In quantum information science, the Bell's states or EPR pairs are specific quantum states of two qubits that represent the simplest examples of quantum entanglement. Due to this superposition, measurement of the qubit will "collapse" it into one of its basis states with a given probability. These mechanisms cannot transmit information faster than the speed of light, a result known as the no-communication theorem.
Bell states
The Bell states are four specific maximally entangled quantum states of two qubits. They are in a superposition of 0 and 1a linear combination of the two states. Their entanglement means the following:
The qubit held by Alice (subscript "A") can be in a superposition of 0 and 1. If Alice measured her qubit in the standard basis, the outcome would be either 0 or 1, each with probability 1/2; if Bob (subscript "B") also measured his qubit, the outcome would be the same as for Alice. Thus, Alice and Bob would each seemingly have random outcome. Through communication they would discover that, although their outcomes separately seemed random, these were perfectly correlated.
This perfect correlation at a distance is 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, following Albert Einstein, Boris Podolsky, and Nathan Rosen in their famous 1935 "EPR paper", there is something missing in the description of the qubit pair given abovenamely this "agreement", called more formally a hidden variable. In his famous paper of 1964, John S. Bell showed by simple probability theory arguments that these correlations (the one for the 0, 1 basis and the one for the +, − basis) cannot both be made perfect by the use of any "pre-agreement" stored in some hidden variablesbut that quantum mechanics predicts perfect correlations. In a more refined formulation known as the Bell–CHSH inequality, it is shown that a certain correlation measure cannot exceed the value 2 if one assumes that physics respects the constraints of local "hidden-variable" theory (a sort of common-sense formulation of how information is conveyed), but certain systems permitted in quantum mechanics can attain values as high as <math>2\sqrt{2}</math>. Thus, quantum theory violates the Bell inequality and the idea of local "hidden variables".
Bell basis
Four specific two-qubit states with the maximal value of <math>2\sqrt{2}</math> are designated as "Bell states". They are known as the four maximally entangled two-qubit Bell states and form a maximally entangled basis, known as the Bell basis, of the four-dimensional Hilbert space for two qubits:
: <math>|\Phi^+\rangle = \frac{1}{\sqrt{2 \big(|0\rangle_A \otimes |0\rangle_B + |1\rangle_A \otimes |1\rangle_B\big) \qquad (1)</math>
: <math>|\Phi^-\rangle = \frac{1}{\sqrt{2 \big(|0\rangle_A \otimes |0\rangle_B - |1\rangle_A \otimes |1\rangle_B\big) \qquad (2)</math>
: <math>|\Psi^+\rangle = \frac{1}{\sqrt{2 \big(|0\rangle_A \otimes |1\rangle_B + |1\rangle_A \otimes |0\rangle_B\big) \qquad (3)</math>
: <math>|\Psi^-\rangle = \frac{1}{\sqrt{2 \big(|0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B\big) \qquad (4)</math>
Creating Bell states via quantum circuits
thumb|right|400px|Quantum circuit to create Bell state <math>|\Phi^+\rangle</math>.Although there are many possible ways to create entangled Bell states through quantum circuits, the simplest takes a computational basis as the input, and contains a Hadamard gate and a CNOT gate (see picture). As an example, the pictured quantum circuit takes the two qubit input <math>|00\rangle</math> and transforms it to the first Bell state <math>|\Phi^+\rangle.</math> Explicitly, the Hadamard gate transforms <math>|00\rangle</math> into a superposition of <math>(|0\rangle|0\rangle + |1\rangle|0\rangle) \over \sqrt{2}</math>. This will then act as a control input to the CNOT gate, which only inverts the target (the second qubit) when the control (the first qubit) is 1. Thus, the CNOT gate transforms the second qubit as follows <math>\frac{(|00\rangle + |11\rangle)}{\sqrt{2} } = |\Phi^+\rangle</math>.
For the four basic two-qubit inputs, <math>|00\rangle, |01\rangle, |10\rangle, |11\rangle</math>, the circuit outputs the four Bell states (listed above). More generally, the circuit transforms the input in accordance with the equation
<math display="block">|\beta(x,y)\rangle = \left ( \frac{|0,y\rangle + (-1)^x|1,\bar{y}\rangle}{\sqrt{2 \right ),</math>
where <math>\bar{y}</math> is the negation of <math>y</math>. 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 <math>n</math> variables, one can distinguish between at most <math>2^{n+1} - 1</math> classes out of <math>4^n</math> Bell states using linear optical techniques.
Bell state correlations
Independent measurements made on two qubits that are entangled in Bell states perfectly anti-correlate if each qubit is measured in the relevant basis. For the <math>|\Phi^+\rangle</math> state, this means measuring the first qubit in basis <math>b_1</math> and the second qubit in basis <math>b_2 = Z.X.b_1</math>. If an experimenter chose to measure both qubits in a <math>|\Phi^-\rangle</math> Bell state using the same basis, the qubits would appear positively correlated when measuring in the <math>\{|0\rangle,|1\rangle\}</math> basis, anti-correlated in the <math>\{|+\rangle,|-\rangle\}</math> basis (|00\rangle - |11\rangle)</math>
<math>= \frac{1}{2\sqrt{2 ((|+\rangle_A + |-\rangle_A)(|+\rangle_B + |-\rangle_B) - (|+\rangle_A - |-\rangle_A)(|+\rangle_B - |-\rangle_B))</math>
<math>= \frac{1}{2\sqrt{2 (|++\rangle + |+-\rangle + |-+\rangle + |--\rangle - |++\rangle + |+-\rangle + |-+\rangle - |--\rangle)
</math>
<math>= \frac{1}{\sqrt{2 (|+-\rangle + |-+\rangle)
</math>
, and partially (probabilistically) correlated in other bases.
The <math>|\Psi^-\rangle</math> correlations can be understood by measuring both qubits in the same basis and observing perfectly anti-correlated results. <math>|\Psi^+\rangle</math> can be understood by measuring the first qubit in basis <math>b_1</math>, the second qubit in basis <math>b_2 = Z.b_1</math>, and observing perfectly aiti-correlated results.
thumb|Relationship between the correlated bases of two qubits in the <math>|\Phi^-\rangle</math> state.
{| class="wikitable"
|-
! Bell state !! Basis <math>b_2</math> to obtain perfect anti-correlation
|-
| <math>|\Phi^+\rangle</math>|| <math>Z.X.b_1</math>
|-
| <math>|\Phi^-\rangle</math>|| <math>X.b_1</math>
|-
| <math>|\Psi^+\rangle</math> || <math>Z.b_1</math>
|-
| <math>|\Psi^-\rangle</math> || <math>b_1</math>
|}
Applications
Superdense coding
Superdense coding allows two individuals to communicate two bits of classical information by only sending a single qubit. The basis of this phenomenon is the entangled states or Bell states of a two qubit system. In this example, Alice and Bob are very far from each other, and have each been given one qubit of the entangled state.
<math>|\psi \rangle = \frac{|00\rangle + |11\rangle}{\sqrt{2</math>.
In this example, Alice is trying to communicate two bits of classical information, one of four two bit strings: <math>'00', '01', '10',</math>or <math>'11'</math>. If Alice chooses to send the two bit message <math>'01'</math>, she would perform the <math>X</math> gate to her qubit. Similarly, if Alice wants to send <math>'10'</math>, she would apply the phase flip <math>Z</math>; if she wanted to send <math>'11'</math>, she would apply the <math>iY</math>gate to her qubit; and finally, if Alice wanted to send the two bit message <math>'00'</math>, she would do nothing to her qubit. Alice performs these quantum gate transformations locally, transforming the initial entangled state <math>|\psi\rangle</math> into one of the four Bell states.
The steps below show the necessary quantum gate transformations, and resulting Bell states, that Alice needs to apply to her qubit for each possible two bit message she desires to send to Bob.
<math>00: I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \longrightarrow |\psi \rangle = \frac{|00\rangle + |11\rangle}{\sqrt2}\equiv |{\Phi^+}\rangle</math>
<math>01: X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\longrightarrow |\psi \rangle = \frac{|01\rangle + |10\rangle}{\sqrt2}\equiv |{\Psi^+}\rangle</math>
<math>10: Z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\longrightarrow |\psi \rangle = \frac{|00\rangle - |11\rangle}{\sqrt2}\equiv |{\Phi^-}\rangle</math>
<math>11: iY = ZX = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\longrightarrow |\psi \rangle = \frac{|01\rangle - |10\rangle}{\sqrt2}\equiv |{\Psi^-}\rangle</math>.
After Alice applies the desired transformations to her qubit, she sends it to Bob. Bob then performs a measurement on the Bell state, which projects the entangled state onto one of the four two-qubit basis vectors, one of which will coincide with the original two bit message Alice was trying to send.
Quantum teleportation
Quantum teleportation is the transfer of a quantum state over a distance. It is facilitated by entanglement between A, the giver, and B, the receiver of this quantum state. This process has become a fundamental research topic for quantum communication and computing. More recently, scientists have been testing its applications in information transfer through optical fibers. The process of quantum teleportation is defined as the following:
Alice and Bob share an EPR pair and each took one qubit before they became separated. Alice must deliver a qubit of information to Bob, but she does not know the state of this qubit and can only send classical information to Bob.
It is performed step by step as the following:
- Alice sends her qubits through a CNOT gate.
- Alice then sends the first qubit through a Hadamard gate.
- Alice measures her qubits, obtaining one of four results, and sends this information to Bob.
- Given Alice's measurements, Bob performs one of four operations on his half of the EPR pair and recovers the original quantum state.