PT 
 
 
 -Symmetric Qubit-System States in the Probability Representation of Quantum Mechanics

PT-symmetric qubit-system states are considered in the probability representation of quantum mechanics. The new energy eigenvalue equation for probability distributions identified with qubit and qutrit states is presented in an explicit form. A possibility to test PT-symmetry and its violation by measuring the probabilities of spin projections for qubits in three perpendicular directions is discussed.


Introduction
In the conventional formulation of quantum mechanics, the pure states of physical systems are identified with state vectors |ψ belonging to a Hilbert space H [1] and with the wave functions [2,3] ψ(x) = x|ψ satisfying the Schrödinger equation [2] determined by the Hamiltonian operatorĤ acting in the Hilbert space and being the Hermitian operator, i.e.,Ĥ † =Ĥ. The eigenfunctions of the Hamiltonian describe the energy levels of quantum systems. The eigenfunctions of the Hamiltonian are associated with the stationary states of the quantum systems. For qubits (two-level atoms or spin-1/2 systems), the Hilbert space H is a two-dimensional space, and the Hamiltonian operator H is represented by the Hermitian 2 × 2-matrix with real eigenvalues and orthogonal eigenvectors. Additionally, in conventional quantum mechanics, the superposition principle means that, for any two arbitrary state vectors |ψ 1 and |ψ 2 , there exists the vector |ψ = c 1 |ψ 1 + c 2 |ψ 2 , with c 1 and c 2 being the complex numbers; this vector |ψ describes the state of the system which does exist [1].
Recently, the problem of non-Hermitian Hamiltonian operators with real eigenvalues was discussed in [4][5][6][7][8] in connection with P T -symmetry (parity-time) of the physical problems, where the eigenvectors and eigenvalues of non-Hermitian Hamiltonians with such symmetries are associated with specific properties of the physical systems (see, for example, [9,10]), including the dynamical Casimir effect reviewed in [11,12] where the non-Hermiticity of the Hamiltonian is related to accounting for the dissipation processes [13]. The problems of non-Hermitian quantum mechanics were studied in [14][15][16][17][18]. The differences between conventional quantum mechanics and non-Hermitian quantum mechanics in the Hilbert-space representation of pure and mixed quantum states were analyzed in [19]. The applications of the P T -symmetry approach in quantum optics and physics of oscillators were discussed in [20]. The anti-P T -symmetry approach in qubit states was presented in [21]. An exactly solvable pseudo-Hermitian system with SU(1, 1) symmetry was studied in [22].
where α, β, γ, and δ are complex numbers. The matrix H of the Hamiltonian on the basis of two vectors |1 and |2 , such that 1|1 = 2|2 = 1 and 1|2 = 0, has the form H = α γ δ β ; (2) complex eigenvalues of this matrix read If Im (α + β) = 0 and the number [(α − β)/2] 2 + γδ is a real number s 2 ≥ 0, the eigenvalues λ 1 and λ 2 of the matrix H and its eigenvectors are expressed in terms of the matrix elements of the matrix H as follows: Two nonnegative normalization constants N 1,2 satisfy the equation where eigenvalues λ 1,2 are determined by Equation (3).
To construct the probability representation of non-Hermitian Hamiltonian eigenvectors |ψ 1 and |ψ 2 , we review the probability representation of qubit states (spin-1/2 states, two-level atom states) [40][41][42][43][44][45][46]. For Hermitian HamiltoniansĤ † =Ĥ, the matrix elements of matrix H in Equation (2) satisfy the conditions Im α = Im β = 0 and δ = γ * . In this case, the eigenvalues of matrix H satisfy the equalities Im λ 1 = Im λ 2 = 0. The energy levels E 1 = λ 1 and E 2 = λ 2 of two-level atom are expressed in terms of the matrix elements of Hamiltonian (2), i.e., real numbers α and β and complex numbers γ and δ = γ * , as follows: In the case of Hermitian matrix H, the eigenvectors (4) are expressed in terms of the matrix elements of matrix H: they read where nonnegative normalization constants N 1,2 are expressed in terms of the matrix elements of Hamiltonian matrix H, i.e., The eigenvectors (7) satisfy the orthogonality condition ψ E 1 |ψ E 2 = 0 and the normalization conditions ψ E 1 |ψ E 1 = ψ E 2 |ψ E 2 = 1. The density 2 × 2-matrices of pure states of the two-level atom with Hermitian Hamiltonian has the form determined by the density operator with the density matrix ρ E 1,2 .
As it was shown in [40][41][42][43][44][45][46], the physical meanings of the matrix elements of density operator (9) can be associated with three dichotomic probability distributions (p 1 , 1 − p 1 ), (p 2 , 1 − p 2 ), and (p 3 , 1 − p 3 ) of spin-1/2 projections m = ±1/2 in three perpendicular directions in the space, i.e., x, y, and z directions. One can check that the traces of the product of ρ E 1,2 with three density matrices expressed in terms of three Pauli matrices 1 2 + 1 2 Tr ρ E 1, 2 σ x,y,z , are equal to three probabilities According to Born's rule, the trace of the density matrix ρ E 1, 2 with the other density matrix ρ j = |ψ j ψ i |, i.e., Tr ρ E 1, 2 ρ j , is equal to the probability p E 1, 2 j to obtain the properties of the state |ψ j in the state ρ E 1, 2 . For the pure state, this probability is p For mixed states with density matrices ρ A and ρ B , Born's rule also provides the expression of analogous probabilities P A B . The probabilities P A B are the probabilities of the results obtained following the procedure: (i) one needs to obtain the properties of the system in the state ρ A ; (ii) for that, one performs the measurement for this system prepared in the state ρ B ; (iii) finally, one employs Born's rule, which provides the equality P A B = Tr (ρ A ρ B ). In view of this method, it was found [40][41][42][43][44][45][46] that any density 2 × 2-matrix of the qubit state can be mapped onto three probability distributions; i.e., the qubit-state density matrix reads Thus, an arbitrary qubit state is determined by three probabilities 0 ≤ p 1 , p 2 , p 3 ≤ 1 satisfying the nonnegativity condition of the density operator Here, numbers p 1 , p 2 , and p 3 are the probabilities of the result of the experiment with qubit state, where one measures spin projections m = +1/2 in three perpendicular directions x, y, and z. The form of the density matrix (10) demonstrates that any qubit state is identified with the probability distributions of dichotomic random variables, which we call the probability representation of quantum state.

Schrödinger's Equation for Energy Levels in the Probability Representation
The energy levels of quantum system satisfy the eigenvalue equation for a Hamiltonian operator H acting in a Hilbert spaceĤ We consider the operatorĤ to be non-Hermitian one, i.e.,Ĥ † =Ĥ. Thus, we have the equality In the case of P T -symmetry, one has the property E = E * . Then, in view of Equation (13), for the operatorρ E = |ψ E ψ E |, we obtain the following equations: Vectors |ψ E can be chosen to satisfy the normalization condition Trρ E = ψ E |ψ E = 1.
In [40,41], the qudit-state density matrices were introduced in the probability representation, i.e., as Hermitian trace-one matrices |ψ E ψ E | with nonnegative eigenvalues. The matrix elements of these Hermitian matrices are ρ jk = p (jk) 1 − 1/2 − i p (jk) 2 − 1/2 for j < k and j; k = 1, 2, . . . , N, 3 . The matrix elements ρ jk satisfy the Silvester criterion of the nonnegativity of the eigenvalues of matrix ρ jk ; the nonnegative numbers p (jk) 1,2,3 can be interpreted as the probability distributions of dichotomic random variables. For the states |ψ E , the eigenvalues of matrix ρ jk satisfy this condition. Now we consider the example of pure qubit state in the case of Hermitian HamiltonianĤ. Since the density matrix ρ jk of qubit state (10) is determined by three probability distributions of dichotomic random variables (p 1 , we can obtain the state vectors of the qubit.
For pure states, |ψ = ψ 1 ψ 2 , the vector |ψ can be written in the form [48,51] where the probabilities p j ; j = 1, 2, 3 satisfy the equality We choose the phase of ψ 1 to be equal to zero due to the gauge symmetry property of the wave functions of quantum systems [52].
The probability parametrization of an arbitrary normalized vector |ψ can be used, including the case of eigenvectors of non-Hermitian HamiltonianĤ; this means that one can provide the invertible map of the solutions of Equation (12) onto a set of three probability distributions of dichotomic random variables, using Equation (15). In the case of E = E * , after introducing the Hamiltonian matrix H of the Hamiltonian operatorĤ, relation (14) takes the form of the system of equations for the probabilities where p = (p 1 − 1/2) − i(p 2 − 1/2) and inequality (16) is valid. Equation (12) can be written in the form of a system of linear equations for the four-vector |ρ E , with components ρ E 1 , ρ E 2 , ρ E 3 , and ρ E 4 , expressed in terms of probabilities of dichotomic random variables, as follows: In these equations given in vector forms, vectors |ρ E satisfy Equation (14) written for P T -symmetric systems with Hamiltonians having real eigenvalues.

Probability Representation of the Eigenvalue Equations for Generic Non-Hermitian Hamiltonians of Qubit Systems
Our aim now is to extend the probability representation of the eigenvalue equation for the P T -symmetric qubit system (17) and derive a system of equations for generic non-Hermitian Hamiltonian with 2 × 2-matrix H, employing the vector representation of operatorρ E , where the matrix of this operator is presented as vector |ρ E . We consider an example of the density matrix where p 1 , p 2 , and p 3 are interpreted as probabilities of spin projections m = +1/2 in three perpendicular directions.
Equation (18) take the matrix form where the complex number E is the eigenvalue of non-Hermitian Hamiltonian matrix H. In the case of P T -symmetry E = E * , and Equations (19) and (20) for the eigenvector of non-Hermitian Hamiltonians with real eigenvalues are given by (17). Analogous forms of Equations (19) and (20) in the probability representation of eigenstates of generic Hamiltonian matrix can be written for N-dimensional states |ψ E .

The Schrödinger Equation for States with Eigenvalues of Energy as Equations for Eigenvectors with Components-Probabilities Determining Qubit States
Employing Equations (19) and (20), we obtain a new form of the Schrödinger equation for probabilities p 1 , p 2 , p 3 , (1 − p 1 ), (1 − p 2 ), and (1 − p 3 ) determining stationary states of the spin-1/2 particle. First, we consider the Hermitian Hamiltonian for which in (19) and (20) E = E * , Im H 11 = Im H 22 = 0, and H * 12 = H 21 . Equations (19) and (20) are written for four-vectors. For matrices and vector |ρ E , the condition means that the following relation: is valid. For generic complex Hamiltonian matrix H, the relation corresponds to Equation (20).  (12) and (13) in the form of eigenvalue equations for the four-vector |ρ E yield the system of complex conjugate eigenvalue equations for probabilities p 1 , p 2 , and p 3 . Thus, we obtained the system of linear equations for the spectrum of complex 2 × 2-matrix H in the form of equations for the four-vector |ρ E , with components expressed in terms of three probability distributions (p 1 , 1 − p 1 ), (p 2 , 1 − p 2 ) , and (p 3 , 1 − p 3 ) of three dichotomic random variables. This result can be extended to the case of arbitrary qudit states. In the next section, we consider the qutrit state.

A Probability Representation for the Non-Hermitian Hamiltonian Eigenvalue Equation of the Qutrit State
Our goal in this section is to demonstrate the extension of the approach to other states; for this, we construct the probability representation of eigenvectors for qudit system on the example of the qutrit state. As it was shown in [51,53], an arbitrary N×N matrix ρ, such that ρ = ρ † , Tr ρ = 1, with nonzero eigenvalues, has matrix elements which can be parameterized as follows: ρ jj = p (jj) 1,2,3 are probabilities of dichotomic random variables satisfying the Silvester criterion for the N×N matrix ρ.
The density matrix ρ for the qutrit state can be mapped onto nine-vector ρ| with components The presented probability-representation construction is valid for an arbitrary 3×3-matrix ρ, such that ρ = ρ † , Tr ρ = 1, with nonnegative eigenvalues. The tool we used is the application of the density matrix representation |ψ ψ| of eigenvectors |ψ in the case of the Hamiltonian 3×3-matrix.

. Numbers ρ
The eigenvalue equations for non-Hermitian Hamiltonian 3×3-matrix of the three-level atom H = H † in the vector form for the nine-vector |ρ Ej ; j = 1, 2, . . . , 9 read Since the vector components ρ Ej are expressed in terms of dichotomic probabilities p and explicit probabilistic form of Equation (26) is where p 3 . Thus, for the three-level atom, we can obtain the energy spectrum for an arbitrary non-Hermitian Hamiltonian with 3×3-density matrix H by solving Equations (27) and (28)  and complex numbers p (13) , p (13) , and p (23) expressed as linear combinations of dichotomic probabilities p (jk) 1, 2 , namely, The result obtained demonstrates that for P T -symmetric systems the probability description of Hamiltonian spectrum is described by Equations (27) and (28), with E = E * . It is worth pointing out that specific behavior of the three-level atom with a non-Hermitian Hamiltonian was experimentally studied in [27].

An Example of a Non-Hermitian Hamiltonian with P T -Symmetry
The particular case of non-Hermitian Hamiltonian was considered in [4,5,19]. Here, we consider a particular example of the Hamiltonian with z = 1 + i; i.e., we have the energy levels where s > 1, and there exists the matrix a, which provides the matrix equality The normalized eigenvectors |1 and |2 of the Hamiltonian H in Equation (29) For s → ∞, |1 → 1 √ 2 The density matrices of pure states |1 and |2 read The probability parameters p 1 (s), p 2 (s), and p 3 (s), describing the qubit states ρ 1 and ρ 2 , are for pure state |1 , p One can see that for the spin-1/2 state (qubit state), we have the following interpretation of the written states |1 and |2 : in these states, the probability to have the projection m = +1/2 in the z direction is equal to 1/2. However, there are different probabilities to have the spin projection equal to m = +1/2 in the x and y directions. In the state ρ 1 , we have the probability p 1 (s) as the probability of the spin projection m = +1/2 on the x axis and p 2 (s) as the probability of the spin projection m = +1/2 on the y axis. Thus, we have both states with specific symmetry properties with respect to the two directions determined by the x and y axes. Born's rule provides the dependence of the probability Tr (ρ 1 ρ 2 ) = s −2 .
If one has spin-1/2 states, which are eigenstates |ρ E 1, 2 of the P T -symmetric Hamiltonian (32), and measures the probabilities to get spin projections m = +1/2 on three perpendicular directions p E 1 1 , p , p E 2 2 , and p E 2 3 in these states, there exists a difference in the behavior of spin-1/2 states for eigenstates of any Hermitian Hamiltonian and the non-Hermitian Hamiltonian we are now discussing. For spin-1/2 states defined as eigenstates of any Hermitian Hamiltonian, the probabilities P E 1 1 , P E 1 2 , P E 1 3 , P E 2 1 , P E 2 2 , and P E 2 3 of spin projections m = +1/2 on three perpendicular directions must satisfy the equality In view of Born's rule, Tr ρ E 1 ρ E 2 = W E 2 E 1 is the probability given by the scalar product of the wave functions | ψ E 1 |ψ For two density matrices of the form (10), one has For P T -symmetric Hamiltonian, the written trace of the product of these two density matrices, expressed in terms of probabilities p E 1 1 (s), p E 1 2 (s), p E 1 3 (s), p E 2 1 (s), p E 2 2 (s), and p E 2 3 (s), is different from zero. For considered example, Tr ρ E 1 ρ E 2 = W E 2 E 1 depends on the parameter s and is equal to zero only at s → ∞. This difference can be a tool for detecting the P T -symmetric behavior of qubit (spin or two-level atom) systems.
In the limit s → 1 (exceptional point), W E 2 E 1 → 1. Thus, the P T -symmetric behavior can be detected, if one measures the probabilities of spin projections m = +1/2 in three directions, x, y, and z, in the two states with the given qubit state vectors |ψ E 1 and |ψ E 2 .

The Superposition Principle and P T -Symmetric States
According to the superposition principle in conventional quantum mechanics, the state |χ , being a superposition of existing states |ψ 0 and |ϕ 0 of the form where c 1 and c 2 are complex numbers, does exist. Let us consider two qubit states |ψ 0 = 1 0 and |ϕ 0 = 0 1 . The states |ψ E 1 and |ψ E 2 are normalized superposition states We see that these states are superpositions of two states with spin projections on the z axis m = +1/2 for the state |ψ 0 and m = −1/2 for the state |ϕ 0 . According to the superposition principle of quantum mechanics, these states exist. On the other hand, these states are the eigenstates of the P T -symmetric Hamiltonian H = 1 + i s s 1 − i . Consequently, one can produce a measurement of spin projections in the z direction and other directions in the states |ψ E 1 and |ψ E 2 given by (37) and experimentally obtain the probabilities to get positive values of the projections. The experimental value of the probability W E 2 E 1 given by Born's rule characterizes the presence of the P T -symmetry in the qubit states depending on the parameter s of the non-Hermitian matrix H.

Qubit States with Broken P T -Symmetry
The discussed approach provides the possibility to understand how to detect the states with broken P T -symmetry Hamiltonians, using the probability representation of these states. We address the question: How can one find experimental characteristics of qubit states, which are the eigenstates of non-Hermitian Hamiltonians with complex eigenvalues? To consider the case of system with broken P T -symmetry Hamiltonian, we formulate the following problem.
Take two qubit vectors |ψ = ψ 1 ψ 2 and |ϕ = ϕ 1 ϕ 2 . Let us construct the matrices ρ ψ = |ψ ψ| and ρ ϕ = |ϕ ϕ| and assume that the vector |ψ is the eigenvector of a complex matrix H with a complex eigenvector E 1 along with the vector |ϕ , which is the eigenvector of a complex matrix H with another complex eigenvector E 2 . Then we have the relations H|ψ ψ| = E 1 |ψ ψ| , H|ϕ ϕ| = E 2 |ϕ ϕ| , (38) It is easy to have the situation where E 2 = E * 1 , if the eigenvalues are not real. Now we consider the example of the real parameter s in (29) in the case of inequality 0 < s < 1.
In this case, one has two complex eigenvalues of the matrix H, which are In this case, the eigenvectors are For any state |ψ , the density matrix ρ = |ψ ψ|, and for states (39) we have two density matrices Then, in view of (10), we obtain the probability representations of two states, which are eigenstates of non-Hermitian Hamiltonian with broken P T -symmetry. The probabilities p (1,2) 1,2,3 determining the density matrices ρ 1 and ρ 2 (40) are expressed as follows: The specific properties of the states with density matrices (33) are that Tr (ρ 1 ρ 2 ) = 0 and one of the probability distributions p (1,2) 2 (s) corresponds to the completely chaotic distribution (1/2, 1/2). Thus, such states can be detected as existing superposition states by measuring the spin-1/2 positive projections; i.e., in the case of experimental obtaining the probabilities of the form (41), the latter ones can witness the presence of non-Hermitian Hamiltonian violating the P T -symmetry.
In the generic case of non-Hermitian Hamiltonian H with complex eigenvalues E 1 and E 2 , which are given by four real parameters and violate the P T -symmetry, one can also construct the probability representation of two eigenvectors of such a Hamiltonian. One can check that, in this case, the density matrices of the eigenstates of such Hamiltonians violating the P T -symmetry are not orthogonal, and Born's probability given by scalar product of normalized eigenvectors ψ E 1 |ψ E 2 2 = W 2 1 depends on the Hamilton parameters and is not equal to zero. This case can be also detected while measuring the probabilities of spin projections, because the results for W E 2 E 1 , where E 1 and E 2 are complex numbers, are different for the situations with the P T -symmetry and with Hermitian Hamiltonians.

Conclusions
To conclude, we point out the main results of our work. In Hermitian and non-Hermitian Hamiltonian systems with corresponding real eigenvalues, the pure states describing the eigenvectors of such Hamiltonians can be expressed in terms of probabilities of dichotomic random variables. We demonstrated this result on the example of qubit systems. We wrote the Schrödinger equation for such a system for the eigenvector of Hamiltonian H in a new form of the eigenvalue equation for the Hamiltonian H = H ⊗ 1, where components of the eigenvector are expressed in the form of probabilities of classical-like dichotomic variables.
For qubit states (spin-1/2 states), these probabilities are the probabilities to have the spin projection m = +1/2 in three perpendicular directions x, y, and z. For systems with Hermitian Hamiltonians, these probabilities satisfies the conditions of orthogonalities of two-dimensional eigenvectors of the 2 × 2 matrix H. For P T -symmetric Hamiltonians, which are non-Hermitian ones, the probabilities are such that the trace of pure-state density matrices ρ 1 = |ψ E 1 ψ E 1 | and ρ 2 = |ψ E 2 ψ E 2 |, i.e., k 12 = Tr (ρ 1 ρ 2 ), where E 1 and E 2 are real eigenvalues of the matrix H, is not equal to zero, and the value k 12 characterizes properties of the P T -symmetric system. The nonorthogonality of the non-Hermitian Hamiltonian eigenvectors associated with P T -symmetry properties of quantum systems was mentioned, e.g., in [19,27].
The systems with broken P T -symmetry, for which the Hamiltonian has complex eigenvalues, we also expressed the complex state vectors of the Hamiltonian in terms of probabilities of dichotomic random variables given in terms of the probabilities to have spin projection m = +1/2 in three perpendicular directions x, y, and z. In principle, these probabilities and corresponding means of the spin projections can be measured experimentally, and qubit systems with P T -symmetry can be compared with systems characterized by Hermitian Hamiltonians. Additionally, qubit systems with broken P T -symmetry can be compared with P T -symmetric ones, in view of values of the probabilities and corresponding parameters determined by Born's rule. The result can be extended to the case of P T -symmetric qudit systems, and the probability representation of qudit systems with generic complex Hamiltonian can be obtained by employing the approach demonstrated in this paper.