Bell state

The Bell states are a concept in quantum information science and represent the simplest possible examples of entanglement. They are named after John S. Bell, as they are the subject of his famous Bell inequality. An EPR pair is a pair of qubits which jointly are in a Bell state, that is, entangled with each other. Unlike classical phenomena such as the nuclear, electromagnetic, and gravitational fields, entanglement is invariant under distance of separation and is not subject to relativistic limitations such as speed of light.

The Bell states

A Bell state is defined as a maximally entangled quantum state of two qubits. The qubits are usually thought to be spatially separated. Nevertheless they exhibit perfect correlations which cannot be explained without quantum mechanics.

To explain, let us first look at the Bell state $|\Phi^+\rangle$:

$|\Phi^+\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B + |1\rangle_A \otimes |1\rangle_B).$

This expression means the following: The qubit held by Alice (subscript "A") can be 0 as well as 1. If Alice measured her qubit the outcome would be perfectly random, either possibility having probability 1/2. But if Bob then measured his qubit, the outcome would be the same as the one Alice got. So, if Bob measured, he would also get a random outcome on first sight, but if Alice and Bob communicated they would find out that, although the outcomes seemed random, they are correlated.

So far, this is nothing special: Maybe the two particles "agreed" in advance, when the pair was created (before the qubits were separated), which outcome they would show in case of a measurement.

Hence, followed Einstein, Podolsky, and Rosen in 1935 in their famous "EPR paper", there is something missing in the description of the qubit pair given above — namely this "agreement", called more formally a hidden variable.

But quantum mechanics allows qubits to be in quantum superposition — i.e. in 0 and 1 simultaneously (this is misleading), e.g. in either of the states $|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ or $|-\rangle = \frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)$. If Alice and Bob chose to measure in this basis, i.e. check whether their qubit were $|+\rangle$ or $|-\rangle$, they will find the same correlations as above. That is because the Bell state can be formally rewritten as follows:

$|\Phi^+\rangle = \frac{1}{\sqrt{2}} (|+\rangle_A \otimes |+\rangle_B + |-\rangle_A \otimes |-\rangle_B).$

Note that this is still the same state.

John S. Bell showed in his famous paper of 1964 by using simple probability theory arguments that these correlations cannot be perfect in case of "pre-agreement" stored in some hidden variables — but that quantum mechanics predict perfect correlations. In a more formal and refined formulation known as the Bell-CHSH inequality, this would be stated such that a certain correlation measure cannot exceed the value 2 according to reasoning assuming local "hidden variable" theory (sort of common-sense) physics, but quantum mechanics predicts $2\sqrt{2}$.

There are three other states of two qubits which lead to this maximal value of $2\sqrt{2}$ and the four are known as the four maximally entangled two-qubit states or Bell states:

$|\Phi^+\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B + |1\rangle_A \otimes |1\rangle_B)$

$|\Phi^-\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B - |1\rangle_A \otimes |1\rangle_B)$

$|\Psi^+\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |1\rangle_B + |1\rangle_A \otimes |0\rangle_B)$

$|\Psi^-\rangle = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B).$

Bell state measurement

The Bell measurement is an important concept in quantum information science: It is a joint quantum-mechanical measurement of two qubits that determines in which of the four Bell states the two qubits are in.

If the qubits were not in a Bell state before, they get projected into a Bell state (according to the projection rule of quantum measurements), and as Bell states are entangled, a Bell measurement is an entangling operation.

Bell-state measurement is the crucial step in quantum teleportation. The result of a Bell-state measurement is used by one's co-conspirator to reconstruct the original state of a teleported particle from half of an entangled pair (the "quantum channel") that was previously shared between the two ends.

Experiments which utilize so-called "linear evolution, local measurement" techniques cannot realize a complete Bell state measurement. Linear evolution means that the detection apparatus acts on each particle independently from the state or evolution of the other, and local measurement means that each particle is localized at a particular detector registering a "click" to indicate that a particle has been detected. Such devices can be constructed, for example, from mirrors, beam splitters, and wave plates, and are attractive from an experimental perspective because they are easy to use and have a high measurement cross-section.

For entanglement in a single qubit variable, only three distinct classes out of four Bell states are distinguishable using such linear optical techniques. This means two Bell states cannot be distinguished from each other, limiting the efficiency of quantum communication protocols such as teleportation. If a Bell state is measured from this ambiguous class, the teleportation event fails.

Entangling particles in multiple qubit variables, such as (for photonic systems) polarization and a two-element subset of orbital angular momentum states, allows the experimenter to trace over one variable and achieve a complete Bell state measurement in the other. Leveraging so-called hyper-entangled systems thus has an advantage for teleportation. It also has advantages for other protocols such as superdense coding, in which hyper-entanglement increases the channel capacity.

In general, for hyper-entanglement in n variables, one can distinguish between at most 2^{n + 1} − 1 classes out of 4ⁿ Bell states using linear optical techniques.

Bell measurements of ion qubits in ion trap experiments, the distinction of all four states is possible.

See also

• Bell test experiments

References

• Nielsen, Michael A.; Chuang, Isaac L. (2000), Quantum computation and quantum information, Cambridge University Press, ISBN 978-0-521-63503-5 , pp. 25.
• Kaye, Phillip; Laflamme, Raymond; Mosca, Michele (2007), An introduction to quantum computing, Oxford University Press, ISBN 978-0-198-57049-3 , pp. 75.