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One-qubit purity in terms of the discrete Wigner transform
Pureza de un qubit en términos de la transformada discreta de Wigner
One-qubit purity in terms of the discrete Wigner transform
CIENCIA ergo-sum, Revista Científica Multidisciplinaria de Prospectiva, vol. 27, núm. 1, marzo-junio 2020
Universidad Autónoma del Estado de México
Recepción: 17 de septiembre de 2018
Aprobación: 29 de mayo de 2019
Abstract: An explanation and an illustration of the meaning of a discrete phase-space is given. The class of a discrete Wigner transform (DWT) for the specific case of a one-qubit state is introduced. We derive the one-qubit state formalism around its formulation in terms of the DWT in detail. A novel structure of a one-qubit purity in terms of the DWT is introduced. We find a criterion for stating when a one-qubit state is either mixed or pure.
Keywords: Discrete phase space, Hilbert space, unbiased bases, discrete Wigner transform, purity.
Resumen: Se proporciona e ilustra una explicación del significado de espacio fase discreto dirigida a un lector no especialista. Se presenta también la clase de la transformada discreta de Wigner (TDW) para el caso específico de un estado de un qubit. Asimismo, derivamos detalladamente el formalismo involucrado en la formulación del estado de un qubit en términos de la TDW. En este contexto, se introduce una estructura novedosa de la pureza de un qubit en términos de la TDW y se halla un criterio para decidir cuando el estado de un qubit es puro o mixto.
Palabras clave: espacio fase discreto, espacio de Hilbert, bases imparciales, transformada discreta de Wigner, pureza.
Introduction
The real-valued Wigner function W(q, p) play the role of a quasi-probability distribution for continuous-variable quantum systems in continuous coordinates (q) versus momentum (p) (phase) space (Wigner, 1932; Hillary et al., 1984). In spite of the fact that W(q, p) allows us to calculate properties of a system through phase-space integrals weighted by it, however this cannot be interpreted as the positive-valued probability of simultaneously measuring observables and with eigenvalues p_{0} and q_{0}. In fact, W(q, p) could be negative in some phase-space regions (from there the term quasi-probability).
Buot (1974), Hannay and Berry (1980) were the first to propose the novel idea of the analogous Wigner function for a discrete (finite-dimensional) Hilbert space. Later on, such findings were rediscovered by Cohen and Scully (1986) and Feynman (1987) who defined a discrete Wigner function W for a single qubit. The above works were extended by Wootters (1987) and Galetti and De Toledo Piza (1988) by introducing a Wigner function for prime-dimensional Hilbert spaces.
These extensions have been employed for teleportation protocols (Koniorczyk et al., 2001; Paz, 2002), quantum algorithms (Bianucci et al., 2002; Miquel et al., 2002), and decoherence (Lopez, 2003).
There is scarce (almost null) information in the literature about the concept of purity of a single qubit from the point of view of the discrete Wigner transform. Within such a formalism it is difficult to find a criteria for stating whether a qubit state is pure. In the present paper we discuss both the discrete Wigner function for a single qubit and introduce the concept of purity for it.
1. One-qubit in terms of a discrete Wigner functio
The conventional definition of continuous phase space is a plane where the horizontal axis denotes the continuous position coordinate q while the vertical axis represents the continuous linear momenta variable p. At this stage it arises the following question: what does a discrete phase space mean? By discrete phase space (DPS) we mean that the coordinate variable q takes a finite set of real values Q = {q_{1}, q_{2},….,q_{n}} while the momenta variable takes also a finite set of real values P = {p_{1}, p_{2},….,p_{n}} in such a way that the DPS is the set {(q_{i}_{,}p_{j})|qiQ, p_{j}P}. We can state that the discrete analogue of phase space in a d-dimensional Hilbert space is a d × d real array. Thus, for the case of one qubit one has d = 2. Consequently a 2-dimensional Hilbert discrete phase space can be represented by the following four real numbers denoted by:
In the continuous phase space (continuous XY-plane) we can draw continuous lines while in the discrete case this is not the case. We define a “line” in the d-dimensional discrete phase space as a set of d points in discrete phase space. Thus, a “line” in the case of a 2-dimensional discrete phase space is a set of 2 points in the discrete phase space of Eq. (1). The discrete phase space of Eq. (1) will contain the following six “lines”
The discrete phase space can then be partitioned into a collection of parallel lines. By a parallel line it is understood disjoint sets of d = 2 phase space points. Such partitions are called striations (Wootters, 1987). According to Eq. (2), in the present case there will be the following d + 1 = 2 + 1 = 3 striations
Wootters’ definition of discrete Wigner functions employs a special set of d + 1 = 2 + 1 = 3 bases for a d-dimensional Hilbert space. Such bases can be defined in terms of eigen-states of generalized Pauli operators. In this way, for the Pauli matrix the respective eigen-states should be:
where we have employed the matrix notation and . In the aboveequation by eigenstates of the Pauli operator σ_{x} we mean the following where we have used that .
For the Pauli matrix the eigen-states are:
Clearly, and _{. }
While for the respective eigen-states will be:
Clearly
A set is a basis for a Hilbert space if any state of the space can be written as where a and b are complex numbers such that |a|^{2} + |b|^{2} = 1. The basis B is orthonormal if where . Let us observe that .
The states given by (5), (6), (7) define the following d + 1 = 2 + 1 = 3 different bases
We must observe that there is a one-to-one correspondence between the striations of Eq. (3) and the bases of Eq. (7), that is
On the other hand, the bases of Eq. (7) are unbiased, that is
2. Discrete Wigner functions
We have the necessary ingredients to define a class of discrete Wigner functions. On one hand, there is a d + 1 = 2 + 1 = 3 mutually unbiased bases {B_{x}, B_{y}, B_{z}} and on the other a set of d + 1 = 2 + 1 = 3 striations {S_{1}, S_{2}, S_{3}} of the d × d = 2 × 2 = 4 phase space into d = 2 parallel lines. We need to choose a one-to-one maps as follows:
Each basis set B_{i} is associated with one striation S_{i}
Each basis vector is associated with a line α_{i}, _{j} (the jth line of the ith striation).
With the above association, the Wigner function W is uniquely defined if we demand that
where are the basis states of Eqs. (5)-(7) and p is the one-qubit density state (Nielsen, & Chuang, 2000). Let us observe that Eq. (10) gives rise to the following Wigner functions associated to the four real numbers of Eq. (1)
where the probabilities satisfy
Eq. (11) define uniquely the Wigner function W in terms of the probabilities
3. Wigner formulation of one-qubit purity
In the literature is scarce the information on the one-qubit purity. We then propose the following formulation of the purity in terms of the Wigner function
where 0 ≤ c_{e} ≤ 1 are real numbers. Eq. (14) can be expanded in terms of the following six equations as follows
We say that one qubit is in a mixedstate if
On the other hand, one qubit is in a pure state if
Conclusions
We have formulated the one-qubit state in terms of the Wigner transform operating on a discrete phase-space. In classical approaches the phase-space is usually understood as the composition of both the continuous spatial coordinates {X} and the continuous momentum space {P}, that is, a {X, P} continuous generalized coordinates space. In the present approach we focus on two spatial coordinates {q_{1}, q_{2}} versus two momenta coordinates {p_{1}, p_{2}} is such a way that the two above sets generates the following four elements set {q_{1}p_{1}, q_{1}p_{2}, q2 p1, q2 p2} ≡ {x11, x12, x21, x22} defining the discrete phase-space of Eq. (1). An example of a one possible phase-space is the following {-17.21, 8.3, -2.1, -0.17}.
The striations of Eq. (3) can be generalized for a prime d-dimensional phase-space. In particular, we have considered a one-qubit state where d = 2. Let us note that for the one-qutrit state one must have d = 3 Eq. (10) defines uniquely a one-qubit state in terms of the Wigner discrete transform W. Such a definition is equivalent to the conventional definition of a one-qubit state if one observe from Eq. (10) that where Trρ = |a_{0}|^{2} + |a_{1}|^{2} = 1. On the other hand, the main achievement of the present work is the formulation of the purity of a one-qubit state in terms of the Wigner discrete transform W through Eq. (14). It is worth to mention that such a formulation is absent in the literature. With Eqs. (14) and (17) we can state a criteria for concluding when a one-qubit is in a pure state.
Prospective analysis
The formulation of a one-qubit state in terms of W is elegant and allows to geometrize the probabilities p(_{i}, _{j}) of Eq. (11). Indeed, the constrain of Eq. (12) on the probabilities p_{i}, _{j}implies that they can be represented in a Bloch sphere.
On the other hand, the understanding of the formulation of the purity of a one-qubit state in terms of the discrete Wigner transform as stated from Eqs. (15)-(17) could help in the future to understand intriguing properties of qubits such as the relation between W and the speed of quantum information processing.
Referencias
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