On the stratifications of 2-qubits X-state space
Arsen Khvedelidze Astghik Torosyan





A. Razmadze Mathematical Institute, 
           Iv.Javakhishvili Tbilisi State University,                Tbilisi, Georgia
Institute of Quantum Physics and Engineering              Technologies, 
           Georgian Technical University, Tbilisi, Georgia
National Research Nuclear University,
           MEPhI (Moscow Engineering Physics Institute),
           Moscow, Russia
Laboratory of Information Technologies,  
           Joint Institute for Nuclear Research,
           Dubna, Russia
          


The 7-dimensional family 𝔓_X of so-called mixed X-states of 2-qubits is considered.  
Two types of stratification of 2-qubits X-state space,  i.e., partitions   of  𝔓_X into orbit types with respect to the adjoint group actions, one of the global unitary group G_X ⊂ SU(4) and another one under the action of the local unitary group LG_X ⊂ G_X,  is described.   
The equations and inequalities in the invariants of the corresponding groups, determining  each stratification component, are given.


On the stratifications of 2-qubits X-state space
    Arsen Khvedelidze1,2,3,4akhved@jinr.ru Astghik Torosyan4astghik@jinr.ru 

    December 30, 2023
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§ INTRODUCTION


The understanding of a symmetry that a physical system possesses, as well as this symmetry's breaking pattern allows us to explain uniquely a wide variety of phenomena in many areas of physics, including elementary particle physics and condensed matter physics <cit.>. The mathematical formulation of symmetries related to the Lie group action consists of the detection of the stratification of the representation space of corresponding symmetry group. Dealing with closed quantum systems the symmetries are realized by the unitary group actions and the quantum state space plays the role of the symmetry group representation space.  Below, having in mind these observations,  we will outline examples of the stratifications occurring for a quantum system composed of a pair of 2-level systems, two qubits. We will analyze symmetries associated with two subgroups of the special unitary group SU(4). More precisely,  we will consider the
7-dimensional subspace 𝔓_X of a generic 2-qubit state space, the family of X-states (for definition see <cit.>, <cit.> and references therein) and reveal two types of its partition into the set of points having the same symmetry type. The  primary stratification originates from the action of the invariance group of X-states, named the global unitary group G__X⊂ SU(4),  whereas  the secondary one is  due to the action of the so-called local group LG__X⊂  G_X of the X-states. 




§ X-STATES AND THEIR SYMMETRIES



The mixed 2-qubit  
X­-states can be defined based on the purely algebraic consideration. The idea is to fix the  subalgebra 
𝔤_X:=𝔰𝔲(2)⊕𝔰𝔲(2) ⊕𝔲(1) ∈𝔰𝔲(4) 
of the algebra 𝔰𝔲(4) 
and define the density matrix of X-states as 

    ϱ_X= 1/4(I + i𝔤_X ) .

In order to coordinatize the X­-state space we use the  tensorial basis for the 
𝔰𝔲(4) algebra, σ_μν =σ_μ⊗σ_ν,  μ,ν =0,1,2,3. It consists of all possible tensor products of two copies of Pauli matrices and a unit 2× 2 matrix, σ_μ=(I, σ_1,σ_2,σ_3) , which we order as follows (see the details in 
<cit.>):

    λ_1, …, λ_15 = 
    i/2(σ_x0,
    σ_y0, σ_z0, σ_0x, σ_0y, σ_0z,σ_xx,
    σ_xy, σ_xz, σ_yx, σ_yy, σ_yz,
    σ_zx, σ_zy, σ_zz) .

In this basis the 7-dimensional subalgebra 
𝔤_X is generated by the subset 
α_X=( λ_3, λ_6, λ_7,  λ_8, λ_10, -λ_11, λ_15), and thus the unit norm X­-state density matrix is given by the decomposition:

    ϱ_X= 1/4(I +2 i∑_λ_k ∈α_X h_kλ_k) .

The real coefficients h_k are subject to the   polynomial inequalities ensuring the semi-positivity of the density matrix, ϱ_X ≥ 0:

    𝔓_X= {h_i ∈ℝ^7  |  (h_3± h_6)^2+(h_8± h_10)^2+(h_7± h_11)^2≤ (1± h_15)^2 } .

Using the definition (<ref>) one can conclude that the X-state space  𝔓_X is invariant under the 7-parameter group, 
G_X :=exp (𝔤_X)∈ SU(4):

    gϱ_X g^†∈𝔓_X ,     ∀ g ∈ G_X .

Group G_X plays the same role for the X­-states as  the special unitary group SU(4) plays for a generic 4-level quantum system, and thus is termed the global unitary group of  X­-states. According to <cit.>,  group G_X admits the representation:

    G_X=P_π(
    [  e^-i ω_15SU(2)               0;               0 e^i ω_15SU(2)^';                 ])P_π ,     
    P_π=(
    1   0   0    0
    
    0   0   0   1
    
    0   0   1   0
    
    0   1   0   0
    ) .

Correspondingly,  the local unitary group of the X­-states is 

    LG_X=P_πexp(ϕ_1/2)
    ⊗exp(ϕ_2/2)P_π ⊂ G_X .






§ GLOBAL ORBITS AND STATE SPACE DECOMPOSITION



Now we give a classification of the global G_X-orbits according to their dimensionality and isotropy group.  
Every density matrix ϱ_X can be diagonalized by some element of the global G_X group. In other words, all global G_X-orbits can be generated from the  density 
matrices, whose eigenvalues form the partially ordered simplex  Δ_3, depicted on Figure <ref>. 


The tangent space to the G_X-orbits is spanned by the subset of linearly independent vectors, built from the  vectors: 
t_k = [λ_k,ϱ_X],  λ_k ∈α_X .
The number of independent vectors t_k determines the dimensionality  of the G__X-orbits and is given  by the rank of the  7 × 7 Gram matrix:

    𝒢_kl = 1/2(t_k t_l) .

The Gram matrix (<ref>) has three zero eigenvalues and two double multiplicity eigenvalues:

    μ_±   =    -1/8((h_3± h_6)^2+(h_8± h_10)^2+(h_7± h_11)^2) .

Correspondingly, the G_X-orbits  have  dimensionality of  either 4, 2 or 0.
The orbits of maximal dimensionality, ( 𝒪)_=4, are characterized  by non-vanishing 
μ_±≠ 0 and consist of the set of density matrices with a generic spectrum, σ(ϱ_x)= (r_1,r_2,r_3,r_4) .
If the density matrices  obey the equations  

    h_6=± h_3 ,   h_10=± h_8 ,    h_11=± h_7 ,

they belong to the so-called degenerate orbits, 
(𝒪)_±=2 .
The latter are generated from the matrices which have the double degenerate spectrum of the  form, σ(ϱ_x)= (p,p,r_3,r_4) and  σ(ϱ_x)= (r_1,r_2,q,q) respectively. 
Finally, there is a single orbit 
(𝒪)_0= 0, corresponding to the maximally mixed state   ϱ_X = 1/4I. 

Considering the diagonal representative of the generic G__X-orbit one can be convinced that its  isotropy group is  
 
    H=P_π(
    [   e^iωexpiγ_1/2σ_3                  ^;                  _ e^-iωexp iγ_2/2σ_3;                    ]) P_π ,

 while for a diagonal representative
 with a double degenerate spectrum the isotropy group is given by one of two groups:
 
    H_+=P_π(
    [         e^iωSU(2)                 ^;                 _ e^-iωexpiγ_2/2σ_3;                   ]) P_π ,   
     H_-=P_π(
    [ e^iωexpiγ_1/2σ_3                ^;                _     e^-iωSU(2)^';                  ]) P_π .

 For the single, zero dimensional orbit the isotropy group H_0 coincides with the whole invariance group, H_0=G_X. 
Therefore, the isotropy group of any element of 
G__X-orbits belongs to one of these conjugacy classes: [H], [H_±] or [H_0]. Moreover, a straightforward analysis shows that 
[H_+]=[H_-].
Hence, any point ϱ∈𝔓_X belongs to one of three above-mentioned types of G__X-orbits[
The orbit type [ϱ] of a point ϱ∈𝔓_X is given by the conjugacy class of the isotropy group of point ϱ, i.e.,  [ϱ]=
[G_ϱ_X] .
], denoted afterwards as [H_t] , t =1,2,3. 
For a given H_t, the associated stratum   𝔓_[H_t] , defined as 
the set of all points whose stabilizer is conjugate to H_t: 

    𝔓_[H_t]: ={
    y ∈ 𝔓_X|  y   H_t}

determines the  sought-for decomposition 
of  the state space 𝔓_X into strata according to the  orbit types:

    𝔓_X =⋃_𝔓_[H_i] .

The  strata  𝔓_(H_i)  are determined by this set of equations and inequalities: 

    (1)   𝔓_[H]:  = 
    {h_i ∈𝔓_X  | μ_+ > 0, μ_- > 0 } ,
       (2)   𝔓_[H_+]∪𝔓_[H_-] := {h_i ∈𝔓_X  | μ_+ = 0, μ_- > 0 }∪{h_i ∈𝔓_X  |  μ_+  > 0, μ_- = 0 } ,
       (3)   𝔓_[H_0]: = {h_i ∈𝔓_X  | μ_+ = 0, μ_- = 0 } .






§ LOCAL ORBITS AND STATE SPACE DECOMPOSITION



Analogously,  one can build up the X­- state space decomposition associated with the local group LG__X action. For this action the dimensionality of LG__X-orbits is given by the rank of the corresponding  2 × 2 Gram matrix constructed out of vectors t_3 and t_6. Since its eigenvalues read:

    μ_1= -1/8((h_8+h_10)^2+(h_7+h_11)^2) ,    μ_2=-1/8((h_8-h_10)^2+(h_7-h_11)^2) ,

the LG_X-orbits are either generic ones with the dimensionality of ( 𝒪_L)_=2, or degenerate (𝒪_L)_±=1, or exceptional ones, (𝒪_L)_0=0.
The LG_X-orbits can be collected into the strata according to their orbit type. There are three types of strata associated with the “local” isotropy subgroups of LH ∈ LG_X. Correspondingly, one can define the following “local” strata of state space: 


  * the generic stratum, 𝔓^L_[I], which  has  a trivial isotropy type, [I],  and is represented by  the inequalities: 
𝔓^L_[I]:  = 
{h_i ∈𝔓_X  | μ_1 > 0, μ_2 > 0 } ,

  * the degenerate stratum, 𝔓^L_[H_L^±], collection of the  orbits whose type is [H_L^±] , with the subgroup either   H_L^+= I×exp(iuσ_3), or  H_L^-= exp(ivσ_3)× I. The stratum defining equations read respectively:

    h_10 = ± h_8 ,      h_11 = ± h_7 ,


  * the exceptional stratum, 𝔓_[LG_X]  of the type [LG_X], determined by the equations:  h_11=h_10=h_8=h_7=0. 

Therefore, the local group  action  prescribes the following stratification of 2-qubit X­-state space: 

    𝔓_X =𝔓_[I]∪𝔓_[H^+_L]∪𝔓_[H^-_L]∪𝔓_[LG_X] .





§ CONCLUDING REMARKS



In the present article we describe the stratification of 2-qubit X­-state space associated with the adjoint action of the global and local unitary groups. The global unitary symmetry is related to the properties  of a system as a whole, while the local symmetries comprise information on the entanglement, cf. <cit.>. In an upcoming publication, based on the introduced stratification of state space, we plan to analyze an interplay between these two symmetries and particularly determine the entanglement/separability characteristics  of every stratum. 




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