Locality and classicality: role of entropic inequalities

Abstract

The use of the so-called entropic inequalities is revisited in the light of new quantum correlation measures, specially nonlocality. We introduce the concept of classicality as the nonviolation of these classical inequalities by quantum states of several multiqubit systems and compare it with the nonviolation of Bell inequalities, that is, locality. We explore—numerically and analytically—the relationship between several other quantum measures and discover the deep connection existing between them. The results are surprising due to the fact that these measures are very different in their nature and application. The cases for $n=2,3,4$ qubits and a generalization to systems with arbitrary number of qubits are studied here when discriminated according to their degree of mixture.

Appendices

Appendix 1: Definition and calculation of quantal measures

For the quantum discord QD, a minimization takes place for two parameters only, whereas nonlocality $\mathrm{MABK}_{N}^{\max }$ requires an exploration among their corresponding unit vectors defining the observers’ settings. In any case, we have performed a twofold search employing (1) an amoeba optimization procedure, where the optimal value is obtained at the risk of falling into a local minimum, and (2) the well-known simulated annealing approach [43]. The advantage of this computational ’duplicity’ is that we can be confident regarding the final results reached. Indeed, the second recipe contains a mechanism that allows for local searches that eventually can escape local optima.

1.1 Quantum discord

Quantum discord [24] constitutes a quantitative measure of the “nonclassicality” of bipartite correlations as given by the discrepancy between the quantum counterparts of two classically equivalent expressions for the mutual information. More precisely, quantum discord is defined as the difference between two ways of expressing (quantum mechanically) such an important entropic quantifier. If $S$ stands for the von Neumann entropy, for a bipartite state $A-B$ of density matrix $\rho$ and reduced (“marginals”) ones $\rho _{A}-\rho _{B}$, the quantum mutual information (QMI) $M_{q}$ reads [24]

$$\begin{aligned} M_{q}(\rho )= S(\rho _{A}) + S(\rho _{B}) - S(\rho ), \end{aligned}$$

(21)

which is to be compared to its associated classical notion $M_\mathrm{class}(\rho )$, that is expressed using conditional entropies. If a complete projective measurement $\Pi _j^B$ is performed on B and (1) $p_i$ stands for $Tr_{AB}\,\Pi _i^B\,\rho$ and (2) $\rho _{A|\vert \Pi _i^B}$ for $[\Pi _i^B\,\rho \,\Pi _i^B/p_i]$, then our conditional entropy becomes

$$\begin{aligned} S(A \vert \,\{ \Pi _j^B \}) =\sum _i\,p_i\, S(\rho _{A|\vert \Pi _i^B}), \end{aligned}$$

(22)

so that $M_\mathrm{class}(\rho )$ adopts the appearance

$$\begin{aligned} M_\mathrm{class}(\rho )_{\{ \Pi _j^B \}} = S(\rho _A)- S(A \vert \,\{ \Pi _j^B \}). \end{aligned}$$

(23)

Now, if we minimize over all possible $\Pi _j^B$ the difference $M_q(\rho )-M_\mathrm{class}(\rho )_{\{ \Pi _j^B \}}$, we obtain the quantum discord $\Delta$, that quantifies nonclassical correlations in a quantum system, including those not captured by entanglement. The most general parameterization of the corresponding local measurements that can be implemented on one qubit (let us call it B) is of the form $\{ \Pi _B^{0^{\prime }}=I_A \otimes |0^{\prime }\rangle \langle 0^{\prime }|, \Pi _B^{1^{\prime }}=I_A \otimes |1^{\prime }\rangle \langle 1^{\prime }|\}$. More specifically, we have

$$\begin{aligned} |0^{\prime }\rangle\leftarrow & {} \cos \alpha ' |0\rangle + e^{i\beta '}\sin \alpha '|1\rangle \\|1^{\prime }\rangle\leftarrow & {} e^{-i\beta '}\sin \alpha '|0\rangle - \cos \alpha ' |1\rangle , \end{aligned}$$

(24)

which is obviously a unitary transformation—rotation in the Bloch sphere defined by angles $(\alpha ',\beta ')$—for the B basis $\{|0\rangle ,|1\rangle \}$ in the range $\alpha ' \in [0,\pi ]$ and $\beta ' \in [0,2\pi )$. The previous computation of the QD has to be carried out numerically, unless the two-qubit states belong to the class of the so-called X-states, where QD is analytic [44]. One notes then that only states with zero $\Delta$ may exhibit strictly classical correlations.

1.2 Geometric quantum discord

Despite increasing evidences for relevance of the quantum discord (Qd) in describing nonclassical resources in information processing tasks, there was until quite recently no straightforward criterion to verify the presence of discord in a given quantum state. Since its evaluation involves an optimization procedure and analytical results are known only in a few cases, such criteria become clearly desirable. Recently, Datta advanced a condition for nullity of quantum discord [45], and progress was also achieved in [25] by introducing an interesting geometric measure of quantum discord (GQD). Let $\chi$ be a generic $\Delta =0-$state. The GQD measure is then given by

$$\begin{aligned} D(\rho )= \mathrm{Min}_{\chi }\left[ ||\rho -\chi ||^2\right] , \end{aligned}$$

(25)

where the minimum is over the set of zero-discord states $\chi$. We deal then with the square of Hilbert–Schmidt norm of Hermitian operators, $||\rho -\chi ||^2= Tr[(\rho -\chi )^2]$. Dakic et al. [25, 28] show how to evaluate this quantity for an arbitrary two-qubit state. Moreover, they demonstrate the their geometric distance contains all relevant information associated with the notion of quantum discord. This was a remarkable feat given that, despite robust evidence for the pertinence of the Qd-notion, its evaluation involves optimization procedures, with analytic results being known only in a few cases.

Now, given the general form of an arbitrary two-qubit state in the Bloch representation

$$\begin{aligned} 4\rho= & {} \mathcal {I} \otimes \mathcal {I} + \sum \limits _{u=1}^{3} x_u \sigma _u \otimes \mathcal {I} + \sum \limits _{u=1}^{3} y_u \mathcal {I} \otimes \sigma _i \nonumber \\&+\sum \limits _{u,v=1}^{3} T_{uv} \sigma _u \otimes \sigma _v, \end{aligned}$$

(26)

with $x_u=Tr(\rho (\sigma _u \otimes \mathcal {I}))$, $y_u=Tr(\rho (\mathcal {I} \otimes \sigma _u))$, and $T_{uv}=Tr(\rho (\sigma _u \otimes \sigma _v))$, it is found in Ref. [25] that a necessary and sufficient criterion for witnessing nonzero quantum discord is given by the rank of the correlation matrix

$$\begin{aligned} \frac{1}{4} \left( \begin{array}{c@{\quad }c@{\quad }c@{\quad }c} 1 &{} y_1 &{} y_2 &{} y_3\\ x_1 &{} T_{11} &{} T_{12} &{} T_{13}\\ x_2 &{} T_{21} &{} T_{22} &{} T_{23}\\ x_3 &{} T_{31} &{} T_{32} &{} T_{33} \end{array} \right) , \end{aligned}$$

(27)

that is, a state $\rho$ of the form (26) exhibits finite quantum discord iff the matrix (27) has a rank greater that two. It is seen that the geometric measure (25) is of the final form [25]

$$\begin{aligned} D(\rho )= & {} \frac{1}{4} \bigg ( ||\mathbf{x}||^2 + || T ||^2 - \lambda _{\max } \bigg )\nonumber \\= & {} \frac{1}{R}-\frac{1}{4}-\frac{1}{4}\bigg ( ||\mathbf{y}||^2 + \lambda _{\max } \bigg ), \end{aligned}$$

(28)

where $||\mathbf{x}||^2=\sum _u x_u^2$, and $\lambda _{\max }$ is the maximum eigenvalue of the matrix $(x_1,x_2,x_3)^t (x_1,x_2,x_3) + TT^t$. Here, the superscript $t$ denotes either vector or matrix transposition. The second expression emphasizes the natural dependence of $D$ on the participation ratio $R=1/Tr(\rho ^2)$. Notice that this measure is intimately connected with the quantities appearing in (27).

1.3 Violation of MABK inequalities

Most of our knowledge on Bell inequalities and their quantum mechanical violation is based on the CHSH inequality [34]. With two dichotomic observables per party, it is the simplest [46] (up to local symmetries) nontrivial Bell inequality for the bipartite case with binary inputs and outcomes. Let $A_1$ and $A_2$ be two possible measurements on A side whose outcomes are $a_j\in \lbrace -1,+1\rbrace$, and similarly for the B side. Mathematically, it can be shown that, following LVM, $|{\mathcal {B}}_\mathrm{CHSH}^\mathrm{LVM}(\lambda )|=|a_1b_1+a_1b_2+a_2b_1-a_2b_2|\le 2$. Since $a_1$($b_1$) and $a_2$($b_2$) cannot be measured simultaneously, instead one estimates after randomly chosen measurements the average value ${\mathcal {B}}_\mathrm{CHSH}^\mathrm{LVM} \equiv \sum _{\lambda } \mathcal {B}_\mathrm{CHSH}^\mathrm{LVM}(\lambda ) \mu (\lambda )= E(A_1,B_1)+E(A_1,B_2)+E(A_2,B_1)-E(A_2,B_2)$, where $E(\cdot )$ represents the expectation value. Therefore, the CHSH inequality reduces to

$$\begin{aligned} |\mathcal {B}_\mathrm{CHSH}^\mathrm{LVM}| \le 2. \end{aligned}$$

(29)

Quantum mechanically, since we are dealing with qubits, these observables reduce to $\mathbf{A_j}(\mathbf{B_j})=\mathbf{a_j}(\mathbf{b_j}) \cdot \sigma$, where $\mathbf{a_j}(\mathbf{b_j})$ are unit vectors in $\mathbb {R}^3$ and $\sigma =(\sigma _x,\sigma _y,\sigma _z)$ are the usual Pauli matrices. Therefore, the quantal prediction for (29) reduces to the expectation value of the operator $\mathcal {B}_\mathrm{CHSH}$

$$\begin{aligned} \mathbf{A_1}\otimes \mathbf{B_1} + \mathbf{A_1}\otimes \mathbf{B_2} + \mathbf{A_2}\otimes \mathbf{B_1} - \mathbf{A_2}\otimes \mathbf{B_2}. \end{aligned}$$

(30)

Tsirelson showed [47–49] that CHSH inequality (29) is maximally violated by a multiplicative factor $\sqrt{2}$ (Tsirelson’s bound) on the basis of quantum mechanics. In fact, it is true that $|Tr(\rho _{AB}\mathcal {B}_\mathrm{CHSH})|\le 2\sqrt{2}$ for all observables $\mathbf{A_1}$, $\mathbf{A_2}$, $\mathbf{B_1}$, $\mathbf{B_2}$, and all states $\rho _{AB}$. Increasing the size of Hilbert spaces on either A and B sides would not give any advantage in the violation of the CHSH inequalities. In general, it is not known how to calculate the best such bound for an arbitrary Bell inequality, although several techniques have been developed [50].

A good witness of useful correlations is, in many cases, the violation of a Bell inequality by a quantum state. Therefore, we shall consider the optimization of the violation of the CHSH inequality over the observer’s settings as a definitive measure for both signaling and quantifying nonlocality in two-qubit systems.

We are going to determine which is the maximum expectation value of the CHSH operator (30) that a two-qubit mixed state $\rho$ with some degree of mixedness, in this case given by the so-called participation ratio $R=1/Tr(\rho ^2)$, may have. Notice that no assumption is needed regarding the state being diagonal or not in the Bell basis. In order to solve the concomitant variational problem (and bearing in mind that $B_\mathrm{CHSH}=Tr(\rho \mathcal {B}_\mathrm{CHSH})$), let us first find the state that extremizes Tr($\rho ^2$) under the constraints associated with a given value of $B_\mathrm{CHSH}$, and the normalization of $\rho$. This variational problem can be cast as

$$\begin{aligned} \delta \big [ Tr(\rho ^2) + \beta Tr(\rho \mathcal {B}_\mathrm{CHSH}) - \alpha Tr(\rho ) \big ]=0, \end{aligned}$$

(31)

where $\alpha$ and $\beta$ are appropriate Lagrange multipliers. After some algebra, we arrive at the result

$$\begin{aligned} B_\mathrm{CHSH}^{\max }=\sqrt{Tr[\mathcal {B}_\mathrm{CHSH}^2]} \cdot \sqrt{ \frac{4-R}{4R} } = 4\cdot \sqrt{ \frac{4-R}{4R} }. \end{aligned}$$

(32)

This result is valid for the range $R\in [2,4]$. In the region $R\in [1,2]$, we obtain

$$\begin{aligned} B_\mathrm{CHSH}^{\max } = \sqrt{\frac{8}{R}}. \end{aligned}$$

(33)

We shall explore nonlocality in the three-qubit case through the violation of the Mermin inequality [51]. This inequality was conceived originally in order to detect genuine three-party quantum correlations impossible to reproduce via LVMs. The Mermin inequality reads as $Tr(\rho \mathcal {B}_\mathrm{Mermin}) \le 2$, where $\mathcal {B}_\mathrm{Mermin}$ is the Mermin operator

$$\begin{aligned} \mathcal {B}_\mathrm{Mermin}=B_{a_{1}a_{2}a_{3}} - B_{a_{1}b_{2}b_{3}} - B_{b_{1}a_{2}b_{3}} - B_{b_{1}b_{2}a_{3}}, \end{aligned}$$

(34)

with $B_{uvw} \equiv \mathbf{u} \cdot \sigma \otimes \mathbf{v} \cdot \sigma \otimes \mathbf{w} \cdot \sigma$ with $\sigma =(\sigma _x,\sigma _y,\sigma _z)$ being the usual Pauli matrices, and $\mathbf{a_j}$ and $\mathbf{b_j}$ unit vectors in $\mathbb {R}^3$. Notice that the Mermin inequality is maximally violated by Greenberger–Horne–Zeilinger (GHZ) states. As in the bipartite case, we shall define the following quantity

$$\begin{aligned} Mermin^{\max } \equiv \max _{\mathbf{a_j},\mathbf{b_j}}\,\,Tr (\rho \mathcal {B}_\mathrm{Mermin}) \end{aligned}$$

(35)

as a measure for the nonlocality of the state $\rho$. While in the bipartite the CHSH inequality was the strongest possible one, this is not the case for three qubits. The Mermin inequality is not the only existing Bell inequality for three qubits, but it constitutes a simple generalization of the CHSH one to the tripartite case. Therefore, it will suffice to use this particular inequality to illustrate the basic results of the present work.

The first Bell inequality for four qubits was derived by Mermin, Ardehali, Belinskii, and Klyshko [52, 53]. It constitutes of four parties with two dichotomic outcomes each, being maximum for the generalized GHZ state $(|0000\rangle + |1111\rangle )/\sqrt{2}$. The Mermin–Ardehali–Belinskii–Klyshko (MABK) inequality reads as $Tr(\rho \mathcal {B}_\mathrm{MABK}) \le 4$, where $\mathcal {B}_\mathrm{MABK}$ is the MABK operator

$$\begin{aligned} \begin{aligned}&B_{1111}-B_{1112} -B_{1121}-B_{1211}-B_{2111}-B_{1122}-B_{1212}\\&\quad -B_{2112}-B_{1221}-B_{2121}-B_{2211}+B_{2222}+B_{2221}\\&\quad +B_{2212}+B_{2122}+B_{1222}, \end{aligned} \end{aligned}$$

(36)

with $B_{uvwx} \equiv \mathbf{u} \cdot \sigma \otimes \mathbf{v} \cdot \sigma \otimes \mathbf{w} \cdot \sigma \otimes \mathbf{x} \cdot \sigma$ with $\sigma =(\sigma _x,\sigma _y,\sigma _z)$ being the usual Pauli matrices. As in previous instances, we shall define the following quantity

$$\begin{aligned} \mathrm{MABK}^{\max } \equiv \max _{\mathbf{a_j},\mathbf{b_j}}\,\,Tr (\rho \mathcal {B}_\mathrm{MABK}) \end{aligned}$$

(37)

as a measure for the nonlocality content for a given state $\rho$ of four qubits. $\mathbf{a_j}$ and $\mathbf{b_j}$ are unit vectors in $\mathbb {R}^3$. MABK inequalities are such that they constitute extensions of previous inequalities with the requirement that generalized GHZ states must maximally violate them. New inequalities for four qubits have appeared recently (see Ref. [37]) that possess some other states required for optimal violation. In the present study, we limit our interest to the MABK inequality, although new ones could be incorporated in order to offer a broader perspective. However, with respect to entanglement, little is know for the quadripartite case, and thus, little comparison can be done.

The optimization is taken over the two observers’ settings $\{\mathbf{a_j},\mathbf{b_j}\}$, which are real unit vectors in $\mathbb {R}^3$. We choose them to be of the form $(\sin \theta _k \cos \phi _k,\sin \theta _k \sin \phi _k,\cos \theta _k)$. With this parameterization, the problem consists in finding the supremum of $Tr(\rho \mathcal {B}_\mathrm{CHSH})$ over the $\{k=1,\ldots ,8\}$ angles of $\{ \mathbf{a_1},\mathbf{b_1},\mathbf{a_2},\mathbf{b_2} \}$ that appear in (30).

Optimization of $\mathrm{Mermin}^{\max }$ (35) (for states diagonal in the Mermin base of maximally correlated states of three qubits $\rho _\mathrm{Mermin}^{(\mathrm diag)}$) is carried out in the same fashion as in the previous bipartite case. Once the observers’ settings $\{\mathbf{a_j},\mathbf{b_j}\}$, which are real unit vectors in $\mathbb {R}^3$, are parameterized in spherical coordinates $(\sin \theta _k \cos \phi _k,\sin \theta _k \sin \phi _k,\cos \theta _k)$, the problem consists in finding the supremum of (35) over the set of $\{k=1,\ldots ,12\}$ possible angles for $\{ \mathbf{a_1},\mathbf{b_1},\mathbf{a_2},\mathbf{b_2},\mathbf{a_3},\mathbf{b_3} \}$ in (34).

Now, in the case of multiqubit systems, one must instead use a generalization of the CHSH inequality to N qubits. This is done in natural fashion by considering an extension of the CHSH or Mermin inequality to the multipartite case. The first Bell inequality (BI) for four qubits was derived by Mermin, Ardehali, Belinskii, and Klyshko [52, 53]. One deals with four parties with two dichotomic outcomes each, the BI being maximum for the generalized GHZ state $(|0000\rangle + |1111\rangle )/\sqrt{2}$. The Mermin–Ardehali–Belinskii–Klyshko (MABK) inequalities are of such nature that they constitute extensions of older inequalities, with the requirement that generalized GHZ states must maximally violate them. To concoct an extension to the multipartite case, we shall introduce a recursive relation that will allow for more parties. This is easily done by considering the operator

$$\begin{aligned} B_{N+1} \propto [(B_1+B_1^{\prime }) \otimes B_N + (B_1-B_1^{\prime }) \otimes B_N^{\prime }] , \end{aligned}$$

(38)

with $B_N$ being the Bell operator for N parties and $B_1=\mathbf{v} \cdot \sigma$, with $\sigma =(\sigma _x,\sigma _y,\sigma _z)$ and $\mathbf{v}$ a real unit vector. The prime on the operator denotes the same expression but with all vectors exchanged. The concomitant maximum value

$$\begin{aligned} \mathrm{MABK}_N^{\max } \equiv \max _{ \mathbf{a_j},\mathbf{b_j} }\,\,Tr (\rho {B_{N}}) \end{aligned}$$

(39)

will serve as a measure for the nonlocality content of a given state $\rho$ of N qubits if $\mathbf{a_j}$ and $\mathbf{b_j}$ are unit vectors in $\mathbb {R}^3$. The nonlocality measure (39) will be maximized by generalized GHZ states, $2^{\frac{N+1}{2}}$ being the corresponding maximum value.

Appendix 2: Generation of two-qubit states with a fixed value of the participation ratio $R$

The two-qubit case ($N=2 \times 2$) is the simplest quantum mechanical system that exhibits the feature of quantum entanglement. One given aspect is that as we increase the degree of mixture, as measured by the so-called participation ratio $R=1/$Tr[$\rho ^2$], the entanglement diminishes (on average). As a matter of fact, if the state is mixed enough, that state will have no entanglement at all. This is fully consistent with the fact that there exists a special class of mixed states which have maximum entanglement for a given $R$ [54] (the maximum entangled mixed states MEMS). These states have been reported to be achieved in the laboratory [55] using pairs of entangled photons. Thus, for practical or purely theoretical purposes, it may happen to be relevant to generate mixed states of two qubits with a given participation ratio $R$.

Here, we describe a numerical recipe to randomly generate two-qubit states, according to a definite measure and with a given, fixed value of $R$. Suppose that the states $\rho$ are generated according to the product measure $\nu = \mu \times {\mathcal {L}}_{N-1}$, where $\mu$ is the Haar measure on the group of unitary matrices $\mathcal {U}(N)$ and the standard normalized Lebesgue measure ${\mathcal {L}}_{N-1}$ on $\mathcal {R}^{N-1}$ provide a reasonable computation of the simplex of eigenvalues of $\rho$. In this case, the numerical procedure we are about to explain owes its efficiency to the following geometrical picture which is valid only if the states are supposed to be distributed according to measure $\nu$). We shall identify the simplex $\Delta$ with a regular tetrahedron of side length 1, in $\mathcal {R}^3$, centered at the origin. Let $\mathbf{r}_i$ stand for the vector positions of the tetrahedron’s vertices. The tetrahedron is oriented in such a way that the vector $\mathbf{r}_4$ points toward the positive $z$-axis and the vector $\mathbf{r_2}$ is contained in the $(x,z)$-semiplane corresponding to positive $x$ values. The positions of the tetrahedron’s vertices correspond to the vectors

$$\begin{aligned} \mathbf{r_1}= & {} \left( -\frac{1}{2\sqrt{3}},-\frac{1}{2},-\frac{1}{4}\sqrt{\frac{2}{3}}\right) \nonumber \\ \mathbf{r_2}= & {} \left( \frac{1}{\sqrt{3}},0,-\frac{1}{4}\sqrt{\frac{2}{3}}\right) \nonumber \\ \mathbf{r_3}= & {} \left( -\frac{1}{2\sqrt{3}},\frac{1}{2},-\frac{1}{4}\sqrt{\frac{2}{3}}\right) \nonumber \\ \mathbf{r_4}= & {} \left( 0,0,\frac{3}{4}\sqrt{\frac{2}{3}}\right) . \end{aligned}$$

(40)

The mapping connecting the points of the simplex $\Delta$ (with coordinates $(\lambda _1,\ldots , \lambda _4)$) with the points $\mathbf r$ within tetrahedron is given by the equations

$$\begin{aligned} \lambda _i \,= & {} \, 2(\mathbf{r}\cdot \mathbf{r}_i ) + \frac{1}{4}\quad i=1, \dots , 4, \\\mathbf{r} \,= & {} \, \sum _{i=1}^4 \lambda _i \mathbf{r}_i \end{aligned}$$

(41)

The degree of mixture is characterized by the quantity $R^{-1} \equiv Tr(\rho ^2) = \sum _i \lambda _i^2$. This quantity is related to the distance $r=\mid \mathbf{r} \mid$ to the center of the tetrahedron $T_{\Delta }$ by

$$\begin{aligned} r^2 = -\frac{1}{8} + \frac{1}{2} \sum _{i=1}^4 \lambda _i^2. \end{aligned}$$

(42)

Thus, the states with a given degree of mixture lie on the surface of a sphere $\Sigma _r$ of radius $r$ concentric with the tetrahedron $T_{\Delta }$. To choose a given $R$ is tantamount to define a given radius of the sphere. There exist three different possible regions (see Fig. 6):

• region I: $r \in [0, h_1]$ ($R \in [4,3]$), where $h_1 \equiv h_c={1 \over 4 }\sqrt{2 \over {3}}$ is the radius of a sphere tangent to the faces of the tetrahedron $T_{\Delta }$. In this case, the sphere $\Sigma _r$ lies completely within the tetrahedron $T_{\Delta }$. Therefore, we only need to generate at random points over its surface. The Cartesian coordinates for the sphere are given by

  $$\begin{aligned} x_1= & {} r \, \sin \theta \, \cos \phi \nonumber \\ x_2= & {} r \, \sin \theta \, \sin \phi \nonumber \\ x_3= & {} r \, \cos \theta , \end{aligned}$$

  (43)

  Denoting rand_u() a random number uniformly distributed between 0 and 1, the random numbers $\phi =2\pi$ rand_u() and $\theta =\arccos (2$ rand_u() $-1)$ (its probability distribution being $P(\theta )=\frac{1}{2}\sin (\theta )$) define an arbitrary state $\rho$ on the surface inside $T_{\Delta }$. The angle $\theta$ is defined between the center of the tetrahedron (the origin) and the vector $\mathbf{r_4}$, and any point aligned with the origin. Substitution of $\mathbf{r}=(x_1,x_2,x_3)$ in (41) provides us with the eigenvalues $\{\lambda _i\}$ of $\rho$, with the desired $R$ as prescribed by the relationship (42). With the subsequent application of the unitary matrices $U$, we obtain a random state $\rho = U D(\Delta ) U^{\dag }$ distributed according to the usual measure $\nu = \mu \times {\mathcal {L}}_{N-1}$.

• region II: $r \in [h_1, h_2]$ ($R \in [3,2]$), where $h_2 \equiv \sqrt{h^{2}_{c}+(\frac{D}{2})^2}={\sqrt{2}\over 4}$ denotes the radius of a sphere which is tangent to the sides of the tetrahedron $T_{\Delta }$. Contrary to the previous case, part of the surface of the sphere lies outside the tetrahedron. This fact means that we are able to still generate the states $\rho$ as before, provided we reject those ones with negative weights $\lambda _i$.

• region III: $r \in [h_2, h_3]$ ($R \in [2,1]$), where $h_3 \equiv \sqrt{h^{2}_{c}+D^2}={\sqrt{6}\over 4}$ is the radius of a sphere passing through the vertices of $T_{\Delta }$. The generation of states is a bit more involved in this case. Again $\phi =2\pi$ rand_u(), but the available angles $\theta$ now range from $\theta _c(r)$ to $\pi$. It can be shown that $w\equiv \cos (\theta _c)$ results from solving the equation $3r^2 w^2 - \sqrt{\frac{3}{2}}r w + \frac{3}{8}-2r^2 = 0$. Thus, $\theta (r)=\arccos (w(r))$, with $w(r)=\cos \theta _c(r) + (1-\cos \theta _c(r))$ rand_u(). Some states may be unacceptable ($\lambda _i<0$) still, but the vast majority are accepted.

Fig. 6

(Color online) Geometric picture of the simplex of eigenvalues (tetrahedron) and a sphere whose radius is related to the degree of mixture $R$. Different growing situations correspond to different regions with definite $R$. See text for details

Combining these three previous regions, we are able to generate arbitrary mixed states $\rho$ endowed with a given participation ratio $R$.