Bell state

Relationship between the two correlated bases of two qubits in the -state ${\displaystyle |\Phi ^{-}\rangle }$

A Bell state is a state of a physical system with several particles in which correlations between the results of simultaneous measurements on several particles are so strong that they cannot be explained in the context of classical physics . In the context of quantum mechanics, however, they can easily be explained by the phenomenon of quantum entanglement , which is particularly pronounced in a Bell state. A theoretical upper limit for the correlations that can be explained with classical physics is given by Bell's inequalitygiven. It must be adhered to in every theory that is based on the conceptual foundations of locality and realism ( local realism ). Hence, the actual existence of Bell states definitely rules out both basic concepts being given in our world.

In a narrower sense, a Bell state is one of four selected states of a two-particle system with two base states for each particle. In these states Bell's inequality is maximally violated. Together they are also known as the Bell Base. The Bell basis plays a major role in the theoretical development of quantum computers . Instead of two entangled particles, one speaks of two entangled qubits , because a particle with a two-dimensional Hilbert space also realizes a qubit.

conditions

Bell base

The Bell basis forms an orthonormal basis of the four-dimensional Hilbert space of the possible two-particle states. If particle A has the base states and , and particle B has the same, then the four Bell states are called as follows: ${\displaystyle |0\rangle _{A}}$ ${\displaystyle |1\rangle _{A}}$

${\displaystyle |\Phi ^{+}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{A}|0\rangle _{B}+|1\rangle _{A}|1\rangle _{B})}$

${\displaystyle |\Phi ^{-}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{A}|0\rangle _{B}-|1\rangle _{A}|1\rangle _{B})}$

${\displaystyle |\Psi ^{+}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{A}|1\rangle _{B}+|1\rangle _{A}|0\rangle _{B})}$

${\displaystyle |\Psi ^{-}\rangle ={\frac {1}{\sqrt {2}}}(|0\rangle _{A}|1\rangle _{B}-|1\rangle _{A}|0\rangle _{B}).}$

All four Bell states are mutually orthogonal in pairs. In each of the states, the entanglement is shown by the fact that each of the particles occupies each of the two basic states with the same probability, and that nevertheless, after one particle has been detected in one of the basic states, the basic state in which the other particle is found is certain is located.

Example spin-½-particle

In the case of spin - particles, the base states are usually referred to as and , which is reminiscent of the two possibilities of orienting the spin along a given axis. The fourth state of the Bell basis is then the singlet state ; H. the state of the total spin : ${\displaystyle {\tfrac {1}{2}}}$ ${\displaystyle \left|{\mathord {\uparrow }}\right\rangle }$ ${\displaystyle \left|{\mathord {\downarrow }}\right\rangle }$ ${\displaystyle m=\pm {\tfrac {1}{2}}}$ ${\displaystyle S=0}$

${\displaystyle |\Psi ^{-}\rangle ={\frac {1}{\sqrt {2}}}(\left|{\mathord {\uparrow }}\right\rangle _{A}\;\left|{\mathord {\downarrow }}\right\rangle _{B}-\left|{\mathord {\downarrow }}\right\rangle _{A}\;\left|{\mathord {\uparrow }}\right\rangle _{B}).}$

Correspondingly, the third state of the Bell basis is the triplet state ( ) with the m-quantum number : ${\displaystyle S=1}$ ${\displaystyle M=0}$

${\displaystyle |\Psi ^{+}\rangle ={\frac {1}{\sqrt {2}}}(\left|{\mathord {\uparrow }}\right\rangle _{A}\;\left|{\mathord {\downarrow }}\right\rangle _{B}+\left|{\mathord {\downarrow }}\right\rangle _{A}\;\left|{\mathord {\uparrow }}\right\rangle _{B}).}$

The first two states of the Bell base are alignment states; H. two orthogonal superpositions of the triplet states ${\displaystyle M=\pm 1}$

${\displaystyle |\Phi ^{+}\rangle ={\frac {1}{\sqrt {2}}}(\left|{\mathord {\uparrow }}\right\rangle _{A}\;\left|{\mathord {\uparrow }}\right\rangle _{B}+\left|{\mathord {\downarrow }}\right\rangle _{A}\;\left|{\mathord {\downarrow }}\right\rangle _{B}).}$

${\displaystyle |\Phi ^{-}\rangle ={\frac {1}{\sqrt {2}}}(\left|{\mathord {\uparrow }}\right\rangle _{A}\;\left|{\mathord {\uparrow }}\right\rangle _{B}-\left|{\mathord {\downarrow }}\right\rangle _{A}\;\left|{\mathord {\downarrow }}\right\rangle _{B}).}$

Bell operator

A Bell operator is an operator that describes the measurement of certain correlations between particles. Some of its eigenvalues ​​- i.e. the possible measured values ​​- lie outside the limits that must exist for such correlations according to classical ideas of space, time and causation. The Bell states defined above are eigenstates of one of Clauser et al. introduced Bell operator, which is also called the CSCH operator (after the first letters of the author's name) .

The construction of the CSCH operator is based on a 1-particle operator which has the eigenvalues ​​for the two base states . A clear example of a photon is a vertical or horizontal polarization filter. It can be expressed by. Such an operator is formed for each of the two entangled particles and denoted by or . In addition, one considers a second basis with corresponding operators (e.g. polarization filter rotated by an angle) for each particle . A bell operator is then ${\displaystyle \pm 1}$ ${\displaystyle {\hat {o}}=|1\rangle \langle 1|-|0\rangle \langle 0|}$ ${\displaystyle {\hat {a}}}$ ${\displaystyle {\hat {b}}}$ ${\displaystyle |0\rangle ',\;|1\rangle '}$ ${\displaystyle {\hat {a}}',\;{\hat {b}}'}$

${\displaystyle {\hat {B}}={\hat {a}}({\hat {b}}+{\hat {b}}')+{\hat {a}}'({\hat {b}}-{\hat {b}}')}$.

Now one chooses the second base "complementary" to the first, i. H. for example

${\displaystyle |0\rangle '={\frac {1}{\sqrt {2}}}\left(|1\rangle -|0\rangle \right)}$

${\displaystyle |1\rangle '={\frac {1}{\sqrt {2}}}\left(|1\rangle +|0\rangle \right)}$,

(Instead of the plus or minus sign, complex phases could also be selected). If the base states are aligned parallel or antiparallel to the z-axis, then the complementary base corresponds without complex phase factors to an orientation of the spins in the x-direction, or, with phase factors, in the direction of any other axis in the xy-plane. ${\displaystyle \mathrm {e} ^{\mathrm {i} \alpha },\;-\mathrm {e} ^{-\mathrm {i} \alpha }}$ ${\displaystyle |0\rangle ,\;|1\rangle }$ ${\displaystyle |0\rangle ',\;|1\rangle '}$

The CSCH operator has the eigenvalue zero for the two states and the eigenvalues for the states . (For the calculation with operator methods, see, alternatively you can calculate this directly by expressing the second base as a linear combination of the first.) The two extreme values ​​lie outside the range on which the measured values ​​of the correlations according to the classical idea (or Bell's inequality) must remain limited, which are measured with the CSCH operator. ${\displaystyle \Phi ^{\pm }}$ ${\displaystyle \pm 2{\sqrt {2}}}$ ${\displaystyle \Psi ^{\pm }}$ ${\displaystyle [-2,\;+2]}$

Maximum entanglement

The Bell states are maximally entangled, since on them all entanglement measures assume the maximum possible value (in Hilbert space ). In particular, the entropy of the state has the maximum value 1. There can also be no state that is more entangled than a Bell state because any other state can be determined deterministically from this through local quantum operations that cannot reinforce the entanglement. ${\displaystyle C^{2}\otimes C^{2}}$

Individual evidence

1. ^ John F Clauser, Michael A Horne, Abner Shimony, Richard A Holt: Proposed experiment to test local hidden-variable theories . In: Physical review letters . tape 23 , no. 15 , 1969, p. 880 ( online [PDF; accessed March 20, 2019]).
2. ^ Samuel L Braunstein, Ady Mann, Michael Revzen: Maximal violation of Bell inequalities for mixed states . In: Physical Review Letters . tape 68 , no. 22 , 1992, pp. 3259 ( online [PDF; accessed March 20, 2019]).
3. ↑ Track-preserving, completely positive linear mappings that can be implemented through operations on the subsystems and classical communication and are often referred to as LOCC (for local operations and classical communication )
4. ^ G. Vidal: On the continuity of asymptotic measures of entanglement . 2002, arxiv : quant-ph / 0203107 . and R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki: Quantum entanglement . In: Rev. Mod. Phys. tape 81 , 2009, p. 865 , p. 902f, sections XIII.A , arxiv : quant-ph / 0702225 .