Entanglement universality of two-qubit X-states
We demonstrate that for every two-qubit state there is a X-counterpart, i.e., a corresponding two-qubit X-state of same spectrum and entanglement, as measured by concurrence, negativity or relative entropy of entanglement. By parametrizing the set of two-qubit X-states and a family of unitary transformations that preserve the sparse structure of a two-qubit X-state density matrix, we obtain the parametric form of a unitary transformation that converts arbitrary two-qubit states into their X-counterparts. Moreover, we provide a semi-analytic 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 X-state density matrices, parametrized by their purity and concurrence, whose elements are in one-to-one correspondence with the points of the concurrence versus purity (CP) diagram for generic two-qubit states.
keywords:Entanglement, Concurrence, Negativity, Relative Entropy of Entanglement, X-states
Pacs:03.65.Ud, 03.67.Mn, 03.65.Aa
remarkRemark \newproofproofProof \newdefinitiondefinitionDefinition \biboptionssort&compress
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 ever-growing 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 two-qubit 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 two-qubit density matrix, but the sparse family that, in the computational basis, can display non-zero entries only along the main- and anti-diagonals; the so-called X-states 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 two-qubit state with a value of entanglement set by any of these measures, there is a corresponding X-state of same spectrum and same entanglement.
Two-qubit X-states generalize many renowned families of entangled two-qubit 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 X-states 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 two-qubit X-states alone is sufficient to access every possible combination of concurrence and purity available to two-qubit states, and conjectured that any generic two-qubit state can be converted into a X-state 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 X-states, thus establishing the top frontier of the relevant entanglement versus mixedness diagrams as comprised by X-states. Our main result extends theirs in implying that X-states not only border such diagrams, but can be put in a many-to-one correspondence with every internal point.
From a pragmatic viewpoint, the interest in this universality property of X-states relies upon their inherent easiness of manipulation, both theoretical and experimental. Owing to the highly sparse form of X-state density matrices written in the computational basis (X-density 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 two-qubit density matrices with X-density matrices is a promising route toward a deeper understanding of mixed-state entanglement. On the experimental side, two-qubit X-states 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 two-qubit X-density matrices, we exploit the luxury of working in a constructive-analytic fashion. Largely, this is enabled by the introduction of a simple parametrization on the set of X-states, which leads to a geometric visualization of separable, entangled and rank-specific X-states in the relevant parameter space. Thanks to this, we are able to explicitly construct a set of two-qubit X-states that can be put in a one-to-one correspondence with the points of the CP-diagram for generic two-qubit states. Most importantly, we parametrize a unitary transformation that maps an arbitrary two-qubit state into a X-state 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 rank-specific two-qubit X-density matrices. Our parametrizations are first put into use in Sec. 3, where we explicitly construct a minimal set of X-states that exhausts the two-qubit CP-diagram. In Sec. 4 our main universality result is established by showing that every X-state can be disentangled with a unitary transformation that preserves the sparse structure of a two-qubit X-density 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 two-qubit X-states
Two-qubit X-states are quantum states of a four-dimensional Hilbert space that do not mix the subspaces and . In the computational basis , they assume the matrix form
with , and . In order to highlight the resemblance of matrix (1) with the alphabet letter ‘X’ (which justifies the nomenclature “X-state”), we replace every vanishing entry of a matrix with a dot. Throughout, every density matrix of the form (1) is referred to as a X-density matrix. More generally, every matrix possessing non-zero terms only along the main- and anti-diagonals is said to be of the X-form.
That any X-density 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 non-negativity of the diagonal entries, whereas the parametrization along the anti-diagonal 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) X-density matrices, (ii) X-density matrices of a fixed rank and (iii) separable X-density matrices.
2.1 Parametrizing two-qubit X-density matrices
The set of two-qubit X-density 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):
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, ,
Now, due to the non-negativity of and , it is clear that the inequalities above are simultaneously satisfied if and only if
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 two-qubit X-density matrices. As we shall see next, such a parametrization enables an appealing geometric visualization of two-qubit X-states and can be easily specialized to parametrize separable and fixed-rank two-qubit X-states.
2.2 Parametrizing two-qubit X-density matrices of a fixed rank
According to the Newton-Girard 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 two-qubit X-density matrices with a fixed rank.
The single zero eigenvalue imposes , and , which implies in the sole saturation of inequality (5c) or, equivalently,
It is thus clear that the set of two-qubit X-density 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 rank-specific parameter choice as a kind of X-state. Accordingly, there are two kinds of rank-1 and rank-3 X-states, three kinds of rank-2 X-states and a single kind of rank-4 X-states.
2.3 Parametrizing two-qubit separable X-density matrices
According to the PPT criterion 96Peres1413 (); 96Horodecki1 (), the set of two-qubit X-density 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
The results of this section are all summarized in Fig. 1, which represent X-states 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 X-states 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 rank-1 or rank-2 separable X-states 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 X-states for full occupancy of the two-qubit CP-diagram
As a first application of the parametrizations obtained in the previous section, we now present a construction of a minimal set of X-states that fully occupy the entangled region of the CP-diagram of generic two-qubit states. We refer to it as a minimal set because its elements are in a one-to-one correspondence with the points of the CP-diagram, in such a way that if a single state is removed from the set, a point of the CP-diagram 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 two-qubit CP-diagram are explicitly obtained. For more information on two-qubit entanglement versus mixedness diagrams, we refer the reader to Refs. 01Munro30302 (); 13Hedemann (); 05Ziman52325 (); 03Wei22110 ().
To present our construction we divide the CP-diagram in three disjoint purity domains whose union equals the interval , where all two-qubit 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 X-states of fixed rank that exhausts the corresponding CP-region. Before stating and proving the theorems, let us briefly outline the procedure by which the proposed parameter values were obtained.
An arbitrary X-state , parametrized as in (1), has its purity and concurrence given by the following formulae:333While 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 two-qubit states to the case of two-qubit X-states — see A, in particular Eq. (67).
which can be specialized to give the purity and concurrence of X-states 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 X-state parameters in favor of and, hence, rewrite as a function of and the remaining X-state 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 X-state 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 X-density matrices of a given rank and (ii) solve the equations and .
For every generic rank-1 state of concurrence , a rank-1 X-state of same concurrence can be constructed from equation (1) by taking
Note that although the choice of parameters (14) leads to rank-1 X-states of the first kind, rank-1 X-states of the second kind can also access every : this is achieved with
A proof of this assertion follows the same steps presented above and will be omitted.
For every generic state of concurrence and purity , rank-2 X-states of same concurrence and purity can be constructed from equation (1) by taking
which complies with , , and, hence, characterizes the resulting states as rank-2 X-states 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 rank-2 X-states also in the CP-region and , hence covering the entire shaded area in Fig. 2(a). Since generic rank-2 states are restricted to the CP-region and (cf. B), we may conclude that the parameters of Eq. (17) lead to X-state counterparts of same concurrence, purity and rank for every two-qubit state of rank-2. Besides, as we demonstrate in C, any pair of rank-2 density matrices of same purity can be related via unitary conjugation. Hence, the X-state counterparts defined by Eq. (17) can be produced by a unitary transformation of a rank-2 two-qubit state of the same purity.
Secondly, in theorem 2 we relied upon rank-2 X-states of the third kind to exhaust the corresponding CP-region. Indeed, rank-2 X-states 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
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 .
For every generic state of concurrence and purity , rank-3 X-states of same concurrence and purity can be constructed from equation (1) by taking
As shown in B, the concurrence of generic two-qubit states with purities is limited to the interval . In this case, some analysis of Eq. (22) reveals that , rendering from Eq. (21) well-defined. Direct computation of the coefficients , and gives
which complies with , , and, hence, characterizes the resulting states as rank-3 X-states of the first kind. To see that , note the following:
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 .
However, since in this purity range the concurrence of generic two-qubit states goes up to , such a choice does not fill the entire CP-region in the purity interval , as shown with shading in Fig. 2(b). Although a different choice of rank-3 parameters could be tailored to exhaust that region, for now we shall leave its occupancy for the choice of rank-2 parameters of Eq. (17), as shown with hatching in Fig. 2(a).
Second, although the choice of parameters of Eq. (21) yields rank-3 X-states of the first kind, it is also possible to access every and with rank-3 X-states of the second kind. This can be achieved, for example, with
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 non-hatched 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 rank-3 two-qubit state of purity we have constructed a X-state 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 rank-1 and rank-2 states. For example, consider the two-qubit density matrix of rank-3
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 X-state 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,
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 CP-diagram of two-qubit states do exist, has the advantage of being highly sparse and formed exclusively by rank-deficient X-states.
Let us conclude this section with a word of caution: the exhaustion of the CP-diagram 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 two-qubits 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 concurrence-like-entanglement for a fixed purity, they may (and do) lack some values of negativity-like-entanglement.
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 CP-diagram 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 concurrence-entangled than the maximally concurrence-entangled states of same purity, they are more negativity-entangled 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 X-state entanglement universality: there are more than necessary X-states to visit every point of a generic two-qubit CP-diagram. As we have seen, though, our produced X-state 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 X-state 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 X-state counterparts for any two-qubit 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 X-state counterparts as X-counterparts. Of course, the requirement of coherent preparation implies in preservation of mixedness (e.g. purity or von Neumann entropy), so that X-counterparts 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 two-qubit 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 non-ascending order, and is the improper orthogonal matrix
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
Thus, for a fixed spectrum, the maximally entangled mixed state (under the three considered measures) is a X-state.
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 X-state can be disentangled via a unitary transformation that preserves the X-form, 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 X-counterpart of can be prepared by composing two unitary evolutions, and , as in
where denotes a unitary transformation that initiates a X-form preserving and disentangling transformation of , which is aborted when the instantaneous X-state 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 X-form Preservation
We start by considering the unitary transformation induced by the following unitary matrix with and :
Clearly, consists of two independent elements applied to the subspaces spanned by and . Since a X-state can be seen as two fictitious qubits living in each of these subspaces, conjugation of an arbitrary X-state with will necessarily preserve the X-form.444Note, however, that does not induce the most general unitary transformation that preserves the X-form. 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 X-form with unitary transformations that are not of the X-form. 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 X-state change under the unitary transformation induced by . Let and be two X-states with parameters and , defined according to Eq. (1). Then, some straightforward (however tedious) computation gives
where, for brevity, we have defined , ,
Moreover, the following conservation laws can be easily established from the invariance of the trace and the determinant of a matrix unitarily conjugated:
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 X-state. 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
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
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 .
In this case, the result of the minimization is , from which follows that (44) reduces to the always-true inequalities and .
In a nutshell, algebraically, inequalities (42) can be satisfied by choosing and as follows:555We need not to consider the case since every with such property is automatically separable [cf. Fig. 1(c)].
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 X-state :
Let be the set of parameters specifying an arbitrary entangled X-state , 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 X-state whose parameters and are given by equation (45).
If , make
and such that if . Otherwise (), is such that
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 well-defined and actually implement the desired transformation.
4.2 Continuity of Entanglement
Now that we have established that an arbitrary two-qubit entangled X-state can be disentangled by conjugation with , we shall consider the entanglement dynamics of that state evolving under the action of the strongly continuous one-parameter unitary group , where and is the (X-formed) 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 X-form 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 X-state . 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:
4.2.1 Continuity of Concurrence
For definiteness, suppose that the initial entangled X-state 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 positive666Note 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 non-positivity 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
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
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 X-state of concurrence . For example, if is such that , then the appropriate can be found by solving the following transcendental equation for :
Numerically, this can be efficiently solved.
4.2.2 Continuity of Negativity
Once again, we first consider an initial entangled X-state 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 negative777In this case, the third term in the minimization of Eq. (56) is an increasing function in , reaching its minimum value when . The non-negativity 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
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 X-state such that , one can derive the following negativity formula, also obviously continuous in :
We conclude by noting that Eqs. (57) and (58) allow the determination of the value of that leads to a X-state of negativity . If has , for example, then can be obtained by numerically solving the following transcendental equation for :
4.2.3 Continuity of Relative Entropy of Entanglement
Since a closed-form for the relative entropy of entanglement of a two-qubit X-state is still unknown (cf. A), we resort to a powerful continuity property of the relative entropy of entanglement888An 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 two-qubit density matrices and such that , the following Fannes-type inequality holds: