an argument purporting to show that ``relative facts do not exist'' and, consequently, ``Relational Quantum Mechanics is incompatible with quantum mechanics''. The argument is based on a GHZ-like contradiction between constraints satisfied by measurement outcomes in an extended Wigner's friend scenario. Here we presen...

We introduce a formalism for the evaluation of counterfactual queries in the framework of quantum causal models, by generalising the three-step procedure of abduction, action, and prediction in Pearl's classical formalism of counterfactuals. To this end, we define a suitable extension of Pearl's notion of a "classical structural causal model", whic...

We address various aspects of a widely used tool in quantum information theory: the Choi-Jamiolkowski isomorphism [A. Jamiolkowski, Rep. Math. Phys., 3, 275 (1972)]. We review different versions of the isomorphism, their properties and propose a unified description that combines them all. To this end, we identify the physical reason for the appeara...

A recent paper by two of us and co-workers, based on an extended Wigner's friend scenario, demonstrated that certain empirical correlations predicted by quantum theory (QT) violate inequalities derived from a set of metaphysical assumptions we called "Local Friendliness" (LF). These assumptions are strictly weaker than those used for deriving Bell...

In a recent work, Bong et al. [Nat. Phys. 16, 1199 (2020)] proved a no-go theorem demonstrating a contradiction between a set of assumptions called "Local Friendliness" (LF) and certain quantum phenomena on an extended version of the Wigner's friend scenario. The LF assumptions can be understood as the conjunction of two independent assumptions: Ab...

In a recent work, it was shown by one of us (EGC) that Bell-Kochen-Specker inequality violations in phenomena satisfying the no-disturbance condition (a generalisation of the no-signalling condition) cannot in general be explained with a faithful classical causal model---that is, a classical causal model that satisfies the assumption of no fine-tun...

We provide a new formulation of the Local Friendliness no-go theorem of Bong et al. [Nat. Phys. 16, 1199 (2020)] from fundamental causal principles, providing another perspective on how it puts strictly stronger bounds on quantum reality than Bell's theorem. In particular, quantum causal models have been proposed as a way to maintain a peaceful coe...

We provide a new formulation of the Local Friendliness no-go theorem of Bong et al [Nat. Phys. 16, 1199 (2020)] from fundamental causal principles, providing another perspective on how it puts strictly stronger bounds on quantum reality than Bell's theorem. In particular, quantum causal models have been proposed as a way to maintain a peaceful coex...

In a recent no-go theorem [Bong et al., Nature Physics (2020)], we proved that the predictions of unitary quantum mechanics for an extended Wigner’s friend scenario are incompatible with any theory satisfying three metaphysical assumptions, the conjunction of which we call “Local Friendliness”: Absoluteness of Observed Events, Locality and No-Super...

The Wigner’s friend paradox illuminates the quantum measurement problem. We derive—and study, in a series of entangled-photon experiments—a new and robust no-go theorem based on the paradox, with stronger constraints than Bell's theorem.

Does quantum theory apply at all scales, including that of observers? New light on this fundamental question has recently been shed through a resurgence of interest in the long-standing Wigner’s friend paradox. This is a thought experiment addressing the quantum measurement problem—the difficulty of reconciling the (unitary, deterministic) evolutio...

In a recent no-go theorem [Bong et al, Nature Physics (to appear)], we proved that the predictions of unitary quantum mechanics for an extended Wigner's friend scenario are incompatible with any theory satisfying three metaphysical assumptions, the conjunction of which we call "Local Friendliness": Absoluteness of Observed Events, Locality and No-S...

The standard formulation of thermostatistics, being based on the Boltzmann-Gibbs distribution and logarithmic Shannon entropy, describes idealized uncorrelated systems with extensive energies and short-range interactions. In this Rapid Communication, we use the fundamental principles of ergodicity (via Liouville's theorem), the self-similarity of c...

In a measurement-device-independent or quantum-refereed protocol, a referee can verify whether two parties share entanglement or Einstein-Podolsky-Rosen (EPR) steering without the need to trust either of the parties or their devices. The need for trusting a party is substituted by a quantum channel between the referee and that party, through which...

In a recent work, it was shown by one of us (EGC) that Bell-Kochen-Specker inequality violations in phenomena satisfying the no-disturbance condition (a generalisation of the no-signalling condition) cannot in general be explained with a faithful classical causal model---that is, a classical causal model that satisfies the assumption of no fine-tun...

Does quantum theory apply to observers? A resurgence of interest in the long-standing Wigner's friend paradox has shed new light on this fundamental question. Brukner introduced a scenario with two separated but entangled friends. Here, building on that work, we rigorously prove that if quantum evolution is controllable on the scale of an observer,...

The standard formulation of thermostatistics, featuring the Boltzmann-Gibbs-Shannon distribution and logarithmic entropy, only applies to idealized uncorrelated systems with extensive energies. In this letter, we use the fundamental principles of ergodicity, i.e. Liouville's theorem for equilibrium conditions, the self-similarity of correlations, a...

In a measurement-device-independent or quantum-refereed protocol, a referee can verify whether two parties share entanglement or Einstein-Podolsky-Rosen (EPR) steering without the need to trust either of the parties or their devices. The need for trusting a party is substituted by a quantum channel between the referee and that party, through which...

Nonlocality and contextuality are at the root of conceptual puzzles in quantum mechanics, and they are key resources for quantum advantage in information-processing tasks. Bell nonlocality is best understood as the incompatibility between quantum correlations and the classical theory of causality, applied to relativistic causal structure. Contextua...

“Bell’s theorem” can refer to two different theorems that John Bell proved, the first in 1964 and the second in 1976. His 1964 theorem is the incompatibility of quantum phenomena with the joint assumptions of locality and predetermination. His 1976 theorem is their incompatibility with the single property of local causality. This is contrary to Bel...

Contextuality is a phenomenon at the heart of the quantum mechanical departure from classical behavior, and has been recently identified as a resource in quantum computation. Experimental demonstration of contextuality is thus an important goal. The traditional form of contextuality—as violation of a Kochen–Specker inequality—requires a quantum sys...

This is an Editorial on "Classification of all alternatives to the Born rule in terms of informational properties" by Thomas D. Galley and Lluis Masanes, published in Quantum 1, 15 (2017). The full text of the editorial can be found here: https://doi.org/10.22331/qv-2017-07-14-2

Nonlocality and contextuality are at the root of conceptual puzzles in quantum mechanics, and are key resources for quantum advantage in information-processing tasks. Bell nonlocality is best understood as the incompatibility between quantum correlations and the classical theory of causality, applied to relativistic causal structure. Contextuality,...

We derive an inequality that is necessary and sufficient to show Einstein-Podolsky-Rosen (EPR) steering in a scenario employing only correlations between two arbitrary dichotomic measurements on each party. Thus the inequality is a complete steering analogy of the Clauser-Horne-Shimony-Holt (CHSH) inequality, a generalization of the result of Caval...

We address Gillis' recent criticism [arXiv:1506.05795] of a series of papers (by different combinations of the present authors) on formulations of Bell's theorem. Those papers intended to address an unfortunate gap of communication between two broad camps in the quantum foundations community that we identify as "operationalists" and "realists". Her...

In light of a recent reformulation of Bell's theorem from causal principles by Wiseman and the author, I argue that the conflict between quantum theory and relativity brought up by Bell's work can be softened by a revision of our classical notions of causation. I review some recent proposals for a quantum theory of causation that make great strides...

We derive a new inequality that is necessary and sufficient to show EPR-steering in a scenario employing only correlations between two arbitrary dichotomic measurements on each party. Thus the inequality is a complete steering analogy of the CHSH inequality, a generalisation of the result of Cavalcanti et al, JOSA B, 32(4), A74 (2015). We show that...

The fully symmetric Gaussian tripartite entangled pure states will not exhibit two-mode Einstein Podolsky-Rosen (EPR)-steering. This means that any two participants cannot share quantum secrets using the security of one-sided device independent quantum key distribution (1SDI-QKD) without involving the third. They are restricted at most to standard...

Weak measurement of a quantum system followed by postselection based on a subsequent strong measurement gives rise to a quantity called the weak value: a complex number for which the interpretation has long been debated. We analyse the procedure of weak measurement and postselection, and the interpretation of the associated weak value, using a theo...

The year 2015 marks 80 years since Einstein, Podolsky, and Rosen argued for the incompleteness of quantum mechanics, in the process describing one of the most counterintuitive characteristics of quantum mechanics: distant measurements can influence the character of local quantum states. This led to reactions from Schrödinger, Bohr, and John Bell. T...

The Clauser–Horne–Shimony–Holt (CHSH) inequality and its permutations are necessary and sufficient criteria for Bell nonlocality in the simplest Bell-nonlocality scenario: two parties, two measurements per party and two outcomes per measurement. Here we derive an inequality for Einstein–Podolsky–Rosen (EPR)-steering that is an analog of the CHSH, i...

"Bell's theorem" can refer to two different theorems that John Bell proved, the first in 1964 and the second in 1976. His 1964 theorem is the incompatibility of quantum phenomena with the joint assumptions of Locality and Predetermination. His 1976 theorem is their incompatibility with the single property of Local Causality. This is contrary to Bel...

The Clauser-Horne-Shimony-Holt (CHSH) inequality (and its permutations), are necessary and sufficient criteria for Bell nonlocality in the simplest Bell-nonlocality scenario: 2 parties, 2 measurements per party and 2 outcomes per measurement. Here we derive an inequality for EPR-steering that is an analogue of the CHSH, in that it is necessary and...

Bellʼs 1964 theorem causes a severe problem for the notion that correlations require explanation, encapsulated in Reichenbachʼs principle of common cause. Despite being a hallmark of scientific thought, dropping the principle has been widely regarded as much less bitter medicine than the perceived alternative—dropping relativistic causality. Recent...

We derive entropic Einstein-Podolsky-Rosen (EPR) steering inequalities from entropic uncertainty relations. This allows us to formulate inequalities that test for EPR steering correlations which are symmetric between parties, or occur across multiple degrees of freedom.

We analyze two classes of Einstein-Podolsky-Rosen (EPR)-steering inequalities, the violation of which can be used to demonstrate EPR-steering with an entangled two-qubit Werner state: linear inequalities and quadratic inequalities. We discuss how post-selection of results (by appeal to the fair sampling assumption) can compromise the rigor of these...

We use entropic uncertainty relations to formulate inequalities that witness Einstein-Podolsky-Rosen (EPR)-steering correlations in diverse quantum systems. We then use these inequalities to formulate symmetric EPR-steering inequalities using the mutual information. We explore the differing natures of the correlations captured by one-way and symmet...

Various protocols exist by which a referee can be convinced that two observers share an entangled resource. Such protocols typically specify the types of communication allowed, and the degrees of trust required, between the referee and each observer. Here it is shown that the need for any degree of trust of the observers by the referee can be compl...

Nonlinear modifications to the laws of quantum mechanics have been proposed as a possible way to consistently describe information processing in the presence of closed timelike curves. These have recently generated controversy due to possible exotic information-theoretic effects, including breaking quantum cryptography and radically speeding up bot...

Duality quantum computer can accommodate both unitary and non-unitary operations. In this paper, we derive the mathematical theory of duality quantum computer in terms of density matrix using the duality mode of quantum computers. We give a solution ...

Demonstrating nonclassical effects over longer and longer distances is essential for both quantum technology and fundamental science. The main challenge is loss of photons during propagation, because considering only those cases where photons are detected opens a "detection loophole" in security whenever parties or devices are untrusted. Einstein-P...

We analyze the security and feasibility of a protocol for Quantum Key Distribution (QKD), in a context where only one of the two parties trusts his measurement apparatus. This scenario lies naturally between standard QKD, where both parties trust their measurement apparatuses, and Device-Independent QKD (DI-QKD), where neither does, and can be a na...

Whether for quantum cryptography or for testing fundamental properties of the universe, it will be essential to demonstrate nonclassical effects over longer and longer distances. The barrier to doing so is loss of photons during propagation, because considering only those cases where a photon is detected opens a “detection loophole” in the security...

Wiseman and co-workers (Phys. Rev. Lett. 98, 140402, 2007) proposed a distinction between the nonlocality classes of Bell's nonlocality, steering and entanglement based on whether or not an overseer trusts each party in a bipartite scenario where they are asked to demonstrate entanglement. Here we extend that concept to the multipartite case and de...

Tests of local hidden-variable theories using measurements with continuous-variable (CV) outcomes are developed, and a comparison of different methods is presented. As examples, we focus on multipartite entangled Greenberger-Horne-Zeilinger and cluster states. We suggest a physical process that produces the states proposed here, and investigate exp...

In a recent Letter, Bennett and coworkers [1] argue that proofs of exotic quantum effects using closed timelike curves (CTC's) based on the work of Deutsch [2], or other nonlinear quantum dynamics, suffer from a fallacy that they call the "linearity trap," by which one cannot assume that a classical mixture of input states will lead to the same mix...

I apply some of the lessons from quantum theory, in particular from Bell’s theorem, to a debate on the foundations of decision theory and causation. By tracing a formal analogy between the basic assumptions of causal decision theory (CDT)—which was developed partly in response to Newcomb’s problem— and those of a local hidden variable theory in the...

The 1964 theorem of John Bell shows that no model that reproduces the predictions of quantum mechanics can simultaneously satisfy the assumptions of locality and determinism. On the other hand, the assumptions of \emph{signal locality} plus \emph{predictability} are also sufficient to derive Bell inequalities. This simple theorem, previously noted...

We obtain criteria for entanglement and the EPR paradox for spin-entangled particles and analyse the effects of decoherence caused by absorption and state purity errors. For a two qubit photonic state, entanglement can occur for all transmission efficiencies. In this case, the state preparation purity must be above a threshold value. However, Bohm’...

We show that arbitrary functions of continuous variables can be used to generate tests of local hidden variable theories. The effect of nonideal detectors and noise is included, revealing that optimized functional inequalities are robust.

We show that arbitrary functions of continuous variables, e.g., position and momentum, can be used to generate tests that distinguish quantum theory from local hidden variable theories. By optimizing these functions, we obtain more robust violations of local causality than obtained previously. We analytically calculate the optimal function and incl...

In the early stages of the famous debates between Einstein and Bohr on the foundations of quantum mechanics, Einstein attempted to devise thought experiments aiming at showing that arbitrarily accurate predictions of incompatible physical quantities was possible, in violation of Heisenberg's uncertainty principle. Bohr was able to defend quantum me...

We propose three criteria for identifying continuous variable entanglement between two many-particle systems with no restrictions on the quantum state of the local oscillators used in the measurements. Mistakenly asserting a coherent state for the local oscillator can lead to incorrectly identifying the presence of entanglement. We demonstrate this...

In recent decades there has been a resurge of interest in the foundations of quantum theory, partly motivated by new experimental techniques, partly by the emerging field of quantum information science. Old questions, asked since the seminal article by Einstein, Podolsky and Rosen (EPR), are being revisited. The work of John Bell has changed the di...

We consider tests for macroscopic, or S-scopic, entanglement, using EPR inequalities. We then test an S-scopic local realism by deriving Bell inequalities applicable to either moments or binned correlations of multi-partite observables.

In recent decades there has been a resurge of interest in the foundations of quantum theory, partly motivated by new experimental techniques, partly by the emerging field of quantum information science. Old questions, asked since the seminal article by Einstein, Podolsky and Rosen (EPR), are being revisited. The work of John Bell has changed the di...

We consider macroscopic, mesoscopic, and “S-scopic” quantum superpositions of eigenstates of an observable and develop some signatures for their existence. We define the extent, or size S of a superposition, with respect to an observable x̂, as being the range of outcomes of x̂ predicted by that superposition. Such superpositions are referred to as...

Quantum trajectory equations are stochastic equations for the state of an open quantum system conditioned on a monitoring i.e. a continuous-in-time measurement of a bath to which it is coupled. They are closely related to classical stochastic equations for classical probability distributions called filtering equations (e.g. the Kalman filter). Give...

We address the important issue of how to unambiguously identify signatures of quantum entanglement in the presence of decoherence and loss. We analyse entanglement, EPR and Bell violation experiments, with both state impurity and detector inefficiency effects included. We also find a new class of Bell inequality, applicable to either discrete or co...

We propose to test the inconsistency of local reality with quantum mechanics progressively, by defining the S-scopic EPR paradox, that would ultimately allow test of the EPR paradox for the macroscopically entangled "Schrodinger cat" state. Practical signatures of the S-scopic EPR-entanglement are presented for several experimental prototypes.

We show that the correct formulation of the premises behind the EPR argument is that of a Local Hidden State model, recently introduced as a formalisation of Steering by Wiseman et al. This unifies the two concepts and allows one to define the problem of characterising the set of correlations that demonstrate the EPR paradox, providing a solid basi...

We give an outlook on the future of coherence theory and many-body quantum dynamics as experiments develop in the arena of ultra-cold atoms. Novel results on quantum heating of centre-of-mass temperature in evaporative cooling and simulation methods for long-range interactions are obtained, using positive-P phase-space techniques.

We develop experimental signatures for generalized macroscopic, mesoscopic and S-scopic quantum superpositions. We also examine the extent to which current experiments confirm their existence, and propose new measurements to further signify them. The size S of a quantum superposition (or pure state), with respect to the observable X, is defined as...

We show that the correct formulation of the premises behind the EPR argument [1] is that of a Local Hidden State model, recently introduced as a formalisation of Steering by Wiseman and co-authors [2] This unifies the two concepts and allows one to define the problem of characterising the set of correlations that demonstrate the EPR paradox, provid...

We develop criteria sufficient to enable detection of macroscopic coherence where there are not just two macroscopically distinct outcomes for a pointer measurement, but rather a spread of outcomes over a macroscopic range. The criteria provide a means to distinguish a macroscopic quantum description from a microscopic one based on mixtures of micr...

We propose macroscopic generalizations of the Einstein–Podolsky–Rosen paradox in which the completeness of quantum mechanics is contrasted with forms of macroscopic reality and macroscopic local reality defined in relation to Schrödinger's original 'cat' paradox. Yes Yes

We have measured multiple-ionization yields of He , Ne , Ar , Kr , and Xe targets by 1.0 MeV∕amu H2+ molecular ions for both the dissociative and nondissociative channels. Except for the case of the breakup of the H2+ into two protons, we have also determined the absolute cross sections for the multiple ionization of the targets. The ratios of mult...

We consider multiple ionization of Ne by fast H2+, with and without projectile electron loss. Measurements of the total cross section for the direct ionization channel support the similarity of ionization of Ne by an H2+ projectile that does not break and Ne ionization by an equi-velocity proton. The multiple target ionization with projectile elect...

Absolute dissociative and non-dissociative ionization cross sections of CHn+ (n = 0–4) molecules by protons in the 0.5–3.5 MeV impact energy range have been measured. The present results are in good agreement with the previously published data, where overlap occurs. A decay-route model for CH4 dissociation induced by proton impact is proposed and a...

We measured the Neq+ charge-state yield distribution for Ne atoms multiply ionized by 1-MeV/amu H2+, in coincidence with the final state of the projectile, for both breakup and nonbreakup channels. Measurements with 1-MeV H+ projectiles were also performed. While the H2+ nonbreakup channel produces results similar to equal-velocity proton or electr...

Absolute cross sections, σq+, for multiple ionization of He (q = 1), Ne (q = 1, 2, and 3), Ar (q = 1, 2, 3, and 4), Kr (q = 1, 2, 3, and 4), and Xe (q = 1, 2, 3, 4, and 5) atoms by protons in the 0.75–3.5 MeV impact energy range have been measured. The present results are in good agreement with the previously published data in the case of the singl...

Absolute cross sections for single and multiple ionization of Ne by fast (0.75–3.50 MeV) protons have been measured. The comparison with available theoretical calculations shows that the inclusion of the contribution from time-delayed post-collisional mechanisms is needed for a good description of the experimental results for double and triple ioni...

The underlying electrical mechanism associated with self-sustained arrhythmias such as cardiac flutter and fibrillation is not completely elucidated. Most of the evidence points towards a reentry path, but the hypothesis of firing of an ectopic focus with high frequency is still being considered. This paper evaluates the possibility of distinguishi...

A technique was previously developed, based on magnetic field measurements, to localize hypodermic and sewing needles lost in the human body, with the purpose of surgical extraction. The measurements are performed using a SQUID magnetometer, which detects the magnetic field associated with the remanent magnetization of the needle. The technique all...

this paper, but the obtained results appeared not to be completely reliable, probably due to the observed asymmetry. The recovery method is presently being modified to take into account possible sources of asymmetry, so as to better evaluate the parameters that describe the circulating wave. Although the present work has used a single channel equip...

The aim of this project is to provide strategies for detecting mesoscopic/macroscopic quantum super- positions and Einstein-Podolsky-Rosen paradoxes in mixed states that accurately represent the output of physical systems used to generate squeezing and entanglement. The paradox of macroscopic en- tanglement occurs where we have a quantum superposit...

In Wiseman et. al. (Phys. Rev. Lett. 98, 140402, 2007) the authors proposed a distinction between the nonlocality classes of Bell nonlocality, EPR steering and entanglement based on whether or not an overseer trusts each party in a bipartite scenario where they are asked to demonstrate entanglement. Here we extend that concept to the multipartite c...

This project considers situations in which beyond mean-field effects are important in the dynamics of Bose gases even at zero temperature. Typically we make use of the truncated Wigner method for solving the quantum evolution of a Bose-condensed gas [1]. The inclusion of quantum noise in the intial conditions means that the technique can incorporat...