Spin polarization-scaling quantum maps and channels
Sergey N. Filippov Kamil Yu. Magadov
quant-ph






Moscow Institute of Physics and Technology,
Institutskii Per. 9, Dolgoprudny, Moscow Region 141700, Russia

Valiev Institute of Physics and Technology of Russian
Academy of Sciences, Nakhimovskii Pr. 34, Moscow 117218, Russia

Steklov Mathematical Institute of Russian Academy of
Sciences, Gubkina St. 8, Moscow 119991, Russia



Deceased July 6, 2017.

Moscow Institute of Physics and Technology,
Institutskii Per. 9, Dolgoprudny, Moscow Region 141700, Russia




We introduce a spin polarization-scaling map for spin-j
particles, whose physical meaning is the decrease of spin
polarization along three mutually orthogonal axes. We find
conditions on three scaling parameters under which the map is
positive, completely positive, entanglement breaking,
2-tensor-stable positive, and 2-locally entanglement annihilating.
The results are specified for maps on spin-1 particles. The
difference from the case of spin-1/2 particles is
emphasized.




Spin polarization-scaling quantum maps and channels
    Kamil Yu. Magadov
    
===================================================





§  INTRODUCTION


Quantum states of a spin-j particle are described by
(2j+1)× (2j+1) density matrices ϱ∈𝒮
(ℋ) satisfying the properties ϱ^† =
ϱ, tr[ϱ] = 1, and ⟨φ|ϱ|φ⟩⩾ 0 for all |φ⟩∈ℋ,
dimℋ = 2j+1. Taking into account the
normalization condition, the density matrix ϱ is defined
by (2j+1)^2-1 real parameters, which are usually treated as
components of the generalized Bloch
vector <cit.>.
However, many physical phenomena can be explained and visualized
via a spin polarization vector p∈ℝ with
components p_i =  tr[ϱ J_i], where J_1,J_2,J_3 are
usual (2j+1)-dimensional representations of angular momentum
operators (see, e.g., <cit.>). Angular momentum
operators are Hermitian and satisfy the commutation relation
[J_k,J_l] = i e_klm J_m, where e_klm is the conventional
Levi-Civita symbol and the summation over m being assumed. Note
that the spin-polarization vector p does not contain the
full information about the quantum state if j ⩾ 1.
Despite this fact, it is of great use in quantum physics and
chemistry as its components represent average spin projections
onto three orthogonal axes and are experimentally measurable.
Linear transformations of the spin polarization vector include
rotations and scaling. Rotations are attributed to the unitary
evolution, so we do not consider them in the present paper.
Physically motivated scaling of the spin polarization vector is
described by a map Φ: ℬ(ℋ) ↦ℬ(ℋ) of the following form:

    Φ[X] = 1/2j+1
    tr[X] I + 3/j(j+1)(2j+1)∑_i=1^3 λ_i 
    tr[X J_i] J_i   ,


where I ∈ℬ(ℋ) is the identity
operator and λ_i ∈ℝ. The factors take into
account that tr[I] = 2j+1 and tr[J_k J_l] =
1/3 j(j+1)(2j+1) δ_kl, δ_kl is the
Kronecker delta. The map (<ref>) is
trace-preserving and unital, i.e. tr[ Φ[X] ]
=  tr [X] and Φ[I] = I. Note that the
map (<ref>) differs in general from other
classes of unital maps <cit.>. Physical
meaning of Eq. (<ref>) is the
transformation of the spin polarization

    p_i ↦λ_i p_i,      i=1,2,3.


In case of spin-1/2 particles,
formula (<ref>) transforms into a
well-known Pauli qubit map Φ[X] = 1/2( 
tr[X] I + ∑_i=1^3 λ_i  tr[X σ_i] σ_i
), where (σ_1,σ_2,σ_3) is the conventional
set of Pauli matrices (see,
e.g., <cit.>). The qubit
(j=1/2) map Φ is known to be positive if and only
if |λ_i| ⩽ 1, completely positive if and only if
1 ±λ_3 ⩾ |λ_1 ±λ_2|,
entanglement breaking if and only if |λ_1| + |λ_2| +
|λ_3| ⩽ 1, 2-local-entanglement-annihilating if and
only if λ_1^2+λ_2^2+λ_3^2 ⩽ 1,
2-tensor-stable positive if and only if 1 ±λ_3^2
⩾ |λ_1^2 ±λ_2^2| <cit.>.
Similar characterization for higher spins (j⩾ 1) is
still missing, so the goal of the present paper is to analyze
analogous properties of such maps and illustrate them for qutrits
(j=1).

The paper is organized as follows. In
Sec. <ref>, we analyze positivity of the
map (<ref>). In Sec. <ref>,
the criterion of complete positivity of such a map is presented.
In Sec. <ref>, the entanglement breaking property is
partially characterized. In Sec. <ref>, we review
the positivity and entanglement annihilation behaviour of the map
Φ⊗Φ. In Sec. <ref>, brief
conclusions are presented.



§ POSITIVITY



We will refer to an operator R as positive-semidefinite and
write R ⩾ 0 if ⟨φ| R |φ⟩⩾
0 for all |φ⟩∈ℋ. A map Φ is called
positive if Φ[X] ⩾ 0 for all X ⩾
0 <cit.>.

Let us now analyze positivity of the spin polarization-scaling
map (<ref>).

Since each J_i is a spin projection operator with eigenvalues
j, j-1, …, -j, eigenvalues of the operator a_1 J_1 + a_2
J_2 + a_3 J_3 are | a|{j,j-1,…,-j}, where |
a| = √(a_1^2 + a_2^2 + a_3^2). Therefore, the minimal
eigenvalue of Φ[X] reads

    1/2j+1(  tr[X] -
    3/j+1√(∑_i=1^3 ( λ_i  tr[X J_i]
    )^2)).


Suppose X⩾ 0. As | tr[X J_i]| ⩽
j  tr[X], the minimal value of (<ref>) is
non-negative if 1 - 3j/j+1√(∑_i=1^3 λ_i^2)⩾ 0. Thus, we have found sufficient condition for
positivity of the map Φ.


Spin-polarization-scaling map Φ is positive if ∑_i=1^3
λ_i^2 ⩽( j+1/3j)^2.


The necessary condition for positivity of the
map (<ref>) follows from the particular
form of the positive-semidefinite operator

    X = x_0 I + ∑_i=1^3 x_i J_i,    x_0
    ⩾ 0,    x_i ∈ℝ,    x_0 ⩾ j
    √(∑_i=1^3 x_i^2).


In fact, if X is given by formula (<ref>),
then Φ[X] = x_0 I + ∑_i=1^3 λ_i x_i J_i and
Φ[X] ⩾ 0 if and only if x_0 ⩾ j
√(∑_i=1^3 λ_i^2 x_i^2). Suppose x_0 = j x_1 and
x_2 = x_3 = 0, then Φ[X] ⩾ 0 if |λ_1|
⩽ 1. Similarly, necessary conditions
|λ_2|⩽ 1 and |λ_3|⩽ 1 appear for
choices x_0 = j x_2, x_1 = x_3 = 0 and x_0 = j x_3, x_1 =
x_2 = 0, respectively.


Suppose the spin polarization-scaling map Φ is positive, then
|λ_i| ⩽ 1, i=1,2,3.





§ COMPLETE POSITIVITY



A linear map Φ is called completely positive if Φ⊗ Id_k is positive for all k=1,2,…. Here Id_k
is the identity transformation of k-dimensional operators
ℬ(ℋ_k).


The spin polarization-scaling map Φ is completely positive if
and only if

    I ⊗ I + 3/j(j+1)(
    λ_1 J_1 ⊗ J_1 - λ_2 J_2 ⊗ J_2 + λ_3
    J_3 ⊗ J_3 ) ⩾ 0.





A linear map Φ:ℬ(ℋ_d) ↦ℬ(ℋ_d) is known to be completely positive if
and only if its Choi matrix Ω_Φ = (Φ⊗
Id_d)[|ψ_+⟩⟨ψ_+|] is
positive-semidefinite <cit.> (see
also <cit.>),
where |ψ_+⟩ = 1/√(d)∑_i=1^d |i⟩⊗|i⟩ is the maximally entangled state and
{|i⟩}_i=1^d is some orthonormal basis in
ℋ_d.

For our construction of the Choi operator let us choose
eigenvectors of the operator J_3 as the basis, namely, J_3
|jm⟩ = m |jm⟩ and jm'jm = δ_m'm, d= 2j
+1. Introduce auxiliary operators J_± = J_1 ± i J_2, then
J_±|jm⟩ = √((j ∓ 1)(j ± m + 1))|j m ±
1⟩. Some algebra yields

    (2j+1)^2 (Φ⊗
    Id)[|ψ_+⟩⟨ψ_+|] 
           = ∑_m,m'=-j^j Φ[|jm⟩⟨jm'|] ⊗|jm⟩⟨jm'|
           = ∑_m=-j^j ( I + 3m/j(j+1)λ_3 J_3
    ) ⊗|jm⟩⟨jm|
             + ∑_m=-j^j 3/2j(j+1)√((j-m)(j+m+1))
                  × (λ_1 J_1 - i λ_2 J_2) ⊗|m⟩⟨m+1|
             + ∑_m=-j^j 3/2j(j+1)√((j+m)(j-m+1))
                  × (λ_1
    J_1 + i λ_2 J_2) ⊗|m⟩⟨m-1|
           = I ⊗ I + 3/j(j+1)( λ_1 J_1 ⊗
    J_1 - λ_2 J_2 ⊗ J_2 + λ_3 J_3 ⊗ J_3
    ).


Thus, Φ is completely positive if and only if the
operator (<ref>) is positive-semidefinite.



If j=1/2, then angular momentum operators are given by
matrices

    J_1 = 1/2(
    [ 0 1; 1 0;   ]
    ),    J_2 = 1/2(
    [  0 -i;  i  0;    ]
    ),    J_3 = 1/2(
    [  1  0;  0 -1;    ]
    )


in the basis {|1/21/2⟩,
|1/2 -1/2⟩}. The
condition (<ref>) reduces to 1 ±λ_3
⩾ | λ_1 ±λ_2 |. Geometrically, these
inequalities correspond a tetrahedron with vertices (1,1,1),
(1,-1,-1), (-1,1,-1), and (-1,-1,1) in the parameter space
(λ_1,λ_2,λ_3) <cit.>.





If j = 1, then angular momentum operators are given by matrices

    J_1 = 1/√(2)(
    [ 0 1 0; 1 0 1; 0 1 0;   ]
    ),    J_2 = 1/√(2)(
    [  0 -i  0;  i  0 -i;  0  i  0;    ]
    ), 
           J_3 = (
    [  1  0  0;  0  0  0;  0  0 -1;    ]
    )


in the basis {|11⟩, |10⟩, |1-1⟩}. The
condition (<ref>) reduces to 4-9(λ_1^2 +
λ_2^2 + λ_3^2) + 27 λ_1 λ_2 λ_3
⩾ 0, |λ_i| ⩽2/3, i=1,2,3.
Geometrical figure corresponding to such inequalities is depicted
in Fig. <ref>(a).





§ ENTANGLEMENT BREAKING



A positive-semidefinite operator R ∈ℬ(ℋ_1)
⊗ℬ(ℋ_2) is called separable
(non-entangled) if there exist positive-semidefinite operators
R_1^(k) and R_2^(k) such that R = ∑_k R_1^(k)⊗ R_2^(k) <cit.>. A linear map
Φ: ℋ_1 ↦ℋ_1 is called entanglement
breaking if (Φ⊗ Id) [ϱ] is separable for
all ϱ∈𝒮(ℋ_1 ⊗ℋ_2)
and identity transformation Id: ℋ_2 ↦ℋ_2 <cit.>.
The well-known result is that Φ is entanglement breaking if
and only if the Choi matrix Ω_Φ is separable.

The necessary condition for separability of Ω_Φ is
that Ω_Φ^Γ⩾ 0, where X^Γ =
∑_j,j' I ⊗|j'⟩⟨j| X I ⊗|j'⟩⟨j|
is the partially transposed operator, X∈ℬ(ℋ_1) ⊗ℬ(ℋ_2) <cit.>.
Applying such a condition to the Choi matrix (<ref>)
and taking into account that in conventional basis
{jm}_m=-j^j the matrices J_x^⊤ = J_x, J_y^⊤ =
-J_y, J_z^⊤ = J_z, we obtain the following result.


Suppose the spin polarization-scaling map Φ is completely
positive and entanglement breaking, then

    I ⊗ I + 3/j(j+1)( λ_1 J_1 ⊗ J_1
    ±λ_2 J_2 ⊗ J_2 + λ_3 J_3 ⊗ J_3 )
    ⩾ 0.



Note that the requirement (<ref>) is sufficient for the
channel Φ to be entanglement binding <cit.>.


If j=1/2, then Eq. (<ref>) is equivalent to
|λ_1| + |λ_2| + |λ_3| ⩽ 1.



If j=1, then Eq. (<ref>) is equivalent to
4-9(λ_1^2 + λ_2^2 + λ_3^2) ± 27 λ_1
λ_2 λ_3 ⩾ 0.




§ 2-TENSOR-STABLE PROPERTIES



Some properties of linear maps do not change under tensoring the
map with itself, for instance, Φ⊗Φ is completely
positive if and only if Φ is completely positive. Similarly,
Φ⊗Φ is entanglement breaking if and only if Φ
is entanglement breaking (see,
e.g., <cit.>). However, other
properties of a map can change drastically under tensor power. For
example, the map Φ⊗Φ can be non-positive even if
Φ is positive <cit.>.

A linear map Φ is called 2-tensor-stable positive if Φ⊗Φ is positive <cit.>.


 It is shown in Ref. <cit.>
that the spin polarization-scaling map Φ given by
Eq. (<ref>) for qubits (j=1/2)
is 2-tensor-stable positive if and only if Φ^2 is completely
positive, i.e. 1 ±λ_3^2 ⩾ |λ_1^2 ±λ_2^2|.


For higher spins (j ⩾ 1) the result of
Example <ref> can be extended as follows.


 If the spin polarization-scaling map
Φ is 2-tensor-stable positive, then Φ^2 is completely
positive.



Consider a positive-semidefinite operator
|ψ_+⟩⟨ψ_+|, where |ψ_+⟩ = (2j+1)^-1/2∑_m=-j^j |jm⟩⊗|jm⟩. The action of the
positive map Φ⊗Φ on such an operator reads

    0 ⩽ (2j+1)^2 (Φ⊗Φ)[|ψ_+⟩⟨ψ_+|] 
           = (2j+1)^2 ( Id⊗Φ) ∘ (Φ⊗
    Id)
    [|ψ_+⟩⟨ψ_+|] 
           = ( Id⊗Φ) [ I ⊗ I + 3/j(j+1)
                    ×( λ_1 J_1 ⊗ J_1 -
    λ_2 J_2 ⊗ J_2 + λ_3 J_3 ⊗ J_3 )
    ]
    
           = I ⊗ I + 3/j(j+1)( λ_1^2 J_1 ⊗
    J_1 - λ_2^2 J_2 ⊗ J_2 +
    λ_3^2 J_3 ⊗ J_3 ) 
           = (2j+1)^2 (Φ^2 ⊗ Id)[|ψ_+⟩⟨ψ_+|],


i.e. the Choi matrix Ω_Φ^2 is
positive-semidefinite and Φ^2 is completely positive.


In contrast to the case j=1/2, for higher spins (j
⩾ 1) Proposition <ref> provides the
necessary condition only. For instance, it is not hard to see that
for j=1 there exists a spin polarization-scaling map Φ such
that Φ^2 is completely positive but
(Φ⊗Φ)[|φ⟩⟨φ|]  0 for
the Schmidt-rank-2 state |φ⟩ = 1/√(2)(
|11⟩ + |1-1⟩).

In the case j=1, the map Φ^2 is completely positive if and
only if 4-9(λ_1^4 + λ_2^4 + λ_3^4) + 27
λ_1^2 λ_2^2 λ_3^2 ⩾ 0, which is
depicted in Fig. <ref>(b).

A linear map Φ: ℬ(ℋ) ↦ℬ(ℋ) is called 2-locally entanglement
annihilating <cit.>
if (Φ⊗Φ)[|ψ⟩⟨ψ|] is separable for all
|ψ⟩∈ℋ⊗ℋ. The same line of
reasoning as for 2-tensor-stable positive maps leads to the
following result.


 If the spin polarization-scaling map
Φ is 2-locally entanglement annihilating, then Φ^2 is
entanglement breaking.





§ CONCLUSIONS



We have considered the physically motivated sets of operator maps
for spin systems. The physical meaning of such maps is the
degradation of spin polarization with scaling parameters
λ_1,λ_2,λ_3 along the axes x,y,z,
respectively. We have found conditions (necessary, or sufficient,
or both) under which the spin polarization-scaling map is
positive, completely positive, entanglement breaking,
2-tensor-stable positive, 2-locally entanglement annihilating.
These results can be of use in the analysis of data, where only
spin polarization degrees of freedom are available. The crucial
difference between the cases of spin-1/2 and spin-1
particles is illustrated in a series of examples.


The study is supported by Russian Science Foundation under project
No. 16-11-00084.


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