A **Tsirelson bound** is an upper limit to quantum mechanical correlations between distant events. Given that quantum mechanics is non-local (i.e., that quantum mechanical correlations violate Bell inequalities), a natural question to ask is "how non-local can quantum mechanics be?", or, more precisely, by how much can the Bell inequality be violated within quantum mechanics. The answer is precisely the Tsirelson bound for the particular Bell inequality in question. In general, this bound is lower than the bound that would be obtained if more general theories, only constrained by "no-signalling" (i.e., that they do not permit communication faster than light), were considered, and much research has been dedicated to the question of why this is the case.

- Bound for the CHSH inequality
- Other Bell inequalities
- Derivation from physical principles
- Tsirelson's problem
- See also
- References

The Tsirelson bounds are named after Boris S. Tsirelson (or Cirel'son, in a different transliteration), the author of the article^{ [1] } in which the first one was derived.

The first Tsirelson bound was derived as an upper bound on the correlations measured in the CHSH inequality. It states that if we have four (Hermitian) dichotomic observables , , , (i.e., two observables for Alice and two for Bob) with outcomes such that for all , then

For comparison, in the classical case (or local realistic case) the upper bound is 2, whereas if any arbitrary assignment of is allowed, it is 4. The Tsirelson bound is attained already if Alice and Bob each makes measurements on a qubit, the simplest non-trivial quantum system.

Several proofs of this bound exist, but perhaps the most enlightening one is based on the Khalfin–Tsirelson–Landau identity. If we define an observable

and , i.e., if the observables' outcomes are , then

If or , which can be regarded as the classical case, it already follows that . In the quantum case, we need only notice that , and the Tsirelson bound follows.

Tsirelson also showed that for any bipartite full-correlation Bell inequality with *m* inputs for Alice and *n* inputs for Bob, the ratio between the Tsirelson bound and the local bound is at most where and is the Grothendieck constant of order *d*.^{ [2] } Note that since , this bound implies the above result about the CHSH inequality.

In general, obtaining a Tsirelson bound for a given Bell inequality is a hard problem that has to be solved on a case-by-case basis. It is not even known to be decidable. The best known computational method for upperbounding it is a convergent hierarchy of semidefinite programs, the NPA hierarchy, that in general does not halt.^{ [3] }^{ [4] } The exact values are known for a few more Bell inequalities:

For the Braunstein–Caves inequalities we have that

For the WWŻB inequalities the Tsirelson bound is

For the inequality the Tsirelson bound is not known exactly, but concrete realisations give a lower bound of 0.25087538, and the NPA hierarchy gives an upper bound of 0.25087539. It is conjectured that only infinite-dimensional quantum states can reach the Tsirelson bound.^{ [5] }^{ [6] }

Significant research has been dedicated to finding a physical principle that explains why quantum correlations go only up to the Tsirelson bound and nothing more. Three such principles have been found: no-advantage for non-local computation,^{ [7] } information causality ^{ [8] } and macroscopic locality.^{ [9] } That is to say, if one could achieve a CHSH correlation exceeding Tsirelson's bound, all such principles would be violated. Tsirelson's bound also follows if the Bell experiment admits a strongly positive quansal measure.^{ [10] }

There are two different ways of defining the Tsirelson bound of a Bell expression. One by demanding that the measurements are in a tensor product structure, and another by demanding only that they commute. Tsirelson's problem is the question of whether these two definitions are equivalent. More formally, let

be a Bell expression, where is the probability of obtaining outcomes with the settings . The tensor product Tsirelson bound is then the supremum of the value attained in this Bell expression by making measurements and on a quantum state :

The commuting Tsirelson bound is the supremum of the value attained in this Bell expression by making measurements and such that on a quantum state :

Since tensor product algebras in particular commute, . In finite dimensions commuting algebras are always isomorphic to (direct sums of) tensor product algebras, so only for infinite dimensions it is possible that . Tsirelson's problem is the question of whether for all Bell expressions .

This question was first considered by Boris Tsirelson in 1993, where he asserted without proof that .^{ [11] } Upon being asked for a proof by Antonio Acín in 2006, he realized that the one he had in mind didn't work,^{ [12] } and issued the question as an open problem.^{ [13] } Together with Miguel Navascués and Stefano Pironio, Antonio Acín had developed an hierarchy of semidefinite programs, the NPA hierarchy, that converged to the commuting Tsirelson bound from above,^{ [4] } and wanted to know whether it also converged to the tensor product Tsirelson bound , the most physically relevant one.

Since one can produce a converging sequencing of approximations to from below by considering finite-dimensional states and observables, if , then this procedure can be combined with the NPA hierarchy to produce a halting algorithm to compute the Tsirelson bound, making it a computable number (note that in isolation neither procedure halts in general). Conversely, if is not computable, then . In January 2020, Ji, Natarajan, Vidick, Wright, and Yuen claimed to have proven that is not computable, thus solving Tsirelson's problem.^{ [14] }

Tsirelson's problem has been shown to be equivalent to Connes' embedding problem.^{ [15] }

In quantum mechanics, **bra–ket notation**, or **Dirac notation**, is used ubiquitously to denote quantum states. The notation uses angle brackets, and , and a vertical bar , to construct "bras" and "kets".

In quantum mechanics, the **uncertainty principle** is any of a variety of mathematical inequalities asserting a fundamental limit to the accuracy with which the values for certain pairs of physical quantities of a particle, such as position, *x*, and momentum, *p*, can be predicted from initial conditions.

**Bell's theorem** proves that quantum physics is incompatible with local hidden-variable theories. It was introduced by physicist John Stewart Bell in a 1964 paper titled "On the Einstein Podolsky Rosen Paradox", referring to a 1935 thought experiment that Albert Einstein, Boris Podolsky and Nathan Rosen used to argue that quantum physics is an "incomplete" theory. By 1935, it was already recognized that the predictions of quantum physics are probabilistic. Einstein, Podolsky and Rosen presented a scenario that, in their view, indicated that quantum particles, like electrons and photons, must carry physical properties or attributes not included in quantum theory, and the uncertainties in quantum theory's predictions were due to ignorance of these properties, later termed "hidden variables". Their scenario involves a pair of widely separated physical objects, prepared in such a way that the quantum state of the pair is entangled.

In quantum mechanics, a **density matrix** is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using the Born rule. It is a generalization of the more usual state vectors or wavefunctions: while those can only represent pure states, density matrices can also represent *mixed states*. Mixed states arise in quantum mechanics in two different situations: first when the preparation of the system is not fully known, and thus one must deal with a statistical ensemble of possible preparations, and second when one wants to describe a physical system which is entangled with another, as its state can not be described by a pure state.

In physics, the **CHSH inequality** can be used in the proof of Bell's theorem, which states that certain consequences of entanglement in quantum mechanics can not be reproduced by local hidden variable theories. Experimental verification of violation of the inequalities is seen as experimental confirmation that nature cannot be described by local hidden variables theories. CHSH stands for John Clauser, Michael Horne, Abner Shimony, and Richard Holt, who described it in a much-cited paper published in 1969. They derived the CHSH inequality, which, as with John Bell's original inequality, is a constraint on the statistics of "coincidences" in a Bell test which is necessarily true if there exist underlying local hidden variables. This constraint can, on the other hand, be infringed by quantum mechanics.

**Quantum decoherence** is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wave function is used to explain various quantum effects. As long as there exists a definite phase relation between different states, the system is said to be coherent. A definite phase relationship is necessary to perform quantum computing on quantum information encoded in quantum states. Coherence is preserved under the laws of quantum physics.

In mathematics as well as physics, a linear operator acting on an inner product space is called **positive-semidefinite** if, for every , and , where is the domain of . Positive-semidefinite operators are denoted as . The operator is said to be **positive-definite**, and written , if for all .

In functional analysis and quantum measurement theory, a **positive operator-valued measure** (**POVM**) is a measure whose values are positive semi-definite operators on a Hilbert space. POVMs are a generalisation of projection-valued measures (PVM) and, correspondingly, quantum measurements described by POVMs are a generalisation of quantum measurement described by PVMs.

In quantum mechanics, notably in quantum information theory, **fidelity** is a measure of the "closeness" of two quantum states. It expresses the probability that one state will pass a test to identify as the other. The fidelity is not a metric on the space of density matrices, but it can be used to define the Bures metric on this space.

**Photon polarization** is the quantum mechanical description of the classical polarized sinusoidal plane electromagnetic wave. An individual photon can be described as having right or left circular polarization, or a superposition of the two. Equivalently, a photon can be described as having horizontal or vertical linear polarization, or a superposition of the two.

In theoretical physics, **quantum nonlocality** refers to the phenomenon by which the measurement statistics of a multipartite quantum system do not admit an interpretation in terms of a local realistic theory. Quantum nonlocality has been experimentally verified under different 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.

In non-equilibrium physics, the **Keldysh formalism** is a general framework for describing the quantum mechanical evolution of a system in a non-equilibrium state or systems subject to time varying external fields. Historically, it was foreshadowed by the work of Julian Schwinger and proposed almost simultaneously by Leonid Keldysh and, separately, Leo Kadanoff and Gordon Baym. It was further developed by later contributors such as O. V. Konstantinov and V. I. Perel.

**Amplitude amplification** is a technique in quantum computing which generalizes the idea behind the Grover's search algorithm, and gives rise to a family of quantum algorithms. It was discovered by Gilles Brassard and Peter Høyer in 1997, and independently rediscovered by Lov Grover in 1998.

**Entanglement distillation** is the transformation of *N* copies of an arbitrary entangled state into some number of approximately pure Bell pairs, using only local operations and classical communication.

In quantum mechanics, and especially quantum information theory, the **purity** of a normalized quantum state is a scalar defined as

**Coherent states** have been introduced in a physical context, first as quasi-classical states in quantum mechanics, then as the backbone of quantum optics and they are described in that spirit in the article Coherent states. However, they have generated a huge variety of generalizations, which have led to a tremendous amount of literature in mathematical physics. In this article, we sketch the main directions of research on this line. For further details, we refer to several existing surveys.

Given a Hilbert space with a tensor product structure a **product numerical range** is defined as a numerical range with respect to the subset of product vectors. In some situations, especially in the context of quantum mechanics product numerical range is known as **local numerical range**

The theory of **causal fermion systems** is an approach to describe fundamental physics. It provides a unification of the weak, the strong and the electromagnetic forces with gravity at the level of classical field theory. Moreover, it gives quantum mechanics as a limiting case and has revealed close connections to quantum field theory. Therefore, it is a candidate for a unified physical theory. Instead of introducing physical objects on a preexisting spacetime manifold, the general concept is to derive spacetime as well as all the objects therein as secondary objects from the structures of an underlying causal fermion system. This concept also makes it possible to generalize notions of differential geometry to the non-smooth setting. In particular, one can describe situations when spacetime no longer has a manifold structure on the microscopic scale. As a result, the theory of causal fermion systems is a proposal for quantum geometry and an approach to quantum gravity.

The **time-dependent variational Monte Carlo (t-VMC)** method is a quantum Monte Carlo approach to study the dynamics of closed, non-relativistic quantum systems in the context of the quantum many-body problem. It is an extension of the variational Monte Carlo method, in which a time-dependent pure quantum state is encoded by some variational wave function, generally parametrized as

In quantum information theory and quantum optics, the **Schrödinger–HJW theorem** is a result about the realization of a mixed state of a quantum system as an ensemble of pure quantum states and the relation between the corresponding purifications of the density operators. The theorem is named after physicists and mathematicians Erwin Schrödinger, Lane P. Hughston, Richard Jozsa and William Wootters. The result was also found independently by Nicolas Hadjisavvas building upon work by Ed Jaynes, while a significant part of it was likewise independently discovered by N. David Mermin. Thanks to its complicated history, it is also known by various other names such as the **GHJW theorem**, the **HJW theorem**, and the **purification theorem**.

- ↑ Cirel'son, B. S. (1980). "Quantum generalizations of Bell's inequality".
*Letters in Mathematical Physics*.**4**(2): 93–100. Bibcode:1980LMaPh...4...93C. doi:10.1007/bf00417500. ISSN 0377-9017. S2CID 120680226. - ↑ Boris Tsirelson (1987). "Quantum analogues of the Bell inequalities. The case of two spatially separated domains" (PDF).
*Journal of Soviet Mathematics*.**36**(4): 557–570. doi:10.1007/BF01663472. S2CID 119363229. - ↑ Navascués, Miguel; Pironio, Stefano; Acín, Antonio (2007-01-04). "Bounding the Set of Quantum Correlations".
*Physical Review Letters*.**98**(1): 010401. arXiv: quant-ph/0607119 . Bibcode:2007PhRvL..98a0401N. doi:10.1103/physrevlett.98.010401. ISSN 0031-9007. PMID 17358458. S2CID 41742170. - 1 2 M. Navascués; S. Pironio; A. Acín (2008). "A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations".
*New Journal of Physics*.**10**(7): 073013. arXiv: 0803.4290 . Bibcode:2008NJPh...10g3013N. doi:10.1088/1367-2630/10/7/073013. S2CID 1906335. - ↑ Collins, Daniel; Gisin, Nicolas (2003-06-01). "A Relevant Two Qubit Bell Inequality Inequivalent to the CHSH Inequality".
*Journal of Physics A: Mathematical and General*.**37**(5): 1775–1787. arXiv: quant-ph/0306129 . doi:10.1088/0305-4470/37/5/021. S2CID 55647659. - ↑ K.F. Pál; T. Vértesi (2010). "Maximal violation of the I3322 inequality using infinite dimensional quantum systems".
*Physical Review A*.**82**: 022116. arXiv: 1006.3032 . doi:10.1103/PhysRevA.82.022116. - ↑ Linden, Noah; Popescu, Sandu; Short, Anthony J.; Winter, Andreas (2007-10-30). "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. ISSN 0031-9007. PMID 17995388. - ↑ Pawłowski, Marcin; Paterek, Tomasz; Kaszlikowski, Dagomir; Scarani, Valerio; Winter, Andreas; Żukowski, Marek (2009). "Information causality as a physical principle".
*Nature*.**461**(7267): 1101–1104. arXiv: 0905.2292 . Bibcode:2009Natur.461.1101P. doi:10.1038/nature08400. ISSN 0028-0836. PMID 19847260. S2CID 4428663. - ↑ Navascués, Miguel; Wunderlich, Harald (2009-11-11). "A glance beyond the quantum model".
*Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*.**466**(2115): 881–890. doi: 10.1098/rspa.2009.0453 . ISSN 1364-5021. - ↑ Craig, David; Dowker, Fay; Henson, Joe; Major, Seth; Rideout, David; Sorkin, Rafael D. (2007). "A Bell inequality analog in quantum measure theory".
*Journal of Physics A: Mathematical and Theoretical*.**40**(3): 501–523. arXiv: quant-ph/0605008 . Bibcode:2007JPhA...40..501C. doi:10.1088/1751-8113/40/3/010. ISSN 1751-8113. S2CID 8706909. - ↑ Tsirelson, B. S. (1993). "Some results and problems on quantum Bell-type inequalities" (PDF).
*Hadronic Journal Supplement*.**8**: 329–345. - ↑ Tsirelson, B. "Bell inequalities and operator algebras" . Retrieved 20 January 2020.
- ↑ Tsirelson, B. "Bell inequalities and operator algebras" (PDF). Retrieved 20 January 2020.
- ↑ Z. Ji; A. Natarajan; T. Vidick; J. Wright; H. Yuen (2020). "MIP* = RE". arXiv: 2001.04383 [quant-ph].
- ↑ M. Junge; M. Navascués; C. Palazuelos; D. Pérez-García; V. B. Scholz; R. F. Werner (2011). "Connes' embedding problem and Tsirelson's problem".
*Journal of Mathematical Physics*.**52**(1): 012102. arXiv: 1008.1142 . Bibcode:2011JMP....52a2102J. doi:10.1063/1.3514538. S2CID 12321570.

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