Canonical structures of $A$ and $B$ forms

In their seminal paper (Phys. Rev.121, 920 (1961)) Sudarshan, Mathews and Rau investigated properties of the dynamical $A$ and $B$ maps acting on $n$ dimensional quantum systems. Nature of the dynamical maps in open quantum system evolutions has attracted great deal of attention in the later years. However, the novel paper on the $A$ and $B$ dynamical maps has not received its due attention. In this tutorial article we review the properties of $A$ and $B$ forms associated with the dynamics of finite dimensional quantum systems. In particular we investigate a canonical structure associated with the $A$ form and establish its equivalence with the associated $B$ form. We show that the canonical structure of the $A$ form captures the completely positive (not completely positive) nature of the dynamics in a succinct manner. This feature is illustrated through physical examples of qubit channels.


I. INTRODUCTION
The conceptual formulation of dynamical A and B forms was pioneered by Sudarshan and coworkers 60 years ago [1,2]. The A and B matrices play an important role to identify if the open system dynamics of finite dimensional quantum systems is completely positive or not [3][4][5][6]. In this article we study the dynamical A and B forms in detail. In particular we investigate the canonical structure of the A form and its properties. We show that there is a one-to-one connection between the canonical A form and the B form. We also construct the canonical A form associated with some important physical examples of qubit channels [3].
In Sec. II we introduce the A and B forms and discuss their properties [1,2]. We present the canonical structure of the A form and establish its equivalence with the B form in Sec. III. The canonical structure of the A form is explicitly constructed for several qubit channels in Sec. IV. Concluding remarks are given in Sec V. * tthdrs@gmail.com

II. PROPERTIES OF A AND B MAPS
Consider a n dimensional Hilbert Space H n . State of a quantum system is described by a density matrix ρ ∈ H n , defining properties of which are given by 1. Hermiticity: ρ † = ρ.
For a qubit (two-level quantum system) we have where I denotes 2 × 2 identity matrix, σ = (σ 1 , σ 2 , σ 3 ) are Pauli matrices: and the real qubit state parameters given by p = (p 1 , p 2 , p 3 ) satisfy the condition | p| = p 2 1 + p 2 2 + p 2 3 ≤ 1. Thus, the state space of a qubit corresponds to a unit ball in R 3 . The vector p is called the Bloch vector. Now we consider a linear map A transforming density matrices in H n : A r ′ s ′ ;rs (ρ i ) rs , r ′ , s ′ = 1, 2, . . . , n. (3) • The n 2 × n 2 matrix A is called a trace-preserving positive map if, for every input density matrix, the output ρ f = A(ρ i ) is also a legitimate density matrix [1,2].
Writing the qubit density matrix (1) explicitly (in the standard basis |0 = (1, 0) T , one may identify the action A : ρ i → ρ f = A(ρ i ) (see (3)) as follows: • Unitary dynamics ρ f = U ρ i U † defines a trace-preserving positive map • Matrix transposition given by is an example of trace-preserving positive map. Under the action of a map A : ρ i =⇒ ρ f = A(ρ i ), preservation of hermiticity i.e., (ρ f ) * s ′ r ′ = (ρ f ) r ′ s ′ and the unit trace condition n r ′ =1 (ρ f ) r ′ r ′ = 1 result in the following constraints on the elements of the n 2 × n 2 process matrix A: and where δ rs denotes Kronecker delta symbol.
A realigned process matrix B was defined as [1,2] B r ′ r;s ′ s = A r ′ s ′ ;rs .
so that the hermiticity and unit trace conditions (8), (9) on the A-form can be expressed as Thus a physically valid A-form requires that the corresponding realigned matrix B (see (10)) is a n 2 × n 2 hermitian matrix with trace n.
Furthermore, positivity of the density matrix ρ f = A(ρ i ) ≥ 0 leads to the following constraints on the elements of A and B respectively [1]: In other words, positivity ρ f = A(ρ i ) ≥ 0 of the density matrix requires that B ≥ 0.

It is pertinent to point out that the B-form is represented by a hermitian matrix whereas
A is not; positivity of the B matrix highlights that the output density matrix is legitimate.
For this reason Sudarshan, Mathews and Rau [1] highlighted that the matrix B incorporates the kinematical restrictions on the dynamical law in a succint fashion; we shall call B the dynamical matrix. The A-form was used in Ref. [1] to define a linear map from input to output density operators (where the elements of the input and outpur density matrices are arranged in the form of n 2 component columns). Beyond this initial definition, the A matrix was not recognized to have any clear role. Our focus here is to unravel the A-form to its full potential. We show in the next section that the A matrix introduced in the Sudarshan-Mathew-Rau paper exhibits an elegant canonical structure and it reveals itself as a powerful tool in capturing all the dynamical features reflected by the corresponding B-form [7].

III. CANONICAL STRUCTURE OF THE A-FORM
Consider an orthonormal set We then construct a basis set of n 2 × n 2 matrices and express the A matrix (see (3)) as follows: where the expansion coefficients a µν are given by The matrix elements A r ′ s ′ ;rs of the A matrix are then given by Let us examine the hermiticity preserving condition (8) on the expansion coefficients a αβ : In other words the coefficients a µν , µ, ν = 0, 1, . . . , n 2 − 1 constitute a n 2 × n 2 hermitian matrix, which we denote by A.
Let U be a unitary matrix which diagonalizes A i.e., where λ µ , 0 ≤ µ ≤ n 2 − 1 denote the eigenvalues of A. Thus we obtain Substituting (21) in (16) we obtain the following canonical structure of the A matrix: where we have denoted • Using (22) we can express the matrix elements of A as Substituting (24) in (3) and simplifying, we obtain the following elegant structure for the action of the linear A-map on the column vector consisting of the elements of the input density matrix ρ i : • From (25) the trace preservation condition (9) assumes the form • From (10) and (24) we may identify the elements of the realigned B matrix as which happens to be the spectral decomposition of the dynamical B matrix with λ µ being its eigenvalues.
Highlighting point here is that (27) brings out an explicit connection between the hermitian (coefficient) matrix A (see (16) and (19)) and the dynamical matrix B of Ref. [1]: 1. The eigenvalues of the coefficient matrix A associated with the A-form are identically same as those of B.

2.
A completely positive map requires that the coefficient matrix A is positive (i.e., the eigenvalues λ α are non-negative whenever the map is completely positive).
3. In the case of a completely positive map one may define a set {E α , α = 0, 1, . . . , n 2 −1} of n × n matrices based on the canonical structure (22) of the A-map: Then the transformation ρ i → ρ f = A(ρ i ) gets expressed in terms of the Kraus operator-sum representation [8] i.e., We point out that the operator sum representation (29) was already described (via the spectral decomposition of the dynamical matrix B) by Sudarshan, Mathews and Rau in their 1961 paper [1] and it was independently proposed by Kraus [8] after 10 years. The operators E α (see (28),(29)) associated with a completely positive map are known as Kraus operators in the literature.
Summarizing, in this section we have shown that the canonical structure (22) of the A-form plays a significant role on its own -bringing forth all the required features of the quantum channel -without any necessity to invoke the realigned B-form. In the next section we employ the A-form to elucidate the completely positive or not completely positive behaviour of some familiar qubit channels.

IV. CANONICAL A -FORM OF STANDARD QUBIT MAPS
In this section we illustrate explicit 4 × 4 matrix forms of the canonical A-form and its equivalence with the dynamical matrix B of some standard qubit transformations.
Observe that where Tr [U σ µ ] , µ = 0, 1, 2, 3 is evaluated using (32): Then the 4 × 4 coefficient matrix A U = ((a U ) µν ) (see (35)) associated with A U is given by From (37) it is seen that the coefficient matrix A is a rank-1 positive matrix with eigenvalue 2 and eigenvector X U . The realigned B U matrix matches exactly with the coefficient matrix A U (see (37)) i.e.,

B. Pin map:
Consider a linear A-form mapping every input state ρ i to a fixed output state ρ 0 i.e., Sudarshan, Mathews and Rau presented this map in terms of the B-form, which was termed as relaxation generator. Here we would like to illustrate the canonical structure of the A-form associated with the qubit pin-map.
Let the fixed output density matrix of the qubit be given by The 4 × 4 matrix A pin corresponding to the pin map is identified as follows: Using the orthonormal basis set of matrices { σµ √ 2 , µ = 0, 1, 2, 3} we expand The coefficient matrix A pin = ((a pin ) µν ) is then found to be The eigenvalues of A pin are given by Clearly, the eigenvalues of A pin are all positive ensuring that the pin map is completely positive.
The B matrix associated with the pin map is constructed using the explicit matrix form of A pin (see (41)): Eigenvalues of B pin match with those of the coefficient matrix A pin (see (43)), thus establishing the equivalence between the two.

C. Transpose map:
Consider the transpose map A T : ρ → ρ T on qubit density matrices (see (4)). We obtain the associated 4 × 4 matrix form of A T as Employing the basis set { σµ √ 2 , µ = 0, 1, 2, 3} we express to obtain the following explicit structure for the coefficient matrix A T : The matrix A T is not positive (one of the eigenvalues of A T is -1) which points towards the not-completely positive nature of the transpose map.
From the explicit matrix structure of A T (see (45)) it is easy to see that the realigned dynamical matrix B T ≡ A T . The eigenvalues of B T match with those of the coefficent matrix A T (see (47)).
D. Projection of the Bloch sphere onto its equatorial plane: A map that projects the entire Bloch sphere onto the equatorial plane is defined by the transformation of the Bloch vector (p 1 , p 2 , p 3 ) → (p 1 , p 2 , 0).

This leads to the following linear transformation
We then express A P = 1 2 µ,ν (a P ) µν (σ µ ⊗ σ * ν ) to obtain Negative eigenvalues of A P clearly indicate that projection of the Bloch sphere onto the equatorial plane is not physical as it corresponds to a not completely positive map. The dynamical matrix B P is then obtained using the realignment (B P ) r ′ r;s ′ s = (A P ) r ′ s ′ ;rs : Eigenvalues of B P are same as those of A P .

E. Bit flip channel
A qubit bit flip channel reverses the state of a qubit from |0 to |1 with probability 1 − p, 0 ≤ p ≤ 1; the channel keeps the states unaltered with a probability p. This is a completely positive map equipped with the Kraus operators given by [3] Using the operator-sum representation we construct the associated A matrix: Adopting the matrix basis σµ √ 2 , µ = 0, 1, 2, 3 , as in all other examples studied earlier, we compute the coefficient matrix A BF = (a µν ) associated with A BF : The linear map A PF associated with the phase flip channel is given explicitly as a 4 × 4 matrix form: Then the associated coefficent matrix A BF is found to be based on the Lorentz singular value decomposition [9] of the canonical A-form associated with qubit transformations is being prepared [10] and it will be presented separately as a sequel to the present work.
We dedicate this tutorial article as a mark of our reverence to Professor ECG Sudarshan.