Editing Bell's theorem (section)

==Theorem==
There are many variations on the basic idea, some employing stronger mathematical assumptions than others.<ref name="Stanford">{{Cite SEP|bell-theorem|title=Bell's Theorem|first = Abner | last = Shimony|author-link=Abner Shimony}}</ref> Significantly, Bell-type theorems do not refer to any particular theory of local hidden variables, but instead show that quantum physics violates general assumptions behind classical pictures of nature. The original theorem proved by Bell in 1964 is not the most amenable to experiment, and it is convenient to introduce the genre of Bell-type inequalities with a later example.<ref name="mike-and-ike"/>

Hypothetical characters [[Alice and Bob]] stand in widely separated locations. Their colleague Victor prepares a pair of particles and sends one to Alice and the other to Bob. When Alice receives her particle, she chooses to perform one of two possible measurements (perhaps by flipping a coin to decide which). Denote these measurements by <math>A_0</math> and <math>A_1</math>. Both <math>A_0</math> and <math>A_1</math> are ''binary'' measurements: the result of <math>A_0</math> is either <math>+1</math> or <math>-1</math>, and likewise for <math>A_1</math>. When Bob receives his particle, he chooses one of two measurements, <math>B_0</math> and <math>B_1</math>, which are also both binary.

Suppose that each measurement reveals a property that the particle already possessed. For instance, if Alice chooses to measure <math>A_0</math> and obtains the result <math>+1</math>, then the particle she received carried a value of <math>+1</math> for a property <math>a_0</math>.{{refn|group=note|We are for convenience assuming that the response of the detector to the underlying property is deterministic. This assumption can be replaced; it is equivalent to postulating a joint probability distribution over all the observables of the experiment.<ref>{{Cite journal |last=Fine |first=Arthur |date=1982-02-01 |title=Hidden Variables, Joint Probability, and the Bell Inequalities |url=https://link.aps.org/doi/10.1103/PhysRevLett.48.291 |journal=[[Physical Review Letters]] |language=en |volume=48 |issue=5 |pages=291–295 |doi=10.1103/PhysRevLett.48.291 |bibcode=1982PhRvL..48..291F |issn=0031-9007}}</ref><ref>{{Cite journal |last1=Braunstein |first1=Samuel L. |last2=Caves |first2=Carlton M. |author-link2=Carlton M. Caves |date=August 1990 |title=Wringing out better Bell inequalities |journal=[[Annals of Physics]] |language=en |volume=202 |issue=1 |pages=22–56 |doi=10.1016/0003-4916(90)90339-P|bibcode=1990AnPhy.202...22B }}</ref>}} Consider the combination<math display="block">a_0b_0 + a_0b_1 + a_1b_0-a_1b_1 = (a_0+a_1)b_0 + (a_0-a_1)b_1 \, .</math>Because both <math>a_0</math> and <math>a_1</math> take the values <math>\pm 1</math>, then either <math>a_0 = a_1</math> or <math>a_0 = -a_1</math>. In the former case, the quantity <math>(a_0-a_1)b_1</math> must equal 0, while in the latter case, <math>(a_0+a_1)b_0 = 0</math>. So, one of the terms on the right-hand side of the above expression will vanish, and the other will equal <math>\pm 2</math>. Consequently, if the experiment is repeated over many trials, with Victor preparing new pairs of particles, the absolute value of the average of the combination <math>a_0b_0 + a_0b_1 + a_1b_0-a_1b_1</math> across all the trials will be less than or equal to 2. No ''single'' trial can measure this quantity, because Alice and Bob can only choose one measurement each, but on the assumption that the underlying properties exist, the average value of the sum is just the sum of the averages for each term. Using angle brackets to denote averages<math display="block">| \langle A_0B_0 \rangle + \langle A_0B_1 \rangle + \langle A_1B_0 \rangle - \langle A_1B_1 \rangle | \leq 2 \, .</math>
This is a Bell inequality, specifically, the [[CHSH inequality]].<ref name="mike-and-ike">{{Cite book|last1=Nielsen|first1=Michael A.|last2=Chuang|first2=Isaac L.|title=Quantum Computation and Quantum Information|author-link1=Michael Nielsen |author-link2=Isaac Chuang |title-link=Quantum Computation and Quantum Information |publisher=Cambridge University Press|location=Cambridge|year=2010|edition=2nd|oclc=844974180|isbn=978-1-107-00217-3}}</ref>{{Rp|115}} Its derivation here depends upon two assumptions: first, that the underlying physical properties <math>a_0, a_1, b_0,</math> and <math>b_1</math> exist independently of being observed or measured (sometimes called the assumption of ''realism''); and second, that Alice's choice of action cannot influence Bob's result or vice versa (often called the assumption of ''locality'').<ref name="mike-and-ike" />{{Rp|117}}

Quantum mechanics can violate the CHSH inequality, as follows. Victor prepares a pair of [[qubit]]s which he describes by the [[Bell state]]<math display="block">|\psi\rangle = \frac{|0\rangle \otimes |1\rangle - |1\rangle \otimes |0\rangle}{\sqrt{2}} ,</math>
where <math>|0\rangle</math> and <math>|1\rangle</math> are the eigenstates of one of the [[Pauli matrices]],<math display="block">\sigma_z = \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}.</math>
Victor then passes the first qubit to Alice and the second to Bob. Alice and Bob's choices of possible [[measurement in quantum mechanics|measurements]] are also defined in terms of the Pauli matrices. Alice measures either of the two observables <math>\sigma_z</math> and <math>\sigma_x</math>:<math display="block">A_0 = \sigma_z,\ A_1 = \sigma_x = \begin{pmatrix}0 & 1 \\ 1 & 0 \end{pmatrix};</math>
and Bob measures either of the two observables<math display="block">B_0 = -\frac{\sigma_x + \sigma_z}{\sqrt{2}},\ B_1 = \frac{\sigma_x - \sigma_z}{\sqrt{2}} .</math>
Victor can calculate the quantum expectation values for pairs of these observables using the [[Born rule]]:<math display="block">\langle A_0 \otimes B_0 \rangle = \frac{1}{\sqrt{2}}, \langle A_0 \otimes B_1 \rangle = \frac{1}{\sqrt{2}}, \langle A_1 \otimes B_0 \rangle = \frac{1}{\sqrt{2}}, \langle A_1 \otimes B_1 \rangle = -\frac{1}{\sqrt{2}} \, . </math>
While only one of these four measurements can be made in a single trial of the experiment, the sum<math display="block">\langle A_0 \otimes B_0 \rangle + \langle A_0 \otimes B_1 \rangle + \langle A_1 \otimes B_0 \rangle - \langle A_1 \otimes B_1 \rangle = 2\sqrt{2} </math>
gives the sum of the average values that Victor expects to find across multiple trials. This value exceeds the classical upper bound of 2 that was deduced from the hypothesis of local hidden variables.<ref name="mike-and-ike"/>{{Rp|116}} The value <math>2\sqrt{2}</math> is in fact the largest that quantum physics permits for this combination of expectation values, making it a [[Tsirelson's bound|Tsirelson bound]].<ref>{{Cite book |last=Rau |first=Jochen |url=https://www.worldcat.org/oclc/1256446911 |title=Quantum theory : an information processing approach |date=2021 |publisher=Oxford University Press |isbn=978-0-192-65027-6 |oclc=1256446911}}</ref>{{Rp|page=140}}

[[File:Chsh-illustration.png|thumb|An illustration of the CHSH game: the referee, Victor, sends a bit each to Alice and to Bob, and Alice and Bob each send a bit back to the referee.]]
The CHSH inequality can also be thought of as [[CHSH game|a ''game'' in which Alice and Bob try to coordinate their actions]].<ref>{{cite book|last1=Cleve |first1=R. |author-link1=Richard Cleve |last2=Hoyer |first2=P. |last3=Toner |first3=B. |last4=Watrous |first4=J. |author-link4=John Watrous (computer scientist) |year=2004 |chapter=Consequences and limits of nonlocal strategies |title=Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004. |pages=236–249 |publisher=[[IEEE]] |doi=10.1109/CCC.2004.1313847 |isbn=0-7695-2120-7 |oclc=55954993 |arxiv=quant-ph/0404076 |bibcode=2004quant.ph..4076C|s2cid=8077237 }}</ref><ref>{{Cite journal|last1=Barnum|first1=H.|last2=Beigi|first2=S.|last3=Boixo|first3=S.|last4=Elliott|first4=M. B.|last5=Wehner|first5=S.|date=2010-04-06|title=Local Quantum Measurement and No-Signaling Imply Quantum Correlations|journal=[[Physical Review Letters]]|language=en|volume=104|issue=14|pages=140401|arxiv=0910.3952|bibcode=2010PhRvL.104n0401B|doi=10.1103/PhysRevLett.104.140401|pmid=20481921|s2cid=17298392|issn=0031-9007}}</ref> Victor prepares two bits, <math>x</math> and <math>y</math>, independently and at random. He sends bit <math>x</math> to Alice and bit <math>y</math> to Bob. Alice and Bob win if they return answer bits <math>a</math> and <math>b</math> to Victor, satisfying
<math display="block">x y = a + b \mod 2 \, .</math>
Or, equivalently, Alice and Bob win if the [[logical AND]] of <math>x</math> and <math>y</math> is the [[logical XOR]] of <math>a</math> and <math>b</math>. Alice and Bob can agree upon any strategy they desire before the game, but they cannot communicate once the game begins. In any theory based on local hidden variables, Alice and Bob's probability of winning is no greater than <math>3/4</math>, regardless of what strategy they agree upon beforehand. However, if they share an entangled quantum state, their probability of winning can be as large as<math display="block">\frac{2+\sqrt{2}}{4} \approx 0.85 \, .</math>