Bipartite Entanglement, Partial Transposition and the Uncertainty Principle for Finite-Dimensional Hilbert Spaces
We first show that partial transposition for pure and mixed two-particle states in a discrete -dimensional Hilbert space is equivalent to a change in sign of the momentum of one of the particles in the Wigner function for the state. We then show that it is possible to formulate an uncertainty relation for two-particle Hermitian operators constructed in terms of Schwinger operators, and study its role in detecting entanglement in a two-particle state: the violation of the uncertainty relation for a partially transposed state implies that the original state is entangled. This generalizes a result obtained for continuous-variable systems to the discrete-variable-system case. This is significant because testing entanglement in terms of an uncertainty relation has a physically appealing interpretation. We study the case of a Werner state, which is a mixed state constructed as a convex combination with a parameter of a Bell state and the completely incoherent state, : we find that for , where is obtained as a function of the dimensionality , the uncertainty relation for the partially transposed Werner state is violated and the original Werner state is entangled.
pacs:03.67.-a, 03.67.Mn, 03.65.Ud
Quantum entanglement occurring in multipartite qubit (and qunits, i.e., -dimensional quantum bits) states is a powerful computation and information resource [1, 2]. Entanglement of pure quantum states is well understood, but entanglement of mixed quantum states, states that cannot be represented using a wave function but must be described using a density matrix, is not yet fully understood. For pure bipartite states, Schmidt coefficients relate the degree of entanglement to the von Neumann entropy of the reduced density matrix associated with either of the two subsystems; a pure state with vanishing von Neumann entropy corresponds to a separable state, whereas one with finite von Neumann entropy is entangled, and one with maximum von Neumann entropy is maximally entangled. But no general measure of entanglement of mixed states exists. Even deciding whether a mixed state is entangled or not is not always an easy task for mixed states. A large variety of measures have been studied in the literature to quantify entanglement for a given state, as discussed in Ref. . Entanglement witnesses, tools used to determine whether a state is separable or not, have been proposed . A useful concept in this context is partial transposition (PT) with respect to one of the particles [4, 5, 6]: when the partially transposed state is not a legitimate quantum mechanical (QM) state, the original state is entangled.
It has been noted in Ref.  (see also Refs. [8, 9]) that, for continuous variables, partial transposition of one particle of a bipartite state amounts to a change in sign of the momentum of that particle in the Wigner function (WF) of the state. Various uncertainty relations which must be fulfilled for a legitimate QM state were studied for continuous variables [7, 10], and conditions were found for their validity in a PT state, their violation implying entanglement of the original state.
These results for continuous variables can be illustrated in a particularly clear fashion in the special case of the “normalized version” of the EPR pure state,
Here, is the coordinate of particle and is the density operator of the system. If the resulting is not a legitimate quantum mechanical state, the original state is entangled. The WF is affected by in the coordinate basis as
thus changing the sign of momentum . The WF of state (1.1) is
or, in terms of the individual-particle positions and momenta ,
The WF resulting from the operation is, according to Eq. (1.3), given by
The variances of , and for are
Hence, is arbitrary and may violate the uncertainty principle; similarly for : if the relative canonical variables violate the uncertainty principle, the center-of-mass ones do not, and vice versa. If , the product of these uncertainties is and the uncertainty principle is not violated. Thus, except for this exceptional case, PT on the entangled pure state (1.1) does not lead to a physical state, as it violates the uncertainty principle: this verifies the fact that the original state (1.1) is entangled.
In the above continuous variable example, the violation of the uncertainty principle is a clear consequence of partial transposition of the entangled state in question. Extending the above example to discrete-variable systems defined in a finite-dimensional Hilbert space turns out to be of real value but is non-trivial: this extension is the purpose of the present paper. The extension is facilitated by the generalization of the concept of Wigner function to the discrete variable case, which has been widely studied in the literature (see Ref.  and references cited therein). Here, the formulation presented in Refs. [11, 12] is used to prove that, for the discrete case, PT can be interpreted in terms of a change in sign of the momentum of one of the particles in the Wigner function of the state. Just as for for the continuous case, this statement is appealing, as it gives an intuitive interpretation of PT. We find that the use of Schwinger operators  allows the generalization from the continuous-variable case to the finite-dimensional one. Hermitian operators can be constructed using the Schwinger operators, and an uncertainty relation (UR) can be formulated to detect entanglement of two-particle states: such an UR must be fulfilled for a legitimate QM state. Its violation for a PT ‘state’ signals entanglement in the original state. To the best of our knowledge, this analysis, originally carried out for continuous-variable systems, has not been pursued for the discrete-variable case: in our view, an UR provides a measure having a clear physical interpretation to test entanglement.
This paper is organized as follows. In the next section we use Schwinger operators to formulate an UR for states defined in a discrete -dimensional Hilbert space. The procedure is first illustrated for one-particle states in Sec. 2.1 and is then extended in Sec. 2.2 to two-particle states. In Sec. 2.2 we study the effect of PT on the Wigner function of a two-particle state. In Sec. 3 we study the application of the PT-UR criterion to a Werner state, which is a mixed state constructed as a convex combination with a parameter of a Bell state and the completely incoherent state, and find the values of for which the uncertainty relation for the PT state is violated and the original Werner state is entangled. Conclusions and a summary are presented in Sec. 4. Four appendices present a number of results mentioned in the text without interrupting the main flow of the presentation.
2 Schwinger operators and uncertainty relations
By way of introduction, we first consider just one particle that can be modeled in terms of a discrete -dimensional Hilbert space. We then extend the analysis to two-particle systems, which is the main topic of this paper.
2.1 One particle
Consider a one-particle system with with a discrete, finite set of states. The eigenvalues of observable operators take on a discrete set of values and the quantum description is given in terms of a finite-dimensional Hilbert space. As an example, consider a system with angular momentum , described in a Hilbert space of dimensionality . Another example is the position and momentum observables taken on a discrete lattice of finite dimensionality (see, e.g., Ref. ). The latter case is the one we shall be concerned with in this paper.
The Hilbert space to be considered is thus spanned by distinct states , with . As discussed in A, the periodicity condition is imposed. The Schwinger operators  and are also defined in A, as are the operators and . As performs translations in the variable and in the variable , we designate and as the position and momentum operators. Notice, however, that their commutation relation for finite is quite complicated [e.g., see Ref. , Eq. (20)], and that in the continuous limit their commutator reduces to the standard form [14, 15], .
B shows that under partial transposition of the density matrix in the coordinate representation and for (for , ), the probability distribution of momentum is affected as follows:
Thus, transposition in the coordinate representation has the intuitive physical meaning of changing the sign of momentum in the momentum probability distribution, an effect which corresponds to time-reversal (if no spin is present). Moreover, the Wigner function defined as in Refs. [11, 12] has the property,
as demonstrated in C, thus again exhibiting a change in sign of . The definition of the Wigner function of Refs. [11, 12] requires to be a prime number larger than 2. It turns out that this is the simplest extension of the continuous case to the discrete one that one can study, which can then be extended to the case where is not prime (see, e.g., Ref. [16, 17]). When is a prime number, the integers form a mathematical field playing a role parallel to that of the field of real numbers in the continuous case. Also, in this case, a set of mutually unbiased basis states is known . In what follows, when the Wigner function is not involved, the prime dimensionality requirement is not needed.
The Hermitian operators
which are physical observables, are formed using the Schwinger operators and introduced in A. For these Hermitian operators one finds the uncertainty relation (see, e.g., Ref. , Eq. (8.31), p. 224)
where is the density operator for a state of interest; the expectation values in (1.7cd) are to be evaluated with it. To obtain this relation one requires the following properties for the density operator and the observables:
Note that, quite generally, given a Hermitian operator , one may introduce the “expectation value”
where may not necessarily qualify as a density operator; i.e., it may not fulfill
The general idea in what follows is, for Hermitian operators and , to put the relation
2.2 Two particles
Let us now consider the two-particle case, which is the one of interest to us. Each particle is described in an -dimensional Hilbert space. We shall use Schwinger unitary operators defined for each particle, and relations similar to Eqs. (1.7cegmnfiada) and (1.7cegmnfiadb) of A to introduce the operators and which play the role of “momentum” and “position” for particle . B shows that under partial transposition of particle 1, for (we recall that for , ), the joint probability distribution (jpd) of the two momenta is affected as follows:
Thus, PT of particle 1 in the coordinate basis has the intuitive physical meaning of changing the sign of momentum for particle 1 in the jpd of the two momenta. The Wigner function, defined as in Refs. [11, 12], has the property, shown in C,
thus exhibiting again a change in sign of . Recall that the definition of the Wigner function of Refs. [11, 12] requires to be a prime number larger than 2; see discussion in the previous subsection for one particle.
We define the two-particle operators
We emphasize that the operator of Eq. (1.7cegma) differs from the “collective-coordinate” operator defined in Ref. , Eq. (50), which could be written as and is a variable “conjugate” to in the sense that it fulfills a commutation relation like that of Eq. (1.7cegmnfiac) of A for one particle. As definition (1.7cegma) does not involve square roots, it does not oblige us, in what follows, to consider prime-number dimensionalities only. In terms of these operators we define the Hermitian operators
For these Hermitian operators we have the uncertainty relation
We find that the commutator can be written as
Hence, for a general operator , we test the uncertainty relation
In D we obtain explicit expressions for the quantities entering this inequality for the case of a general “density operator” which is not necessarily positive definite or Hermitian.
3 Applications of the two-particle uncertainty relation
3.1 Werner state
Consider the Werner state ,
where is the pure state
This is a maximally-entangled pure state: the partial density operator of particle 1, obtained by tracing over particle 2, gives , meaning that the states of particle 1 are equally probable, thus conforming with the definition given, e.g., in Ref. . The density matrix elements of the Werner state of Eq. (1.7cegmna) are
and the matrix elements of the PT are
Note that this is Hermitian; however, it is not be positive definite. From D we find
The uncertainty relation (1.7cegmnq), applied to , requires
For , (1.7cegmnfg) gives , an inconclusive result. We therefore concentrate on . When
The criterion (1.7cegmnfh) arising from the present UR detects entanglement for , with becoming closer to 1 — a value which corresponds to the pure maximally entangled state (1.7cegmnb) — as the dimensionality increases. We remark that this does not mean that for other values of the state is not entangled. Our measure establishes that being in the interval is sufficient for the state to be entangled, but not necessary: there might be other values of for which the state is entangled.
There might be other measures that are stronger than the present one, in the sense that they may detect entanglement also for other values of for which the present UR criterion does not. An example we analyzed is provided by the notion of “negativity” introduced in Ref. . For , our uncertainty relation reveals that the original state is entangled for ; it does not yield any information for . On the other hand, the notion of negativity predicts entanglement for . Our result is not inconsistent with the negativity prediction. The reason for this difference is that (1.7cegmnfg) is just one property of a “proper state”: The uncertainty relation is fulfilled in the range , but other properties of a proper state must be violated, in order to break the non-negativity of the “density matrix” for the partially transposed state. Figure 2 plots the negativity function
versus for various values of . Although Eq. (1.7cegmnfij) has been developed here, it conforms to the negativity results of Ref. . The negativity is a stronger measure of entanglement than the one arising from the UR. The strength of the negativity measure is even more pronounced for larger values of ; the reason is that, as gets larger, ever more conditions are needed, in addition to the UR, to determine if the state is entangled. We have not examined uncertainty relations involving other observables to see if the entanglement regions shown in Fig. 1 are enlarged.
From a conceptual point of view, an attractive feature of the formulation presented in the present paper is that the effect of is intuitively simple to describe: changes the sign of the momentum of particle 1 in the jpd of the two momenta and in the Wigner function (in the latter case for prime); as a result, an uncertainty relation is violated for certain values of the parameter , from which we conclude that the original state is entangled. With the present procedure we are testing entanglement by a measure having a clear physical interpretation, just as was suggested in the conclusions of Ref. .
We studied a number of features of bipartate entanglement of discrete dynamical-variable-systems having a Hilbert space with finite dimensionality for each particle. We concentrated on the notion of partial transposition of one of the particles [4, 5, 6]; it is well known that a necessary condition for separability is that the density matrix obtained by partial transposition of the density matrix has only nonnegative eigenvalues, i.e., if the “state” obtained by partial transposition is not a bona fide QM state, the original state is entangled. We showed that the Wigner function, as formulated in Refs. [11, 12] (for prime ), has the intuitive property that under partial transposition the momentum (defined as in these references) of the corresponding particle changes sign [see Eq. (1.7cegl)]. This extends to discrete variables the result found for continuous variables in Ref.  (see also Refs. [8, 9]).
We formulated an uncertainty relation, Eq. (1.7cegmnq), for observables defined in terms of Schwinger operators, for an arbitrary QM state and for an arbitrary dimensionality . This extends the continuous-variables results of Refs. [7, 10]. When the uncertainty relation for the partially transposed state is violated, the original state is entangled. We applied the result to analyze the property of entanglement for a Werner state, constructed as a convex combination with a parameter of a maximally entangled Braunstein-Mann-Revzen pure state and the completely incoherent state . We found the inequality (1.7cegmnfh) which, when fulfilled, signals that the original Werner state is entangled: this occurs for a range of values of the parameter, where is dependent.
Reference  indicates that quantum correlations and entanglement would be best tested by measures having a physical meaning. We believe that the interpretation of partial transposition, in conjunction with the uncertainty relation we presented, meet this requirement.
Appendix A Schwinger operators for one particle
We consider the -dimensional Hilbert space spanned by distinct states , with , which are subject to the periodic condition . These states are designated as the “reference basis” of the space. We follow Schwinger  and introduce the unitary operators and , defined by their action on the states of the reference basis by the equations
The operators and fulfill the periodicity condition
being the unit operator. These definitions lead to the commutation relation
The two operators and form a complete algebraic set, in that only a multiple of the identity commutes with both . As a consequence, any operator defined in our -dimensional Hilbert space can be written as a function of and . We also introduce (i.e., define) the Hermitian operators and , which play the role of “momentum” and “position”, through the equations [14, 15]
What we defined as the reference basis can thus be considered as the “position basis”. With (1.7cegmnfiac) and definitions (1.7cegmnfiada), (1.7cegmnfiadb), the commutator of and in the continuous limit [14, 15] is the standard one, .
Appendix B Proof of Eq. (1.7cegk)
The jpd of the two momenta in the state is given by
The jpd of the two momenta for the PT operator is given by
This proves Eq. (1.7cegk). The above proof applies for , since, for , .
For the case of only one particle, the above result reduces to that of Eq. (1.7a).
Appendix C Proof of Eq. (1.7cegl)
By definition, the Wigner function after is then
where we made the change of variables . This proves Eq. (1.7cegl).
For the case of only one particle, the above result reduces to that of Eq. (1.7b).
Appendix D The quantities entering the uncertainty relation (1.7cegmnq)
We write explicitly the quantities entering the uncertainty relation (1.7cegmnq) for the case of a general “density operator” , not necessarily positive definite or Hermitian: