Entanglement universality of twoqubit Xstates
Abstract
We demonstrate that for every twoqubit state there is a Xcounterpart, i.e., a corresponding twoqubit Xstate of same spectrum and entanglement, as measured by concurrence, negativity or relative entropy of entanglement. By parametrizing the set of twoqubit Xstates and a family of unitary transformations that preserve the sparse structure of a twoqubit Xstate density matrix, we obtain the parametric form of a unitary transformation that converts arbitrary twoqubit states into their Xcounterparts. Moreover, we provide a semianalytic prescription on how to set the parameters of this unitary transformation in order to preserve concurrence or negativity. We also explicitly construct a set of Xstate density matrices, parametrized by their purity and concurrence, whose elements are in onetoone correspondence with the points of the concurrence versus purity (CP) diagram for generic twoqubit states.
keywords:
Entanglement, Concurrence, Negativity, Relative Entropy of Entanglement, XstatesPacs:
03.65.Ud, 03.67.Mn, 03.65.AaremarkRemark \newproofproofProof \newdefinitiondefinitionDefinition \biboptionssort&compress
1 Introduction
Despite our limited understanding of what entanglement is at the most fundamental level, many tasks that feature entanglement as a sine qua non condition have been successfully performed thanks to our evergrowing ability to manipulate quantum systems comprised of interacting subsystems 91Ekert661 (); 07Ursin481 (); 92Bennett2881 (); 08Barreiro282 (); 93Bennett1895 (); 12Ma269 (); 97Shor1484 (); 12Lopez773 (). Ultimately, entanglement is an attribute of quantum states and, as such, practical applications will unavoidably rely upon one’s ability to prepare certain density matrices. In practice, though, depending on the details of a particular implementation and on the types of noise that affect the relevant quantum system, some entangled states may turn out to be very hard to produce, thus limiting the entanglement available to practical applications. As a result, it is natural to ask: how much entanglement is left as we avoid certain density matrices?
In this paper this question is approached in the context of twoqubit states and with a clear specification as to which states are to be avoided. Surprisingly, we find that no entanglement (as quantified by three entanglement measures and with respect to a fixed level of mixedness) is lost as we avoid every twoqubit density matrix, but the sparse family that, in the computational basis, can display nonzero entries only along the main and antidiagonals; the socalled Xstates 07Yu459 (). For entanglement measures we consider concurrence, negativity and relative entropy of entanglement, in terms of which our main result acquires its more precise expression: for every twoqubit state with a value of entanglement set by any of these measures, there is a corresponding Xstate of same spectrum and same entanglement.
Twoqubit Xstates generalize many renowned families of entangled twoqubit states, for example, Bell states 00Nielsen (), Werner states 89Werner4277 (), isotropic states 99Horodecki4206 () and maximally entangled mixed states 00Ishizaka22310 (); 01Verstraete12316 (); 01Munro30302 (); 03Wei22110 (). They were first identified as a class of states of interest in the work of Yu and Eberly 07Yu459 (), where some of their properties in connection with the phenomenon of sudden death of entanglement were investigated. Ever since, the interest in Xstates exceeded its original motivation and has been manifested in many other contexts 05Retzker050504 (); 09Rau412002 (); 12Quesada1322 (); 13Hedemann (); 13Costa (). Particularly relevant for this paper is the work of Hedemann 13Hedemann (), who provided compelling numerical evidence that the set of twoqubit Xstates alone is sufficient to access every possible combination of concurrence and purity available to twoqubit states, and conjectured that any generic twoqubit state can be converted into a Xstate via a unitary transformation that preserves concurrence. Besides proving Hedemann’s conjecture, we demonstrate that it also holds true when entanglement is quantified with negativity or relative entropy of entanglement instead of concurrence.
Also closely related to our purposes is the work of Verstraete et al. 01Verstraete12316 (), where it was shown that, for a fixed set of eigenvalues, the states of maximal concurrence, negativity or relative entropy of entanglement are the same Xstates, thus establishing the top frontier of the relevant entanglement versus mixedness diagrams as comprised by Xstates. Our main result extends theirs in implying that Xstates not only border such diagrams, but can be put in a manytoone correspondence with every internal point.
From a pragmatic viewpoint, the interest in this universality property of Xstates relies upon their inherent easiness of manipulation, both theoretical and experimental. Owing to the highly sparse form of Xstate density matrices written in the computational basis (Xdensity matrices, for short), a great deal of symbolic computations is possible, even in the context of entanglement quantification where one is usually forced to resort to numerical approaches. The possibility of replacing generic twoqubit density matrices with Xdensity matrices is a promising route toward a deeper understanding of mixedstate entanglement. On the experimental side, twoqubit Xstates can be produced and evolved, for example, with standard interactions arising in the context of nuclear magnetic resonance 09Rau412002 (); 00Rau032301 () and with variations of available technology for generating Werner states in optical and atomic implementations 02Zhang062315 (); 04Barbieri177901 (); 04Cinelli022321 (); 04Peters133601 (); 06Agarwal022315 (); 13Jin2830 ().
Throughout, aiming to take full advantage of the highly sparse form of twoqubit Xdensity matrices, we exploit the luxury of working in a constructiveanalytic fashion. Largely, this is enabled by the introduction of a simple parametrization on the set of Xstates, which leads to a geometric visualization of separable, entangled and rankspecific Xstates in the relevant parameter space. Thanks to this, we are able to explicitly construct a set of twoqubit Xstates that can be put in a onetoone correspondence with the points of the CPdiagram for generic twoqubit states. Most importantly, we parametrize a unitary transformation that maps an arbitrary twoqubit state into a Xstate of same entanglement (according to any one of the three considered measures), and show how to set the parameter values to achieve conservation of concurrence or negativity.
Our paper is structured as follows. In order to obtain the constructions that form the core of our work, in Sec. 2 we parametrize separable, entangled and rankspecific twoqubit Xdensity matrices. Our parametrizations are first put into use in Sec. 3, where we explicitly construct a minimal set of Xstates that exhausts the twoqubit CPdiagram. In Sec. 4 our main universality result is established by showing that every Xstate can be disentangled with a unitary transformation that preserves the sparse structure of a twoqubit Xdensity matrix (Sec. 4.1) and that our selected entanglement measures vary continuously during the disentangling process (Sec. 4.2). We summarize our main results and discuss some possible avenues for future research in Sec. 5.
2 Parametrizing twoqubit Xstates
Twoqubit Xstates are quantum states of a fourdimensional Hilbert space that do not mix the subspaces and . In the computational basis , they assume the matrix form
(1) 
with , and . In order to highlight the resemblance of matrix (1) with the alphabet letter ‘X’ (which justifies the nomenclature “Xstate”), we replace every vanishing entry of a matrix with a dot. Throughout, every density matrix of the form (1) is referred to as a Xdensity matrix. More generally, every matrix possessing nonzero terms only along the main and antidiagonals is said to be of the Xform.
That any Xdensity matrix has the form (1) is a direct consequence of the fact that, apart from the decoupling between and , all the inbuilt constraints of (1) are necessary features of a density matrix: the parametrization along the main diagonal establishes only normalization and nonnegativity of the diagonal entries, whereas the parametrization along the antidiagonal establishes only Hermiticity.
However, not every matrix of the form (1) with , and is a density matrix. In what follows we show how to further constrain the ranges of and in order to make the set of matrices of the form (1) with the corresponding parameter ranges to coincide with the set of (i) Xdensity matrices, (ii) Xdensity matrices of a fixed rank and (iii) separable Xdensity matrices.
2.1 Parametrizing twoqubit Xdensity matrices
The set of twoqubit Xdensity matrices is equal to the subset of matrices of the form (1) with parameter values that render it positive semidefinite. For that, we start by considering the characteristic equation for (1):
(2) 
where
(3) 
In the above, the calligraphic letters , , and are functions of the diagonal parameters , and . In fact, and ( and ) give the sum (product) of the diagonal entries of the unnormalized density matrices of the ‘fictitious qubits’ living in the subspaces and , respectively. Explicitly, ,
(4) 
Since the positive semidefiniteness of (1) is equivalent to the set of inequalities 09Bernstein (), we are left with three nonvacuous inequalities
(5a)  
(5b)  
(5c) 
Now, due to the nonnegativity of and , it is clear that the inequalities above are simultaneously satisfied if and only if
(6) 
which summarize necessary and sufficient conditions for the positive semidefiniteness of the form (1). Therefore, the set of these matrices with , , and fully characterizes the set of twoqubit Xdensity matrices. As we shall see next, such a parametrization enables an appealing geometric visualization of twoqubit Xstates and can be easily specialized to parametrize separable and fixedrank twoqubit Xstates.
2.2 Parametrizing twoqubit Xdensity matrices of a fixed rank
According to the NewtonGirard formulae 09Bernstein (), the coefficients of the characteristic equation (2) are the sum of all products of eigenvalues of matrix (1). This observation can be used to parametrize twoqubit Xdensity matrices with a fixed rank.
 Rank1:
 Rank2:
 Rank3:

The single zero eigenvalue imposes , and , which implies in the sole saturation of inequality (5c) or, equivalently,
(9)  Rank4:
It is thus clear that the set of twoqubit Xdensity matrices of a fixed rank is equivalent to the set of matrices (1) with parameter values verifying the corresponding constraint specified above. Throughout, we shall refer to each alternative of rankspecific parameter choice as a kind of Xstate. Accordingly, there are two kinds of rank1 and rank3 Xstates, three kinds of rank2 Xstates and a single kind of rank4 Xstates.
2.3 Parametrizing twoqubit separable Xdensity matrices
According to the PPT criterion 96Peres1413 (); 96Horodecki1 (), the set of twoqubit Xdensity matrices is equal to the set of matrices of the form (1) with parameter values that render itself and its partial transpose positive semidefinite. In Sec. 2.1, we have seen how the positive semidefiniteness of (1) constrains and [cf. Eq. (6)]. In this section, we find analogous constraints for the positive semidefiniteness of the partial transpose of (1).
It suffices to consider the partial transpose over one of the two subsystems, which we choose to be the second. In that case, the partial transpose operation over (1) yields a matrix of the same form, but with and (and also and ) swapped over, which implies that the positive semidefiniteness of the partially transposed matrix is guaranteed by the constraints (6) with and . Clearly, in order to have both (1) and its partial transpose positive semidefinite, and must be chosen according to
(11) 
Therefore, the set of separable twoqubit Xdensity matrices is identical to the set of matrices of the form (1) with parameter values that verify (11).
The results of this section are all summarized in Fig. 1, which represent Xstates with a fixed value of in a parameter space. Each plot corresponds to a different contribution of the parameters , and toward , in such a way that separable and entangled Xstates of all ranks and kinds can be visualized as vertices, sides and interior of a rectangle of side lengths and . Although can assume any real value between and , the figure conveys only cases with . If then , which implies the collapse of the rectangle to the origin of the parameter space. In this particular case, the resulting states are all rank1 or rank2 separable Xstates of the forms or , for and . If , it is no longer possible to establish or , hence no pure states can occur.
3 Minimal set of Xstates for full occupancy of the twoqubit CPdiagram
As a first application of the parametrizations obtained in the previous section, we now present a construction of a minimal set of Xstates that fully occupy the entangled region of the CPdiagram of generic twoqubit states. We refer to it as a minimal set because its elements are in a onetoone correspondence with the points of the CPdiagram, in such a way that if a single state is removed from the set, a point of the CPdiagram is consequently missed. For the reader’s convenience, in A we briefly review some basic aspects of the entanglement measure concurrence, and in B the boundaries of the twoqubit CPdiagram are explicitly obtained. For more information on twoqubit entanglement versus mixedness diagrams, we refer the reader to Refs. 01Munro30302 (); 13Hedemann (); 05Ziman52325 (); 03Wei22110 ().
To present our construction we divide the CPdiagram in three disjoint purity domains whose union equals the interval , where all twoqubit entangled states live 98Zyczkowski883 (). For each of these purity subdomains we prove a theorem whose statement provides parameter values, as functions of the desired purity and concurrence values, that produce a family of Xstates of fixed rank that exhausts the corresponding CPregion. Before stating and proving the theorems, let us briefly outline the procedure by which the proposed parameter values were obtained.
An arbitrary Xstate , parametrized as in (1), has its purity and concurrence given by the following formulae:^{3}^{3}3While the purity formula follows by direct evaluation of for given by Eq. (1), the concurrence formula can be easily obtained from a useful specialization, due to Wang and coworkers 06Wang4343 (), of the standard concurrence formula 98Wootters2245 () for arbitrary twoqubit states to the case of twoqubit Xstates — see A, in particular Eq. (67).
(12)  
(13) 
which can be specialized to give the purity and concurrence of Xstates of a fixed rank by restricting their parameters according to the constraints (7) to (10). From the resulting purity equation for each rank we can eliminate one of the Xstate parameters in favor of and, hence, rewrite as a function of and the remaining Xstate parameters. Then, fixing and , with and representing any possible values of purity and concurrence for the specific rank and the relevant purity subdomain, we obtain a transcendental equation that can be solved for the Xstate parameters.
Although the constructions presented in the following theorems were obtained by solving such transcendental equations, we refrain from presenting the constructive steps that led to them. Instead, we state the obtained parameters values in terms of and and prove that, for any possible pair , they: (i) give origin to valid Xdensity matrices of a given rank and (ii) solve the equations and .
Theorem 1
For every generic rank1 state of concurrence , a rank1 Xstate of same concurrence can be constructed from equation (1) by taking
(14) 
We start by showing that for every the choice of parameters of (14) yields a valid rank1 Xstate. In fact, using (14) to compute the coefficients , and gives
(15) 
which complies with , , and, hence, characterizes the resulting states as rank1 Xstates of the first kind. Finally, substituting (14) in the concurrence formula (13) we obtain .
Note that although the choice of parameters (14) leads to rank1 Xstates of the first kind, rank1 Xstates of the second kind can also access every : this is achieved with
(16) 
A proof of this assertion follows the same steps presented above and will be omitted.
Theorem 2
For every generic state of concurrence and purity , rank2 Xstates of same concurrence and purity can be constructed from equation (1) by taking
(17) 
where
(18) 
As shown in B, the concurrence of generic twoqubit states with purities is limited to the interval , thus and from Eq. (17) are welldefined. Direct computation of the coefficients , and results in
(19) 
which complies with , , and, hence, characterizes the resulting states as rank2 Xstates of the third kind. Finally, note that straightforward evaluation of Eqs. (12) and (13) with the choice of parameters (17) gives, respectively, and .
Regarding theorem 2, two remarks are worth pointing out. First, the parameters of Eq. (17) produce valid rank2 Xstates also in the CPregion and , hence covering the entire shaded area in Fig. 2(a). Since generic rank2 states are restricted to the CPregion and (cf. B), we may conclude that the parameters of Eq. (17) lead to Xstate counterparts of same concurrence, purity and rank for every twoqubit state of rank2. Besides, as we demonstrate in C, any pair of rank2 density matrices of same purity can be related via unitary conjugation. Hence, the Xstate counterparts defined by Eq. (17) can be produced by a unitary transformation of a rank2 twoqubit state of the same purity.
Secondly, in theorem 2 we relied upon rank2 Xstates of the third kind to exhaust the corresponding CPregion. Indeed, rank2 Xstates of the first and second kinds cannot achieve concurrences greater than [dotted line in Fig. 2(a)], being thus unsuitable for the task. This can be easily seen by computing the purity and concurrence for such states and then combining the resulting expressions to get
(20) 
where represents (in the case of the first kind parameters) or (in the case of the second kind parameters). Clearly, the maximal value of is .
Theorem 3
For every generic state of concurrence and purity , rank3 Xstates of same concurrence and purity can be constructed from equation (1) by taking
(21) 
where
(22) 
As shown in B, the concurrence of generic twoqubit states with purities is limited to the interval . In this case, some analysis of Eq. (22) reveals that , rendering from Eq. (21) welldefined. Direct computation of the coefficients , and gives
(23) 
which complies with , , and, hence, characterizes the resulting states as rank3 Xstates of the first kind. To see that , note the following:
(24) 
where the first, second and fourth inequalities follow from the upper bounds for , and , respectively. This particular choice of parameters allows, through the straightforward evaluation of Eqs. (12) and (13), to obtain and .
A few remarks about theorem 3 are due. First, by imposing and to the choice of parameters of Eq. (21), we find that they also produce valid rank3 Xstates in the CPregion and , where
(25) 
However, since in this purity range the concurrence of generic twoqubit states goes up to , such a choice does not fill the entire CPregion in the purity interval , as shown with shading in Fig. 2(b). Although a different choice of rank3 parameters could be tailored to exhaust that region, for now we shall leave its occupancy for the choice of rank2 parameters of Eq. (17), as shown with hatching in Fig. 2(a).
Second, although the choice of parameters of Eq. (21) yields rank3 Xstates of the first kind, it is also possible to access every and with rank3 Xstates of the second kind. This can be achieved, for example, with
(26) 
where
(27) 
A proof of this follows the same steps presented above and will be omitted. We only note that also this choice of parameters can be extended to the purity domain . However, this is only possible for certain values of which (i) do not cover every point already visited by the extension of the parameters of Eq. (21) [shaded and nonhatched region in Fig. 2(b)] and (ii) do not cover every point left unvisited by the extension of the parameters of Eq. (21) [empty area between the solid and dotted lines in Fig. 2(b)].
Third, although for every rank3 twoqubit state of purity we have constructed a Xstate of same purity, rank and concurrence, that does not mean that our construction is related to the input state via a unitary conjugation. As we demonstrate in C, such a conclusion can only be drawn in the case of rank1 and rank2 states. For example, consider the twoqubit density matrix of rank3
(28) 
which has and . The following matrices are rank3 Xstate counterparts constructed according to the parameter values of Eqs. (21) and (26), respectively:
(29) 
Although the three matrices share the same rank, purity and concurrence, each one displays a different set of eigenvalues, being thus impossible to be related via unitary conjugation. Of course, this does not preclude the existence of yet another Xstate counterpart that could be obtained via unitary conjugation of the input density matrix. The existence of such counterparts will be proved in the next section.
As a summary of the main results of this section, we now explicitly state the matrix forms (in the computational basis) of the elements of our minimal set – formed from the parameter choices of Eqs. (14), (17) and (21) – namely,
(30) 
with
(31)  
(32)  
(33) 
where and were defined in Eqs. (18) and (22), respectively. We remark that although many other minimal sets that exhaust the entangled region of the CPdiagram of twoqubit states do exist, has the advantage of being highly sparse and formed exclusively by rankdeficient Xstates.
Let us conclude this section with a word of caution: the exhaustion of the CPdiagram with elements of does not imply that other entanglement versus mixedness diagrams will also be exhausted by . This is illustrated in Fig. 3 in the case of the negativity versus purity diagram (cf. A for the definition and a brief review of the entanglement measure negativity). In this figure, the thick line bounds the negativity of generic twoqubits states of a fixed purity (for more details, see Ref. 03Wei22110 ()). The shading highlights the accessible region to the elements of and it was obtained by numerically computing their negativity. Noticeably, the shading does not completely fill the area below the thick line. This is better understood by recalling that different entanglement measures quantify different “types of entanglement” 09Horodecki865 (). So, while the states of exhaust the possible values of concurrencelikeentanglement for a fixed purity, they may (and do) lack some values of negativitylikeentanglement.
The magnification glass in Fig. 3 also shows that, for , the shading goes beyond the (thin) line generated by the elements of with maximal concurrence per purity (such states formed the border of the CPdiagram in Fig. 2). This is also due to the existence of many types of entanglement and, in particular, to the fact that any two different entanglement measures place different orderings on the set of density matrices 99Eisert145 (); 00Virmani31 (): although the states that generate points above the line are obviously less concurrenceentangled than the maximally concurrenceentangled states of same purity, they are more negativityentangled than the latter. Curiously, though, for , the elements of with maximal concurrence are also the elements of highest negativity.
4 Entanglement Universality via Unitary Evolution
So far, we have established a weak form of Xstate entanglement universality: there are more than necessary Xstates to visit every point of a generic twoqubit CPdiagram. As we have seen, though, our produced Xstate counterparts cannot, in general, be obtained from the input states via a unitary transformation, nor they will exhaust entanglement versus purity diagrams other than the one in which entanglement is quantified by concurrence. Needless to say, however, is that the possibility of coherently producing Xstate counterparts to achieve any value of entanglement, as measured by any entanglement measure, would be very interesting from both fundamental and practical viewpoints.
In this section we considerably strengthen our preliminary universality result to accommodate some of the aforementioned desiderata. Specifically, we claim to be always possible to coherently produce Xstate counterparts for any twoqubit state preserving (not simultaneously) its concurrence, negativity or relative entropy of entanglement (cf. A for a brief review of these entanglement measures). Throughout, we shall refer to these coherently produced Xstate counterparts as Xcounterparts. Of course, the requirement of coherent preparation implies in preservation of mixedness (e.g. purity or von Neumann entropy), so that Xcounterparts will exhaust many types of entanglement versus mixedness diagrams (e.g., all types considered in Ref. 03Wei22110 ()).
An important step toward proving this stronger universality claim was given by Verstraete et al. 01Verstraete12316 (), who showed that concurrence, negativity and relative entropy of entanglement of a generic twoqubit state is maximized by conjugation with a unitary matrix of the form , where and are arbitrary local unitary transformations, is a unitary diagonal matrix, is the unitary matrix such that is the diagonal matrix of eigenvalues of sorted in nonascending order, and is the improper orthogonal matrix
(34) 
From here, it is immediate to find that (up to local unitary transformations), the density matrix of eigenvalues with maximal concurrence, negativity and relative entropy of entanglement is
(35) 
Thus, for a fixed spectrum, the maximally entangled mixed state (under the three considered measures) is a Xstate.
It follows from this observation and from the intermediate value theorem (see, e.g., Ref. 67Apostol ()), that to prove our stronger universality claim it suffices to show that (i) any entangled Xstate can be disentangled via a unitary transformation that preserves the Xform, and (ii) concurrence, negativity and relative entropy of entanglement vary continuously during the referred disentangling evolution. In fact, if (i) and (ii) are true, then the Xcounterpart of can be prepared by composing two unitary evolutions, and , as in
(36) 
where denotes a unitary transformation that initiates a Xform preserving and disentangling transformation of , which is aborted when the instantaneous Xstate reaches either the concurrence, negativity or relative entropy of entanglement of the initial state .
The remainder of this section is devoted to prove assertions (i) and (ii).
4.1 Coherent Disentanglement with Xform Preservation
We start by considering the unitary transformation induced by the following unitary matrix with and :
(37) 
Clearly, consists of two independent elements applied to the subspaces spanned by and . Since a Xstate can be seen as two fictitious qubits living in each of these subspaces, conjugation of an arbitrary Xstate with will necessarily preserve the Xform.^{4}^{4}4Note, however, that does not induce the most general unitary transformation that preserves the Xform. First, the most general element of has parameters (disconsidering an unimportant global phase), whereas each element in (37) has only parameters. Second, even if we employed the most general parametrization, it is not difficult to see that it is possible to preserve the Xform with unitary transformations that are not of the Xform. Two obvious examples are the unitary transformations induced by conjugation with the unitary matrices and .
For what follows, it will prove itself useful to determine how certain parameters of an arbitrary Xstate change under the unitary transformation induced by . Let and be two Xstates with parameters and , defined according to Eq. (1). Then, some straightforward (however tedious) computation gives
(38) 
where, for brevity, we have defined , ,
(39) 
Moreover, the following conservation laws can be easily established from the invariance of the trace and the determinant of a matrix unitarily conjugated:
(40)  
(41) 
where , , and were defined in Eq. (4).
Let us now see how to set the parameters of in order to turn it into a disentangling unitary transformation for any entangled Xstate. From Sec. 2.3, we know that will be a separable density matrix if and only if and are no greater than the minimum between and , cf. Eq. (11). Combined with the conservation law (41), this condition can be rewritten as
(42) 
From a strictly algebraic viewpoint, a simple choice of and that fulfills both inequalities immediately comes out. Consider, first, the case where the input state has , being thus identified with a point in the parameter space of Fig. 4(a). If conjugation with can move that point to the left in order to make , while keeping its ordinate constant, i.e. , then the inequalities (42) become
(43) 
Noticeably, the first inequality is satisfied with saturation, as the minimization yields thanks to the positive semidefiniteness of that requires [cf. Eq. (6)]. For the same reason, the second inequality becomes , whose validity also follows from the positive semidefiniteness of .
Analogously, if the input state has , as illustrated in Fig. 4(b), moving it down such that and , turns inequalities (42) into
(44) 
In this case, the result of the minimization is , from which follows that (44) reduces to the alwaystrue inequalities and .
In a nutshell, algebraically, inequalities (42) can be satisfied by choosing and as follows:^{5}^{5}5We need not to consider the case since every with such property is automatically separable [cf. Fig. 1(c)].
(45) 
Before accepting Eq. (45) as a solution, though, we must check whether conjugation of with can produce states with such parameters values of and . As it turns out, this is feasible for every entangled Xstate :
Theorem 4
Let be the set of parameters specifying an arbitrary entangled Xstate , and the set of associated functions of , and defined in equations (4) and (39). For every such , the following choice of parameters for the unitary matrix of equation (37) produces a Xstate whose parameters and are given by equation (45).
If , make
(46) 
and such that if . Otherwise (), is such that
(47) 
where
(48) 
If , the expressions for and specified above must be interchanged, and the substitutions , and performed in equation (48).
We defer to D a proof that the parameters specified in the theorem are welldefined and actually implement the desired transformation.
4.2 Continuity of Entanglement
Now that we have established that an arbitrary twoqubit entangled Xstate can be disentangled by conjugation with , we shall consider the entanglement dynamics of that state evolving under the action of the strongly continuous oneparameter unitary group , where and is the (Xformed) Hermitian matrix such that . The explicit form of can be promptly obtained from Eq. (37) by performing the replacements for every and with chosen according to theorem 4.
Clearly, as we vary within the range , the resulting states preserve Xform and spectrum, while entanglement varies from the initial value, in , until zero, in . In this section, we show that entanglement, as measured by concurrence, negativity and relative entropy of entanglement, varies continuously with , in such a way that every value of these entanglement measures between zero and the initial value can be reached by suitably choosing . Furthermore, equations linking the value of to any desired value of concurrence and negativity are derived. We conduct separate analysis for each entanglement measure.
Before proceeding with the continuity analysis, a few notational points are worth mentioning. Throughout, we add the subindex to every parameter associated with the Xstate . Accordingly, since is a matrix of the form (37), the conservation laws of Eq. (40) and (41) also hold with the primes replaced with the subindex , namely:
(49)  
(50) 
In addition, the parameters and can be promptly obtained by applying the replacement to Eq. (38), where denotes the value of specified in theorem 4:
(51) 
Identities (49), (50) and (51) will be extensively used in what follows.
4.2.1 Continuity of Concurrence
Using Eqs. (67) and (50) we can express the concurrence of as the following function of :
(52) 
For definiteness, suppose that the initial entangled Xstate is such that . Then, according to Eq. (51), and , in which case the maximization of Eq. (52) can be explicitly evaluated by noticing that the third term is never positive^{6}^{6}6Note that the third term in the maximization of Eq. (52) is a decreasing function in , in such a way that its maximum value is achieved for the minimum value of . Consequently, the nonpositivity of this term is implied by the condition , which can be seen to be always satisfied by adding up and . and the second term is never negative. As a result, the concurrence formula simplifies to
(53) 
which reaches the maximum at (or ), the minimum at (or ), and is clearly a continuous real function of defined on the interval . Thanks to the obvious continuity of in [cf. Eq. (51)], it turns immediate that the concurrence is also continuous in defined on the interval .
A completely analogous argument shows that if is such that , then
(54) 
which is also clearly continuous in .
Apart from establishing the continuity of the concurrence in , Eqs. (53) and (54) allow the determination of the value of that leads to a Xstate of concurrence . For example, if is such that , then the appropriate can be found by solving the following transcendental equation for :
(55) 
Numerically, this can be efficiently solved.
4.2.2 Continuity of Negativity
This closely follows the steps taken for establishing the continuity of the concurrence. Substitution of Eqs. (49) and (50) into (71) gives the negativity of as the following function of :
(56) 
Once again, we first consider an initial entangled Xstate with , for which we have already seen that and . Use of these in the minimization of Eq. (56) leads to an optimization problem that can be trivially solved by noticing that the third term is never negative^{7}^{7}7In this case, the third term in the minimization of Eq. (56) is an increasing function in , reaching its minimum value when . The nonnegativity of this term is thus implied by , which we have already seen to be true. and the second term is never positive. The negativity formula thus becomes
(57) 
which reaches the maximum at (or ), the minimum at and is clearly a continuous real function of defined on the interval . Just as occurred in the analysis of concurrence, the continuity of in implies that also the negativity is continuous in defined on the interval .
For an initial entangled Xstate such that , one can derive the following negativity formula, also obviously continuous in :
(58) 
We conclude by noting that Eqs. (57) and (58) allow the determination of the value of that leads to a Xstate of negativity . If has , for example, then can be obtained by numerically solving the following transcendental equation for :
(59) 
4.2.3 Continuity of Relative Entropy of Entanglement
Since a closedform for the relative entropy of entanglement of a twoqubit Xstate is still unknown (cf. A), we resort to a powerful continuity property of the relative entropy of entanglement^{8}^{8}8An analogous continuity property has been demonstrated in 00Nielsen64301 () for the entanglement of formation. It could have been exploited, in Sec. 4.2.1, to establish the continuity of the concurrence in . derived by Donald and Horodecki 99Donald257 () elaborating on a celebrated inequality due to Fannes 73Fannes291 ().
The general result of Ref. 99Donald257 () implies that, for any pair of twoqubit density matrices and such that , the following Fannestype inequality holds:
(60) 
where