Sakurai's Bell inequality
Encyclopedia
The intention of a Bell inequality is to serve as a test of local realism or local hidden variable theories
as against quantum mechanics
, applying Bell's theorem
, which shows them to be incompatible. Not all the Bell's inequalities that appear in the literature are in fact fit for this purpose. The one discussed here holds only for a very limited class of local hidden variable theories and has never been used in practical experiments. It is, however, discussed by John Bell
in his "Bertlmann's socks" paper (Bell, 1981), where it is referred to as the "Wigner–d'Espagnat inequality" (d'Espagnat, 1979; Wigner, 1970). It is also variously attributed to Bohm (1951?) and Belinfante (1973).
The following is a description taken almost without alteration from an earlier edition of the Bell's theorem
page. Note that the inequality is not really applicable either to electrons or photons, since it builds in no probabilistic properties in the measurement process. Much more realistic hidden variable theories can be devised, modelling spin (or polarisation, in optical Bell tests) as a vector and allowing for the fact that not all emitted particles will be detected.
Pick three arbitrary directions a, b, and c in which Alice and Bob can measure the spins of each
electron they receive. We assume three hidden variables on each electron, for the three direction spins. We furthermore
assume that these hidden variables are assigned to each electron pair in a consistent way at the time they are emitted
from the source, and don't change afterwards. We do not assume anything about the probabilities of the various hidden
variable values.
Alice and Bob are two spatially
separated observers. Between them is an apparatus that continuously produces
pairs of electron
s. One electron in each pair is sent toward Alice, and the other toward Bob. The setup
is shown in the diagram.
(This is just a thought-experiment, remember. Real experiments on pairs of electrons are not feasible and most "Bell test experiments" have instead been based on either the polarisation direction or the frequency and phase of light, assumed to come in particle-like "photons".)
The electron pairs are specially prepared so that if both observers measure the spin
of their
electron along the same axis, then they will always get opposite results. For example, suppose Alice and Bob both
measure the z-component of the spins that they receive. According to quantum mechanics, each of Alice's measurements
will give either the value +1/2 or −1/2, with equal probability. For each result of +1/2 obtained by Alice, Bob's result will inevitably be −1/2, and vice versa.
Mathematically, the state of each two-electron composite system can be described by the state vector
Each ket
is labelled by the direction in which the electron spin points. The above state is known
as a spin singlet. The z-component of the spin corresponds to the operator (1/2)σz, where
σz is the third Pauli matrix
. (The quantum mechanics of spin is discussed in the
article spin (physics)
).
It is possible to explain this phenomenon without resorting to quantum mechanics. Suppose our electron-producing
apparatus assigns a parameter, known as a hidden variable, to each electron. It labels one electron "spin +1/2", and
the other "spin −1/2". The choice of which of the two electrons to send to Alice is decided by some classical random
process. Thus, whenever Alice measures the z-component spin and finds that it is +1/2, Bob will measure −1/2, simply
because that is the label assigned to his electron. This reproduces the effects of quantum mechanics, while preserving
the locality principle.
The appeal of the hidden variables explanation dims if we notice that Alice and Bob are not restricted to measuring the
z-component of the spin. Instead, they can measure the component along any arbitrary direction, and the result of each
measurement is always either +1/2 or −1/2. Therefore, each electron must have an infinite number of hidden
variables, one for each measurement that could possibly be performed.
This is ugly, but not in itself fatal. However, Bell showed that by choosing just three directions in which to perform
measurements, Alice and Bob can differentiate hidden variables from quantum mechanics.
{| border="1" width="250" align=right
! width="80" |Alice
! width="80" |Bob
! width="80" |Probability
|-
! a b c
! a b c
!
|-
! + + +
! − − −
! P1
|-
! + + −
! − − +
! P2
|-
! + − +
! − + −
! P3
|-
! + − −
! − + +
! P4
|-
! − + +
! + − −
! P5
|-
! − + −
! + − +
! P6
|-
! − − +
! + + −
! P7
|-
! − − −
! + + +
! P8
|}
Each row in the table describes one type of electron pair, with their respective hidden variable values and their probabilities P.
Suppose Alice measures the spin in the a direction and Bob measures it in the b direction. Denote the
probability that Alice obtains +1/2 and Bob obtains +1/2 by
P(a+, b+) = P3 + P4
Similarly, if Alice measures spin in a direction and Bob measures in c direction, the probability that both
obtain +1/2 is
Finally, if Alice measures spin in c direction and Bob measures in b direction, the probability that both
obtain the value +1/2 is
P(c+, b+) = P3 + P7
The probabilities P are always non-negative, and therefore:
P3 + P4 ≤ P3 + P4 + P2 + P7
This gives P(a+, b+) ≤ P(a+, c+) + P(c+, b+)
which is a (rather trivial) Bell inequality. It must be satisfied by any hidden variable theory if it is to match the quantum-mechanical prediction in circumstances in which every single particle is detected.
The quantum-mechanical prediction for the above setup is:
P(a+, b+) = 1/2 (sin(a − b)/2)2.
The inequality would require
1/2 (sin 45°)2 ≤ 1/2 (sin 22.5°)2 + 1/2 (sin 22.5°)2
or
0.2500 ≤ 0.1464
which is not true, proving his theorem. The prediction of quantum mechanics does indeed conflict with that of local realism.
In real experiments, given that not all particles are detected, the above test could (as with the practical version of the CHSH Bell test) not legitimately be used unless the assumption is made that the detected particles are a fair sample of those emitted. The failure of this assumption results in the best known "loophole". When there are some non-detections, hidden variable theories exist that, like quantum mechanics, predict violation of the inequality.
Hidden variable theory
Historically, in physics, hidden variable theories were espoused by some physicists who argued that quantum mechanics is incomplete. These theories argue against the orthodox interpretation of quantum mechanics, which is the Copenhagen Interpretation...
as against quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, applying Bell's theorem
Bell's theorem
In theoretical physics, Bell's theorem is a no-go theorem, loosely stating that:The theorem has great importance for physics and the philosophy of science, as it implies that quantum physics must necessarily violate either the principle of locality or counterfactual definiteness...
, which shows them to be incompatible. Not all the Bell's inequalities that appear in the literature are in fact fit for this purpose. The one discussed here holds only for a very limited class of local hidden variable theories and has never been used in practical experiments. It is, however, discussed by John Bell
John Stewart Bell
John Stewart Bell FRS was a British physicist from Northern Ireland , and the originator of Bell's theorem, a significant theorem in quantum physics regarding hidden variable theories.- Early life and work :...
in his "Bertlmann's socks" paper (Bell, 1981), where it is referred to as the "Wigner–d'Espagnat inequality" (d'Espagnat, 1979; Wigner, 1970). It is also variously attributed to Bohm (1951?) and Belinfante (1973).
The following is a description taken almost without alteration from an earlier edition of the Bell's theorem
Bell's theorem
In theoretical physics, Bell's theorem is a no-go theorem, loosely stating that:The theorem has great importance for physics and the philosophy of science, as it implies that quantum physics must necessarily violate either the principle of locality or counterfactual definiteness...
page. Note that the inequality is not really applicable either to electrons or photons, since it builds in no probabilistic properties in the measurement process. Much more realistic hidden variable theories can be devised, modelling spin (or polarisation, in optical Bell tests) as a vector and allowing for the fact that not all emitted particles will be detected.
Derivation of the inequality
The approach of Sakurai (1994) is followed.Pick three arbitrary directions a, b, and c in which Alice and Bob can measure the spins of each
electron they receive. We assume three hidden variables on each electron, for the three direction spins. We furthermore
assume that these hidden variables are assigned to each electron pair in a consistent way at the time they are emitted
from the source, and don't change afterwards. We do not assume anything about the probabilities of the various hidden
variable values.
Alice and Bob are two spatially
Space
Space is the boundless, three-dimensional extent in which objects and events occur and have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum...
separated observers. Between them is an apparatus that continuously produces
pairs of electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s. One electron in each pair is sent toward Alice, and the other toward Bob. The setup
is shown in the diagram.
The electron pairs are specially prepared so that if both observers measure the spin
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
of their
electron along the same axis, then they will always get opposite results. For example, suppose Alice and Bob both
measure the z-component of the spins that they receive. According to quantum mechanics, each of Alice's measurements
will give either the value +1/2 or −1/2, with equal probability. For each result of +1/2 obtained by Alice, Bob's result will inevitably be −1/2, and vice versa.
Mathematically, the state of each two-electron composite system can be described by the state vector
Each ket
Bra-ket notation
Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...
is labelled by the direction in which the electron spin points. The above state is known
as a spin singlet. The z-component of the spin corresponds to the operator (1/2)σz, where
σz is the third Pauli matrix
Pauli matrices
The Pauli matrices are a set of three 2 × 2 complex matrices which are Hermitian and unitary. Usually indicated by the Greek letter "sigma" , they are occasionally denoted with a "tau" when used in connection with isospin symmetries...
. (The quantum mechanics of spin is discussed in the
article spin (physics)
Spin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
).
It is possible to explain this phenomenon without resorting to quantum mechanics. Suppose our electron-producing
apparatus assigns a parameter, known as a hidden variable, to each electron. It labels one electron "spin +1/2", and
the other "spin −1/2". The choice of which of the two electrons to send to Alice is decided by some classical random
process. Thus, whenever Alice measures the z-component spin and finds that it is +1/2, Bob will measure −1/2, simply
because that is the label assigned to his electron. This reproduces the effects of quantum mechanics, while preserving
the locality principle.
The appeal of the hidden variables explanation dims if we notice that Alice and Bob are not restricted to measuring the
z-component of the spin. Instead, they can measure the component along any arbitrary direction, and the result of each
measurement is always either +1/2 or −1/2. Therefore, each electron must have an infinite number of hidden
variables, one for each measurement that could possibly be performed.
This is ugly, but not in itself fatal. However, Bell showed that by choosing just three directions in which to perform
measurements, Alice and Bob can differentiate hidden variables from quantum mechanics.
{| border="1" width="250" align=right
! width="80" |Alice
! width="80" |Bob
! width="80" |Probability
|-
! a b c
! a b c
!
|-
! + + +
! − − −
! P1
|-
! + + −
! − − +
! P2
|-
! + − +
! − + −
! P3
|-
! + − −
! − + +
! P4
|-
! − + +
! + − −
! P5
|-
! − + −
! + − +
! P6
|-
! − − +
! + + −
! P7
|-
! − − −
! + + +
! P8
|}
Each row in the table describes one type of electron pair, with their respective hidden variable values and their probabilities P.
Suppose Alice measures the spin in the a direction and Bob measures it in the b direction. Denote the
probability that Alice obtains +1/2 and Bob obtains +1/2 by
P(a+, b+) = P3 + P4
Similarly, if Alice measures spin in a direction and Bob measures in c direction, the probability that both
obtain +1/2 is
- (3) P(a+, c+) = P2 + P4
Finally, if Alice measures spin in c direction and Bob measures in b direction, the probability that both
obtain the value +1/2 is
P(c+, b+) = P3 + P7
The probabilities P are always non-negative, and therefore:
P3 + P4 ≤ P3 + P4 + P2 + P7
This gives P(a+, b+) ≤ P(a+, c+) + P(c+, b+)
which is a (rather trivial) Bell inequality. It must be satisfied by any hidden variable theory if it is to match the quantum-mechanical prediction in circumstances in which every single particle is detected.
The quantum-mechanical prediction for the above setup is:
P(a+, b+) = 1/2 (sin(a − b)/2)2.
Bell's application of the inequality
Bell (pages 145–150 of his "Bertlmann's socks" article) discusses the application of the inequality to thought-experiment involving the spin of electrons and Stern-Gerlach magnets.The inequality would require
1/2 (sin 45°)2 ≤ 1/2 (sin 22.5°)2 + 1/2 (sin 22.5°)2
or
0.2500 ≤ 0.1464
which is not true, proving his theorem. The prediction of quantum mechanics does indeed conflict with that of local realism.
In real experiments, given that not all particles are detected, the above test could (as with the practical version of the CHSH Bell test) not legitimately be used unless the assumption is made that the detected particles are a fair sample of those emitted. The failure of this assumption results in the best known "loophole". When there are some non-detections, hidden variable theories exist that, like quantum mechanics, predict violation of the inequality.