Spin coherent states, Bell states, spin Hamilton operators, entanglement, Husimi distribution, uncertainty relation and Bell inequality

Bell states [1–7], Dicke states [6, 8, 9] and spin coherent states [10–23] play a central role in quantum computing. The Bell states are fully entangled whereas among quantum states the spin coherent states (also called atomic coherent states or Bloch coherent states) are the “most classical states”. Spin Hamilton operators are provided which admit the Bell states and the Dicke states as eigenvectors. We also show how the Bell states can be constructed from the spin coherent states in C 2 and the Kronecker product. The measure of entanglement for these states is compared. The Husimi distributions are evaluated and discussed. The distances between the Bell states and spin coherent states are derived and shown that the distances cannot be 0. The uncertainty relations for the spin matrices S ₁ and S ₂, Bell states and spin coherent states are derived and compared. Furthermore we look at the Bell inequality for Bell states and spin coherent states. We find for spin coherent states that the Bell inequality can be violated depending on the parameter values. The Bell matrix is expressed with spin matrices and the Rayleigh quotients for spin coherent states and Bell states are derived and compared to the eigenvalues of the Bell matrix. We show that the spin coherent state cannot provide the ground state. With spin coherent states we can define a semi-classical density distribution. The Wehrl entropy for spin coherent states is evaluated.

Let S 1 ( 1 / 2 ) , S 2 ( 1 / 2 ) and S 3 ( 1 / 2 ) be the spin matrices for spin- 1 2

obeying the commutation relations

and

Let | 0 〉 = 1 0 , | 1 〉 = 0 1 be the standard basis in the Hilbert space C 2 . The four Bell states are given by

where |Ψ₋⟩ is the singlet state. The four Bell states form an orthonormal basis in the Hilbert space C 4 . Let |e ₁⟩, |e ₂⟩, |e ₃⟩ and |e ₄⟩ be the standard basis in the Hilbert space C 4 . Then the Bell states are given by

where B is the Bell matrix

where I ₂ is the 2 × 2 identity matrix. The Bell matrix is hermitian and unitary with B ² = I ₄, tr(B) = 0 and det(B) = 1. Hence, the eigenvalues are −1 (twice) and +1 (twice). The Bell matrix cannot be written as a Kronecker product of two 2 × 2 matrices. The Bell matrix can be written as B = exp(K) with the skew hermitian matrix

with the trace equal to 2πi. With K = iH we find a hermitian matrix H. The eigenvalues of K are 0 (twice) and iπ (twice). From the Bell matrix we can form the projection operators Π₊ = (I ₄ + B)/2, Π₋ = (I ₄ − B)/2 with Π₊Π₋ = 0₄ and Π₊ − Π₋ = B.

Let σ 1 = 2 S 1 ( 1 / 2 ) , σ 2 = 2 S 2 ( 1 / 2 ) and σ 3 = 2 S 3 ( 1 / 2 ) be the Pauli spin matrices. Then we have the eigenvalue equations for the Bell states

Consider now entanglement for the Bell states. From the Bell states we can form the four density matrices

The reduced density matrices (taking the partial trace) is the same for all four Bell states, namely

Hence the von Neumann entropy is given by S(ρ) = 1 which indicates that the Bell states are fully entangled. Using the 2-tangle as entanglement measure [24] we also find that the Bell states are completely entangled. Furthermore the Bell states cannot be written as the Kronecker product of two vectors in the Hilbert space C 2 .

The Dicke states in the Hilbert space C 4 are given by

Hence the second Dicke state is the Bell state |Ψ₊⟩.

The hermitian matrix

with γ > 0 admits the Bell states |Ψ₋⟩, |Ψ₊⟩, |Φ₋⟩ and |Φ₊⟩ as eigenvectors with the corresponding eigenvalues

Consider the spin S j ( 1 / 2 ) matrices (j = 1, 2, 3) and the Hamilton operator

with γ = ω ₂/ω ₁. The eigenvalues of K ̂ are given by

with the corresponding normalized eigenvectors

where v ₁ is the singlet state, v ₂ is a Bell state and also a Dicke state. The eigenvectors v ₃ and v ₄ are Dicke states.

The Dicke states are eigenvectors of

Starting point in the construction of the spin coherent states are the spin matrices S 1 ( s ) , S 2 ( s ) , S 3 ( s ) and S + ( s ) ≔ S 1 ( s ) + i S 2 ( s ) , S − ( s ) ≔ S 1 ( s ) − i S 2 ( s ) . Then

is a skew hermitian matrix with tr(K ^(s)(θ, ϕ)) = 0 and 0 ≤ θ < π, 0 ≤ ϕ < 2π. It follows that exp(K ^(s)(θ, ϕ)) is a unitary matrix with det(exp(K ^(s)(θ, ϕ))) = 1. We note that applying disentanglement we have the well-known identity

where z = e^iϕ  tan(θ/2). The spin coherent states are given by

with

We set |Ω⟩ ≡ |θ, ϕ⟩ in the following. The unitary matrix exp(K ^(s)(θ, ϕ)) describes a rotation through an angle θ about an axis n(ϕ) = (sin(ϕ), −cos(ϕ), 0). Note that the eigenvalues of K ^(s)(θ, ϕ) do not depend on ϕ and the eigenvectors do not depend on θ. This is true for all spin s. The overlap between two spin coherent states is given by

For s = 3/2 the eigenvalues of K ^(3/2)(θ, ϕ) are λ ₁ = −3iθ/2, λ ₂ = −iθ/2, λ ₃ = iθ/2 and λ ₄ = 3iθ/2 with the corresponding normalized eigenvectors

The four vectors form an orthonormal basis in the Hilbert space C 4 . The unitary matrix exp(K ^(3/2)) is given by

The spin coherent state for spin s = 3 2 can also be written as

where z = e^iϕ  tan(θ/2) (0 ≤ θ < π, 0 ≤ ϕ < 2π). With the standard basis

we can write

i.e.

Consider now the spin coherent states for s = 3/2 and entanglement. We note that the spin- 1 2 coherent state is given by

Then the Kronecker product [21] provides the normalized state in the Hilbert space C 4

Hence |Ω⟩ cannot be written as a Kronecker product of the spin coherent state of spin- 1 2 . Note, however, that the Bell states can be constructed from the spin coherent states for C 2 and the Kronecker product. Let

Then

and the first Bell states is given by

with the measure dΩ given by

Analogously we can find the other three Bell sates.

Consider now entanglement of the spin coherent state via the density matrix and partial trace. Let N = 1 + 3 z z ̄ + 3 ( z z ̄ ) 2 + ( z z ̄ ) 3 ≡ ( 1 + x 2 + y 2 ) 3 with z = x + i y ( x , y ∈ R ) . The density matrices are given by

and

with (.) = (θ/2). Utilizing the partial trace the reduced density matrices are

and

Obviously the trace is given by 1 for the two matrices. The determinant for |Ω⟩⟨Ω| _R is given by

Hence the determinant is equal to 0 for θ = 0 and 1/16 for θ = π/2. The eigenvalues of |Ω⟩⟨Ω| _R are given by

For θ = 0 we have the eigenvalues λ ₁(0) = 1 and λ ₂(0) = 0 and the state is not entangled. For θ = π/2 we have λ 1 ( π / 2 ) = 1 / 2 + 3 / 4 , λ 2 ( π / 2 ) = 1 / 2 − 3 / 4 and the state is entangled but not fully entangled.

Consider now the Bell states and the spin coherent states for spin − 3 2 and the Husimi distribution. We obtain

Hence we find

From these results it follows that the distance between the Bell states and the spin coherent states cannot be 0

The shortest distance for ‖|Φ₊⟩ − |Ω⟩‖², ‖|Φ₋⟩ − |Ω⟩‖² is 2 − 2 , the shortest distance for ‖|Ψ₊⟩ − |Ω⟩‖² is 2 − 3 and the shortest distance for ‖|Ψ₋⟩ − |Ω⟩‖² is 2 − 8 / ( 3 7 ) .

Consider now the uncertainty relation for Bell states, spin coherent states for s = 3/2 and the spin matrices S 1 ( 3 / 2 ) , S 2 ( 3 / 2 ) . Let S 1 ( s ) , S 2 ( s ) and S 3 ( s ) be the spin matrices with spin-s and the commutation relations

Let |ψ⟩ be a normalized state in the Hilbert space C 2 s + 1 . One defines the variance as

The uncertainty relation for all s is

With s = 3/2 we have ⟨ ψ | S 3 ( 3 / 2 ) | ψ ⟩ = 0 for all four Bell states and obtain for |Φ₊⟩, |Φ₋⟩ that 3/4 > 0. For |Ψ₊⟩ and the singlet state |Ψ₋⟩ we find 21 / 4 > 0 . For s = 3/2 and the spin coherent state we obtain

Note that the standard form of the uncertainty relation we obtain by taking the square root on both sides. The nonnegative term sin⁴(θ) sin²(ϕ) cos²(ϕ) takes a maximum for θ = π/2, ϕ = π/4, namely 1/4. For θ = 0 and ϕ arbitrary we find an equality for the uncertainty relation.

Consider now the Bell inequality, Bell states and spin coherent states. Let σ ₁ and σ ₂ be the Pauli spin matrices and

Note that ( A ̂ 1 ) 2 = ( A ̂ 2 ) 2 = ( B ̂ 1 ) 2 = ( B ̂ 2 ) 2 = I 2 , where I ₂ is the 2 × 2 identity matrix. We evaluate

with |ψ⟩ replaced by the Bell states. We obtain the well-known results

If we find θ, ϕ such that

then the Bell inequality is violated. For the absolute value we obtain

With ϕ = 0, θ/2 = π/4 we obtain 3 / 2 ≈ 2.121 and the Bell inequality is violated for this value. The normalized state for ϕ = 0, θ/2 = π/4 is given by