Quantum foundations

Last updated

Quantum foundations is a discipline of science that seeks to understand the most counter-intuitive aspects of quantum theory, reformulate it and even propose new generalizations thereof. Contrary to other physical theories, such as general relativity, the defining axioms of quantum theory are quite ad hoc, with no obvious physical intuition. While they lead to the right experimental predictions, they do not come with a mental picture of the world where they fit.

Contents

There exist different approaches to resolve this conceptual gap:

Research in quantum foundations is structured along these roads.

Non-classical features of quantum theory

Quantum nonlocality

Two or more separate parties conducting measurements over a quantum state can observe correlations which cannot be explained with any local hidden variable theory. [1] [2] Whether this should be regarded as proving that the physical world itself is "nonlocal" is a topic of debate, [3] [4] but the terminology of "quantum nonlocality" is commonplace. Nonlocality research efforts in quantum foundations focus on determining the exact limits that classical or quantum physics enforces on the correlations observed in a Bell experiment or more complex causal scenarios. [5] This research program has so far provided a generalization of Bell's theorem that allows falsifying all classical theories with a superluminal, yet finite, hidden influence. [6]

Quantum contextuality

Nonlocality can be understood as an instance of quantum contextuality. A situation is contextual when the value of an observable depends on the context in which it is measured (namely, on which other observables are being measured as well). The original definition of measurement contextuality can be extended to state preparations and even general physical transformations. [7]

Epistemic models for the quantum wave-function

A physical property is epistemic when it represents our knowledge or beliefs on the value of a second, more fundamental feature. The probability of an event to occur is an example of an epistemic property. In contrast, a non-epistemic or ontic variable captures the notion of a “real” property of the system under consideration.

There is an on-going debate on whether the wave-function represents the epistemic state of a yet to be discovered ontic variable or, on the contrary, it is a fundamental entity. [8] Under some physical assumptions, the Pusey–Barrett–Rudolph (PBR) theorem demonstrates the inconsistency of quantum states as epistemic states, in the sense above. [9] Note that, in QBism [10] and Copenhagen-type [11] views, quantum states are still regarded as epistemic, not with respect to some ontic variable, but to one's expectations about future experimental outcomes. The PBR theorem does not exclude such epistemic views on quantum states.

Axiomatic reconstructions

Some of the counter-intuitive aspects of quantum theory, as well as the difficulty to extend it, follow from the fact that its defining axioms lack a physical motivation. An active area of research in quantum foundations is therefore to find alternative formulations of quantum theory which rely on physically compelling principles. Those efforts come in two flavors, depending on the desired level of description of the theory: the so-called Generalized Probabilistic Theories approach and the Black boxes approach.

The framework of generalized probabilistic theories

Generalized Probabilistic Theories (GPTs) are a general framework to describe the operational features of arbitrary physical theories. Essentially, they provide a statistical description of any experiment combining state preparations, transformations and measurements. The framework of GPTs can accommodate classical and quantum physics, as well as hypothetical non-quantum physical theories which nonetheless possess quantum theory's most remarkable features, such as entanglement or teleportation. [12] Notably, a small set of physically motivated axioms is enough to single out the GPT representation of quantum theory. [13]

L. Hardy introduced the concept of GPT in 2001, in an attempt to re-derive quantum theory from basic physical principles. [13] Although Hardy's work was very influential (see the follow-ups below), one of his axioms was regarded as unsatisfactory: it stipulated that, of all the physical theories compatible with the rest of the axioms, one should choose the simplest one. [14] The work of Dakic and Brukner eliminated this “axiom of simplicity” and provided a reconstruction of quantum theory based on three physical principles. [14] This was followed by the more rigorous reconstruction of Masanes and Müller. [15]

Axioms common to these three reconstructions are:

An alternative GPT reconstruction proposed by Chiribella et al. [16] [17] around the same time is also based on the

The use of purification to characterize quantum theory has been criticized on the grounds that it also applies in the Spekkens toy model. [18]

To the success of the GPT approach, it can be countered that all such works just recover finite dimensional quantum theory. In addition, none of the previous axioms can be experimentally falsified unless the measurement apparatuses are assumed to be tomographically complete.

Categorical quantum mechanics or process theories

Categorical Quantum Mechanics (CQM) or Process Theories are a general framework to describe physical theories, with an emphasis on processes and their compositions. [19] It was pioneered by Samson Abramsky and Bob Coecke. Besides its influence in quantum foundations, most notably the use of a diagrammatic formalism, CQM also plays an important role in quantum technologies, most notably in the form of ZX-calculus. It also has been used to model theories outside of physics, for example the DisCoCat compositional natural language meaning model.

The framework of black boxes

In the black box or device-independent framework, an experiment is regarded as a black box where the experimentalist introduces an input (the type of experiment) and obtains an output (the outcome of the experiment). Experiments conducted by two or more parties in separate labs are hence described by their statistical correlations alone.

From Bell's theorem, we know that classical and quantum physics predict different sets of allowed correlations. It is expected, therefore, that far-from-quantum physical theories should predict correlations beyond the quantum set. In fact, there exist instances of theoretical non-quantum correlations which, a priori, do not seem physically implausible. [20] [21] [22] The aim of device-independent reconstructions is to show that all such supra-quantum examples are precluded by a reasonable physical principle.

The physical principles proposed so far include no-signalling, [22] Non-Trivial Communication Complexity, [23] No-Advantage for Nonlocal computation, [24] Information Causality, [25] Macroscopic Locality, [26] and Local Orthogonality. [27] All these principles limit the set of possible correlations in non-trivial ways. Moreover, they are all device-independent: this means that they can be falsified under the assumption that we can decide if two or more events are space-like separated. The drawback of the device-independent approach is that, even when taken together, all the afore-mentioned physical principles do not suffice to single out the set of quantum correlations. [28] In other words: all such reconstructions are partial.

Interpretations of quantum theory

An interpretation of quantum theory is a correspondence between the elements of its mathematical formalism and physical phenomena. For instance, in the pilot wave theory, the quantum wave function is interpreted as a field that guides the particle trajectory and evolves with it via a system of coupled differential equations. Most interpretations of quantum theory stem from the desire to solve the quantum measurement problem.

Extensions of quantum theory

In an attempt to reconcile quantum and classical physics, or to identify non-classical models with a dynamical causal structure, some modifications of quantum theory have been proposed.

Collapse models

Collapse models posit the existence of natural processes which periodically localize the wave-function. [29] Such theories provide an explanation to the nonexistence of superpositions of macroscopic objects, at the cost of abandoning unitarity and exact energy conservation.

Quantum measure theory

In Sorkin's quantum measure theory (QMT), physical systems are not modeled via unitary rays and Hermitian operators, but through a single matrix-like object, the decoherence functional. [30] The entries of the decoherence functional determine the feasibility to experimentally discriminate between two or more different sets of classical histories, as well as the probabilities of each experimental outcome. In some models of QMT the decoherence functional is further constrained to be positive semidefinite (strong positivity). Even under the assumption of strong positivity, there exist models of QMT which generate stronger-than-quantum Bell correlations. [31]

Acausal quantum processes

The formalism of process matrices starts from the observation that, given the structure of quantum states, the set of feasible quantum operations follows from positivity considerations. Namely, for any linear map from states to probabilities one can find a physical system where this map corresponds to a physical measurement. Likewise, any linear transformation that maps composite states to states corresponds to a valid operation in some physical system. In view of this trend, it is reasonable to postulate that any high-order map from quantum instruments (namely, measurement processes) to probabilities should also be physically realizable. [32] Any such map is termed a process matrix. As shown by Oreshkov et al., [32] some process matrices describe situations where the notion of global causality breaks.

The starting point of this claim is the following mental experiment: two parties, Alice and Bob, enter a building and end up in separate rooms. The rooms have ingoing and outgoing channels from which a quantum system periodically enters and leaves the room. While those systems are in the lab, Alice and Bob are able to interact with them in any way; in particular, they can measure some of their properties.

Since Alice and Bob's interactions can be modeled by quantum instruments, the statistics they observe when they apply one instrument or another are given by a process matrix. As it turns out, there exist process matrices which would guarantee that the measurement statistics collected by Alice and Bob is incompatible with Alice interacting with her system at the same time, before or after Bob, or any convex combination of these three situations. [32] Such processes are called acausal.

See also

Related Research Articles

<span class="mw-page-title-main">Einstein–Podolsky–Rosen paradox</span> Historical critique of quantum mechanics

The Einstein–Podolsky–Rosen (EPR) paradox is a thought experiment proposed by physicists Albert Einstein, Boris Podolsky and Nathan Rosen which argues that the description of physical reality provided by quantum mechanics is incomplete. In a 1935 paper titled "Can Quantum-Mechanical Description of Physical Reality be Considered Complete?", they argued for the existence of "elements of reality" that were not part of quantum theory, and speculated that it should be possible to construct a theory containing these hidden variables. Resolutions of the paradox have important implications for the interpretation of quantum mechanics.

<span class="mw-page-title-main">Quantum entanglement</span> Correlation between quantum systems

Quantum entanglement is the phenomenon of a group of particles being generated, interacting, or sharing spatial proximity in such a way that the quantum state of each particle of the group cannot be described independently of the state of the others, including when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between classical and quantum physics: entanglement is a primary feature of quantum mechanics not present in classical mechanics.

The de Broglie–Bohm theory, also known as the pilot wave theory, Bohmian mechanics, Bohm's interpretation, and the causal interpretation, is an interpretation of quantum mechanics. It postulates that in addition to the wavefunction, an actual configuration of particles exists, even when unobserved. The evolution over time of the configuration of all particles is defined by a guiding equation. The evolution of the wave function over time is given by the Schrödinger equation. The theory is named after Louis de Broglie (1892–1987) and David Bohm (1917–1992).

Bell's theorem is a term encompassing a number of closely related results in physics, all of which determine that quantum mechanics is incompatible with local hidden-variable theories, given some basic assumptions about the nature of measurement. "Local" here refers to the principle of locality, the idea that a particle can only be influenced by its immediate surroundings, and that interactions mediated by physical fields cannot propagate faster than the speed of light. "Hidden variables" are putative properties of quantum particles that are not included in quantum theory but nevertheless affect the outcome of experiments. In the words of physicist John Stewart Bell, for whom this family of results is named, "If [a hidden-variable theory] is local it will not agree with quantum mechanics, and if it agrees with quantum mechanics it will not be local."

In physics, a hidden-variable theory is a deterministic physical model which seeks to explain the probabilistic nature of quantum mechanics by introducing additional variables.

In physics, the principle of locality states that an object is influenced directly only by its immediate surroundings. A theory that includes the principle of locality is said to be a "local theory". This is an alternative to the concept of instantaneous, or "non-local" action at a distance. Locality evolved out of the field theories of classical physics. The idea is that for a cause at one point to have an effect at another point, something in the space between those points must mediate the action. To exert an influence, something, such as a wave or particle, must travel through the space between the two points, carrying the influence.

A Bell test, also known as Bell inequality test or Bell experiment, is a real-world physics experiment designed to test the theory of quantum mechanics in relation to Albert Einstein's concept of local realism. Named for John Stewart Bell, the experiments test whether or not the real world satisfies local realism, which requires the presence of some additional local variables to explain the behavior of particles like photons and electrons. As of 2015, all Bell tests have found that the hypothesis of local hidden variables is inconsistent with the way that physical systems behave.

In the interpretation of quantum mechanics, a local hidden-variable theory is a hidden-variable theory that satisfies the principle of locality. These models attempt to account for the probabilistic features of quantum mechanics via the mechanism of underlying, but inaccessible variables, with the additional requirement that distant events be statistically independent.

In quantum mechanics, the Kochen–Specker (KS) theorem, also known as the Bell–KS theorem, is a "no-go" theorem proved by John S. Bell in 1966 and by Simon B. Kochen and Ernst Specker in 1967. It places certain constraints on the permissible types of hidden-variable theories, which try to explain the predictions of quantum mechanics in a context-independent way. The version of the theorem proved by Kochen and Specker also gave an explicit example for this constraint in terms of a finite number of state vectors.

<span class="mw-page-title-main">Greenberger–Horne–Zeilinger state</span> "Highly entangled" quantum state of 3 or more qubits

In physics, in the area of quantum information theory, a Greenberger–Horne–Zeilinger state is a certain type of entangled quantum state that involves at least three subsystems. The four-particle version was first studied by Daniel Greenberger, Michael Horne and Anton Zeilinger in 1989, and the three-particle version was introduced by N. David Mermin in 1990. Extremely non-classical properties of the state have been observed. GHZ states for large numbers of qubits are theorized to give enhanced performance for metrology compared to other qubit superposition states.

In mathematical physics, Gleason's theorem shows that the rule one uses to calculate probabilities in quantum physics, the Born rule, can be derived from the usual mathematical representation of measurements in quantum physics together with the assumption of non-contextuality. Andrew M. Gleason first proved the theorem in 1957, answering a question posed by George W. Mackey, an accomplishment that was historically significant for the role it played in showing that wide classes of hidden-variable theories are inconsistent with quantum physics. Multiple variations have been proven in the years since. Gleason's theorem is of particular importance for the field of quantum logic and its attempt to find a minimal set of mathematical axioms for quantum theory.

In theoretical physics, quantum nonlocality refers to the phenomenon by which the measurement statistics of a multipartite quantum system do not allow an interpretation with local realism. Quantum nonlocality has been experimentally verified under a variety of physical assumptions. Any physical theory that aims at superseding or replacing quantum theory should account for such experiments and therefore cannot fulfill local realism; quantum nonlocality is a property of the universe that is independent of our description of nature.

The Leggett–Garg inequality, named for Anthony James Leggett and Anupam Garg, is a mathematical inequality fulfilled by all macrorealistic physical theories. Here, macrorealism is a classical worldview defined by the conjunction of two postulates:

  1. Macrorealism per se: "A macroscopic object, which has available to it two or more macroscopically distinct states, is at any given time in a definite one of those states."
  2. Noninvasive measurability: "It is possible in principle to determine which of these states the system is in without any effect on the state itself, or on the subsequent system dynamics."

The Leggett inequalities, named for Anthony James Leggett, who derived them, are a related pair of mathematical expressions concerning the correlations of properties of entangled particles. They are fulfilled by a large class of physical theories based on particular non-local and realistic assumptions, that may be considered to be plausible or intuitive according to common physical reasoning.

In quantum information theory, quantum discord is a measure of nonclassical correlations between two subsystems of a quantum system. It includes correlations that are due to quantum physical effects but do not necessarily involve quantum entanglement.

<span class="mw-page-title-main">Quantum Bayesianism</span> Interpretation of quantum mechanics

In physics and the philosophy of physics, quantum Bayesianism is a collection of related approaches to the interpretation of quantum mechanics, the most prominent of which is QBism. QBism is an interpretation that takes an agent's actions and experiences as the central concerns of the theory. QBism deals with common questions in the interpretation of quantum theory about the nature of wavefunction superposition, quantum measurement, and entanglement. According to QBism, many, but not all, aspects of the quantum formalism are subjective in nature. For example, in this interpretation, a quantum state is not an element of reality—instead, it represents the degrees of belief an agent has about the possible outcomes of measurements. For this reason, some philosophers of science have deemed QBism a form of anti-realism. The originators of the interpretation disagree with this characterization, proposing instead that the theory more properly aligns with a kind of realism they call "participatory realism", wherein reality consists of more than can be captured by any putative third-person account of it.

Quantum contextuality is a feature of the phenomenology of quantum mechanics whereby measurements of quantum observables cannot simply be thought of as revealing pre-existing values. Any attempt to do so in a realistic hidden-variable theory leads to values that are dependent upon the choice of the other (compatible) observables which are simultaneously measured. More formally, the measurement result of a quantum observable is dependent upon which other commuting observables are within the same measurement set.

<span class="mw-page-title-main">Sandu Popescu</span> British physicist

Sandu Popescu is a Romanian-British physicist working in the foundations of quantum mechanics and quantum information.

A generalized probabilistic theory (GPT) is a general framework to describe the operational features of arbitrary physical theories. A GPT must specify what kind of physical systems one can find in the lab, as well as rules to compute the outcome statistics of any experiment involving labeled preparations, transformations and measurements. The framework of GPTs has been used to define hypothetical non-quantum physical theories which nonetheless possess quantum theory's most remarkable features, such as entanglement or teleportation. Notably, a small set of physically motivated axioms is enough to single out the GPT representation of quantum theory.

Robert W. Spekkens is a Canadian theoretical quantum physicist working in the fields of quantum foundations and quantum information.

References

  1. ↑ Bell, J. S. (1964). "On the Einstein Podolsky Rosen Paradox" (PDF). Physics Physique Физика . 1 (3): 195–200. doi: 10.1103/PhysicsPhysiqueFizika.1.195 .
  2. ↑ Mermin, N. David (July 1993). "Hidden Variables and the Two Theorems of John Bell". Reviews of Modern Physics . 65 (3): 803–15. arXiv: 1802.10119 . Bibcode:1993RvMP...65..803M. doi:10.1103/RevModPhys.65.803. S2CID   119546199.
  3. ↑ Werner, R. F. (2014). "Comment on 'What Bell did'". Journal of Physics A . 47 (42): 424011. Bibcode:2014JPhA...47P4011W. doi:10.1088/1751-8113/47/42/424011. S2CID   122180759.
  4. ↑ Żukowski, M.; Brukner, Č. (2014). "Quantum non-locality—it ain't necessarily so...". Journal of Physics A . 47 (42): 424009. arXiv: 1501.04618 . Bibcode:2014JPhA...47P4009Z. doi:10.1088/1751-8113/47/42/424009. S2CID   119220867.
  5. ↑ Fritz, T. (2012). "Beyond Bell's Theorem: Correlation Scenarios". New Journal of Physics . 14 (10): 103001. arXiv: 1206.5115 . Bibcode:2012NJPh...14j3001F. doi: 10.1088/1367-2630/14/10/103001 .
  6. ↑ Bancal, Jean-Daniel; Pironio, Stefano; Acín, Antonio; Liang, Yeong-Cherng; Scarani, Valerio; Gisin, Nicolas (2012). "Quantum nonlocality based on finite-speed causal influences leads to superluminal signaling". Nature Physics . 8 (12): 867–870. arXiv: 1110.3795 . Bibcode:2012NatPh...8..867B. doi: 10.1038/nphys2460 .
  7. ↑ Spekkens, R. W. (2005). "Contextuality for preparations, transformations, and unsharp measurements". Physical Review A . 71 (5): 052108. arXiv: quant-ph/0406166 . Bibcode:2005PhRvA..71e2108S. doi:10.1103/PhysRevA.71.052108. S2CID   38186461.
  8. ↑ Harrigan, N.; R. W. Spekkens (2010). "Einstein, Incompleteness, and the Epistemic View of Quantum States". Foundations of Physics . 40 (2): 125–157. arXiv: 0706.2661 . Bibcode:2010FoPh...40..125H. doi:10.1007/s10701-009-9347-0. S2CID   32755624.
  9. ↑ Pusey, M. F.; Barrett, J.; Rudolph, T. (2012). "On the reality of the quantum state". Nature Physics . 8 (6): 475–478. arXiv: 1111.3328 . Bibcode:2012NatPh...8..476P. doi:10.1038/nphys2309. S2CID   14618942.
  10. ↑ Fuchs, C. A. (2010). "QBism, the Perimeter of Quantum Bayesianism". arXiv: 1003.5209 [quant-ph].
  11. ↑ Schlosshauer, M.; Kofler, J.; Zeilinger, A. (2013). "A snapshot of foundational attitudes toward quantum mechanics". Studies in History and Philosophy of Science Part B . 44 (3): 222–230. arXiv: 1301.1069 . Bibcode:2013SHPMP..44..222S. doi:10.1016/j.shpsb.2013.04.004. S2CID   55537196.
  12. ↑ Barnum, H.; Barrett, J.; Leifer, M.; Wilce, A. (2012). S. Abramsky and M. Mislove (ed.). Teleportation in General Probabilistic Theories. AMS Proceedings of Symposia in Applied Mathematics. American Mathematical Society, Providence.
  13. 1 2 Hardy, L. (2001). "Quantum Theory From Five Reasonable Axioms". arXiv: quant-ph/0101012 .
  14. 1 2 Dakic, B.; Brukner, Č. (2011). "Quantum Theory and Beyond: Is Entanglement Special?". In H. Halvorson (ed.). Deep Beauty: Understanding the Quantum World through Mathematical Innovation. Cambridge University Press. pp. 365–392.
  15. ↑ Masanes, L.; Müller, M. (2011). "A derivation of quantum theory from physical requirements". New Journal of Physics . 13 (6): 063001. arXiv: 1004.1483 . Bibcode:2011NJPh...13f3001M. doi:10.1088/1367-2630/13/6/063001. S2CID   4806946.
  16. ↑ Chiribella, G.; D'Ariano, G. M.; Perinotti, P. (2011). "Informational derivation of Quantum Theory". Phys. Rev. A . 84 (1): 012311. arXiv: 1011.6451 . Bibcode:2011PhRvA..84a2311C. doi:10.1103/PhysRevA.84.012311. S2CID   15364117.
  17. ↑ D'Ariano, G. M.; Chiribella, G.; Perinotti, P. (2017). Quantum Theory from First Principles: An Informational Approach. Cambridge University Press. ISBN   9781107338340. OCLC   972460315.
  18. ↑ Appleby, M.; Fuchs, C. A.; Stacey, B. C.; Zhu, H. (2017). "Introducing the Qplex: a novel arena for quantum theory". European Physical Journal D . 71 (7): 197. arXiv: 1612.03234 . Bibcode:2017EPJD...71..197A. doi:10.1140/epjd/e2017-80024-y. S2CID   119240836.
  19. ↑ Coecke, Bob; Kissinger, Aleks (2017). Picturing Quantum Processes: a first course in quantum theory and diagrammatic reasoning. Cambridge, United Kingdom: Cambridge University Press. ISBN   978-1-316-21931-7. OCLC   983730394.
  20. ↑ Rastall, Peter (1985). "Locality, Bell's theorem, and quantum mechanics". Foundations of Physics . 15 (9): 963–972. Bibcode:1985FoPh...15..963R. doi:10.1007/bf00739036. S2CID   122298281.
  21. ↑ Khalfin, L.A.; Tsirelson, B. S. (1985). Lahti; et al. (eds.). Quantum and quasi-classical analogs of Bell inequalities. Symposium on the Foundations of Modern Physics. World Sci. Publ. pp. 441–460.
  22. 1 2 Popescu, S.; Rohrlich, D. (1994). "Nonlocality as an axiom". Foundations of Physics . 24 (3): 379–385. doi:10.1007/BF02058098. S2CID   120333148.
  23. ↑ Brassard, G; Buhrman, H; Linden, N; Methot, AA; Tapp, A; Unger, F (2006). "Limit on Nonlocality in Any World in Which Communication Complexity Is Not Trivial". Physical Review Letters . 96 (25): 250401. arXiv: quant-ph/0508042 . Bibcode:2006PhRvL..96y0401B. doi:10.1103/PhysRevLett.96.250401. PMID   16907289. S2CID   6135971.
  24. ↑ Linden, N.; Popescu, S.; Short, A. J.; Winter, A. (2007). "Quantum Nonlocality and Beyond: Limits from Nonlocal Computation". Physical Review Letters . 99 (18): 180502. arXiv: quant-ph/0610097 . Bibcode:2007PhRvL..99r0502L. doi:10.1103/PhysRevLett.99.180502. PMID   17995388.
  25. ↑ Pawlowski, M.; Paterek, T.; Kaszlikowski, D.; Scarani, V.; Winter, A.; Zukowski, M. (October 2009). "Information Causality as a Physical Principle". Nature . 461 (7267): 1101–1104. arXiv: 0905.2292 . Bibcode:2009Natur.461.1101P. doi:10.1038/nature08400. PMID   19847260. S2CID   4428663.
  26. ↑ Navascués, M.; H. Wunderlich (2009). "A Glance Beyond the Quantum Model". Proc. R. Soc. A . 466 (2115): 881–890. arXiv: 0907.0372 . doi: 10.1098/rspa.2009.0453 .
  27. ↑ Fritz, T.; Sainz, A. B.; Augusiak, R.; Brask, J. B.; Chaves, R.; Leverrier, A.; Acín, A. (2013). "Local orthogonality as a multipartite principle for quantum correlations". Nature Communications . 4: 2263. arXiv: 1210.3018 . Bibcode:2013NatCo...4.2263F. doi:10.1038/ncomms3263. PMID   23948952. S2CID   14759956.
  28. ↑ Navascués, M.; Guryanova, Y.; Hoban, M. J.; Acín, A. (2015). "Almost Quantum Correlations". Nature Communications . 6: 6288. arXiv: 1403.4621 . Bibcode:2015NatCo...6.6288N. doi:10.1038/ncomms7288. PMID   25697645. S2CID   12810715.
  29. ↑ Ghirardi, G. C.; A. Rimini; T. Weber (1986). "Unified dynamics for microscopic and macroscopic systems". Physical Review D . 34 (2): 470–491. Bibcode:1986PhRvD..34..470G. doi:10.1103/PhysRevD.34.470. PMID   9957165.
  30. ↑ Sorkin, R. D. (1994). "Quantum Mechanics as Quantum Measure Theory". Mod. Phys. Lett. A . 9 (33): 3119–3128. arXiv: gr-qc/9401003 . Bibcode:1994MPLA....9.3119S. doi:10.1142/S021773239400294X. S2CID   18938710.
  31. ↑ Dowker, F.; Henson, J.; Wallden, P. (2014). "A histories perspective on characterizing quantum non-locality". New Journal of Physics . 16 (3): 033033. Bibcode:2014NJPh...16c3033D. doi: 10.1088/1367-2630/16/3/033033 . hdl: 10044/1/60886 .
  32. 1 2 3 Oreshkov, O.; Costa, F.; Brukner, C. (2012). "Quantum correlations with no causal order". Nature Communications . 3: 1092–. arXiv: 1105.4464 . Bibcode:2012NatCo...3.1092O. doi: 10.1038/ncomms2076 . PMC   3493644 . PMID   23033068.
This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.