A geometric approach to the distribution of quantum states in bipartite physical systems
Abstract
Any set of pure states living in an given Hilbert space possesses a natural and unique metric –the Haar measure– on the group of unitary matrices. However, there is no specific measure induced on the set of eigenvalues of any density matrix . Therefore, a general approach to the global properties of mixed states depends on the specific metric defined on . In the present work we shall employ a simple measure on that has the advantage of possessing a clear geometric visualization whenever discussing how arbitrary states are distributed according to some measure of mixedness. The degree of mixture will be that of the participation ratio and the concomitant maximum eigenvalue . The cases studied will be the qubitqubit system and the qubitqutrit system, whereas some discussion will be made on higherdimensional bipartite cases in both the domain and the domain.
pacs:
03.65.Ud; 03.67.Bg; 03.67.Mn; 89.70.CfI Introduction
The amount of entanglement and the purity of quantum states of composite systems exhibit a dualistic relationship. As the degree of mixture increases, quantum states tend to have a smaller amount of entanglement. In the case of twoqubits systems, states with a large enough degree of mixture are always separable ZHS98 (). A detailed knowledge of the relation between the degree of mixture and the amount of entanglement is essential in order to understand the limitations that mixture imposes on quantum information processes such as quantum teleportation or quantum computing. To study the relationship between entanglement and mixture we need quantitative measures for these two quantities. The entanglement of formation provides a natural quantitative measure of entanglement with a clear physical motivation. As for mixedness, there are several measures of mixture that can be useful within the present context. The von Neumann measure
(1) 
is important because of its relationship with the thermodynamic entropy. On the other hand, the so called participation ratio,
(2) 
is particularly convenient for calculations ZHS98 (); MJWK01 (). Another measure for mixedness can be found in the maximum eigenvalue of a density matrix , which is in turn a monotonically increasing function for the Renyi entropy .
Given a particular way to explore the space of both pure and mixed states in bipartite systems, it is possible to provide a clear physical geometric insight into the problem of how states distribute according to their degree of mixture, which is the main subject of the present work.
This paper is organized as follows: in Section II we introduce the nature of the measures defined on the set of eigenvalues and unitary matrices. Section III analyzes 2x2 systems and how their concomitant states are distributed according to and . Section IV provides a further analytic approach for 2x3 and additional bipartite systems. In Section V we discuss the implications of the nonuniqueness of a general measure for mixed states and the corresponding geometric implications into our problem. Finally, some conclusions are drawn in Section VI.
Ii Measures on the set of eigenvalues and unitary matrices
In order to perform a survey of the properties of arbitrary (pure and mixed) states of the concomitant statespace , it is necessary to introduce an appropriate measure on this space. Such a measure is needed to compute volumes within , as well as to determine what is to be understood by a uniform distribution of states on . The natural measure that we are going to adopt here was first considered in Refs. ZHS98 (); Z99 (). An arbitrary (pure or mixed) state of a quantum system described by an dimensional Hilbert space can always be expressed as the product of three matrices,
(3) 
Here is an unitary matrix and is an diagonal matrix whose diagonal elements are , with , and . The group of unitary matrices is endowed with a unique, uniform measure: the Haar measure PZK98 (). On the other hand, the simplex , consisting of all the real uples appearing in (3), is a subset of a dimensional hyperplane of . Consequently, the standard normalized Lebesgue measure on provides a natural measure for . The aforementioned measures on and lead then to a natural measure on the set of all the states of our quantum system ZHS98 (); Z99 (); PZK98 (), namely,
(4) 
All our present considerations are based on the assumption that the uniform distribution of states of a quantum system is the one determined by the measure (4). Thus, in our numerical computations, we are going to randomly generate states according to the measure (4). The quantities computed with a Monte Carlo procedure have an associated error which is on the type , where is the number of generated states, is the value corresponding to the Student distribution with degrees of freedom, computed with a certain desired accuracy , and is the usual computed standard deviation. Therefore, if we seek a result with an error say less than units, we have to generate a number of points around 10 or 100 million. If not stated explicitly, from now on all quantities computed are exact up to the last digit.
The applications that have appeared so far in quantum information theory, in the form of dense coding, teleportation, quantum cryptography and specially in algorithms for quantum computing (quantum error correction codes for instance), deal with finite numbers of qubits. A quantum gate which acts upon these qubits or even the evolution of that system is represented by a unitary matrix , with being the dimension of the associated Hilbert space . The state describing a system of qubits is given by a hermitian, positivesemidefinite () matrix, with unit trace. In view of these facts, it is natural to think that an interest has appeared in the quantification of certain properties of these systems, most of the times in the form of the characterization of a certain state , described by matrices of finite size. Natural applications arise when one tries to simulate certain processes through random matrices, whose probability distribution ought to be described accordingly.
ii.1 The Haar measure
As stated before, in the space of pure states, with , there is a natural candidate measure, induced by the Haar measure on the group of unitary matrices. In mathematical analysis, the Haar measure Haar33 () is known to assign an “invariant volume” to what is known as subsets of locally compact topological groups. In origin, the main objective was to construct a measure invariant under the action of a topological group Mehta90 (). Here we present the formal definition Conway90 (): given a locally compact topological group (multiplication is the group operation), consider a algebra generated by all compact subsets of . If is an element of and is a set in , then the set also belongs to . A measure on will be letfinvariant if for all and . Such an invariant measure is the Haar measure on (it happens to be both left and right invariant). In other words Haarsimetria (), the Haar measure defines the unique invariant integration measure for Lie groups. It implies that a volume element d is identified by defining the integral of a function over as , being left and right invariant
(5) 
The invariance of the integral follows from the concomitant invariance of the volume element d. It is plain, then, that once d is fixed at a given point, say the unit element , we can move performing a left or right translation. Suppose that the map defines the action of a left translation. We have , with being the coordinates in the vicinity of . Assume, also, that dd defines the volume element spanned by the differentials d, d, …, d at point . It follows then that the volume element at point is given by ddd, where is the Jacobian of the previous map evaluated at the unit element : . In a right or left translation, both dd and are multiplied by the same Jacobian determinant, preserving invariance of d. The Lie groups also allow an invariant metric and d is just the volume element dd.
We do not gain much physical insight with these definitions of the Haar measure and its invariance unless we identify with the group of unitary matrices , the element with a unitary matrix and with subsets of the group of unitary matrices , so that given a reference state and a unitary matrix , we can associate a state to . Physically what is required is a probability measure invariant under unitary changes of basis in the space of pure states, that is,
(6) 
These requirements can only be met by the Haar measure, which is rotationally invariant.
Iii Analytical approach for 2x2 systems. Generation of states
The twoqubits case () is the simplest quantum mechanical system that exhibits the feature of quantum entanglement. The relationship between entanglement and mixedness has been described intensively in the literature. One given aspect is that as we increase the degree of mixture, as measured by the so called participation ratio Tr[], 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 MJWK01 () (the maximum entangled mixed states MEMS). These states have been recently reported to be achieved in the laboratory MEMSexp () using pairs of entangled photons. Thus for practical or purely theoretical purposes, it may happen to be relevant to generate mixed states of twoqubits with a given participation ratio . It may represent an excellent tool in the simulation of algorithms in a given quantum circuit: as the input pure states go through the quantum gates, they interact with the environment, so that they become mixed with some . This degree of mixture , which varies with the number of iterations, can be used as a probe for the evolution of the degradation of the entanglement present between any two qubits in the circuit. Different evolutions of the degree of mixture on the output would shed some light on the optimal architecture of the circuit that has to perform a given algorithm.
Here we describe a numerical recipe to randomly generate twoqubit states, according to a definite measure, and with a given, fixed value of . Suppose that the states are generated according to the product measure (4), where is the Haar measure on the group of unitary matrices and the Lebesgue measure on provides a reasonable measure for the simplex of eigenvalues of . 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 (4). We shall identify the simplex with a regular tetrahedron of side length 1, in , centred at the origin. Let stand for the vector positions of the tetrahedron’s vertices. The tetrahedron is oriented in such a way that the vector points towards the positive axis and the vector is contained in the semiplane corresponding to positive values. The positions of the tetrahedron’s vertices correspond to the vectors
(7) 
The mapping connecting the points of the simplex (with coordinates ) with the points within tetrahedron is given by the equations
(8)  
(9) 
The degree of mixture is characterized by the quantity . This quantity is related to the distance to the centre of the tetrahedron by
(10) 
Thus, the states with a given degree of mixture lie on the surface of a sphere of radius concentric with the tetrahedron . To choose a given is tantamount to define a given radious of the sphere. There exist three different possible regions (see Fig. 1):

region I: (), where is the radius of a sphere tangent to the faces of the tetrahedron . In this case the sphere lies completely within the tetrahedron . Therefore we only need to generate at random points over its surface. The cartesian coordinates for the sphere are given by
(11) Denoting rand_u() a random number uniformly distributed between 0 an 1, the random numbers rand_u() and rand_u() (its probability distribution being ) define an arbitrary state on the surface inside . The angle is defined between the centre of the tetrahedron (the origin) and the vector , and any point aligned with the origin. Substitution of in (8) provides us with the eigenvalues of , with the desired as prescribed by the relationship (10). With the subsequent application of the unitary matrices we obtain a random state distributed according to the usual measure .

region II: (), where denotes the radius of a sphere which is tangent to the sides of the tetrahedron . 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 as before, provided we reject those ones with negative weights .

region III: (), where is the radius of a sphere passing through the vertices of . The generation of states is a bit more involved in this case. Again rand_u(), but the available angles now range from to . It can be shown that results from solving the equation . Thus, , with rand_u(). Some states may be unacceptable () still, but the vast majority are accepted.
Combining these three previous regions, we are able to generate arbitrary mixed states endowed with a given participation ratio .
iii.1 Rdomain
In this case the degree of mixture is characterized by the quantity . This quantity is related to the distance to the centre of the tetrahedron by
(12) 
Thus, the states with a given degree of mixture lie on the surface of a sphere of radius concentric with the tetrahedron .
The volume associated with states endowed with a value of lying within a small interval is clearly associated with the volume of the subset of points in whose distances to the centre of are between and , with . Let denote the sphere of radius concentric with . The volume is then proportional to the area of the part of which lies within . In order to compute the aforementioned area, it is convenient to separately consider three different ranges for the radius .
Let us first consider the range of values , where is the radius of a sphere tangent to the faces of the tetrahedron . In this case the sphere lies completely within the tetrahedron . Thus, the area we are interested in is just the area of the sphere,
(13) 
We now consider a second range of values of the radius, , where denotes the radius of a sphere which is tangent to the sides of the tetrahedron . In this case, the area of the portion of which lies within is
(14) 
Finally, we consider the range of values , where is the radius of a sphere passing through the vertices of . This cas is, by far, the most intricate one, where many methods borrowed from spherical trigonometry are employed. In this case the area of the part of the sphere lying within is
(15) 
where
(16)  
(17) 
The quantities appearing in the right hand sides of the above expressions are defined by
(18) 
and
(19)  
(20) 
Using the relation between and the participation rate ,
(21) 
we analytically obtained the probability of finding a quantum state with a participation rate ,
(22) 
iii.2 domain
Coming back to twoqubits, the quantity is not appropriately suited to discuss the limit . However, does exhibit a nice behaviour when . Indeed, we have
(23) 
where
(24) 
is the maximum eigenvalue of the density matrix . Hence, in the limit , the entropies (when properly behaving) depend only on the largest eigenvalue of the density matrix. For example, in the limit , the Rényi entropy reduces to
(25) 
It is worth realizing that the largest eigenvalue itself constitutes a legitimate measure of mixture. Its extreme values correspond to (i) pure states () and (ii) the identity matrix (). It is also interesting to mention that, for states diagonal in the Bell basis, the entanglement of formation is completely determined by (This is not the case, however, for general states of twoqubits systems).
In terms of the geometric representation of the simplex , the set of states with a given value of their maximum eigenvalue is represented by the tetrahedron determined by the four planes
(26) 
The four vertices of this tetrahedron are given by the intersection points of each one of the four possible triplets of planes that can be selected among the four alluded to planes.
For the accessible states with a given degree of mixture are on the surface of a small tetrahedron concentric with the tetrahedron . We are going to characterize each tetrahedron (representing those states with a given value of ) by the distance between (i) the common centre of and and (ii) each vertex of . The volume associated with states with a value of belonging to a given interval is proportional to the area of the portion of lying within .
Following a similar line of reasoning as the one pursued in the case , we consider three ranges of values for . The first range of values is given by . The particular value corresponds to a tetrahedron whose vertices are located at the centres of the faces of . Within the aforementioned range of values, is lies completely within . Consequently, coincides with the area of ,
(27) 
The second range of values corresponds to . The area of the part of lying within is now
(28) 
Finally, the third range of values we are going to consider is . In this case we have
(29) 
In a similar way as in the case, the above expressions for lead to the analytical form of the probability (density) of finding a twoqubits state with a given value of its greatest eigenvalue,
(30) 
Remarkably enough, as tends to infinity all discontinuities in the derivative of disappear. In the domain the distribution is completely smooth, as opposed to the domain.
Iv Analytical approach for 2x3 systems and higher systems
iv.1 Rdomain
Previously, we obtained the distribution vs. for twoqubits (), under the assumption that they are distributed according to measure (4). Through a useful analogy, we have mapped the problem into a geometrical one in regarding interior and common sections of two geometrical bodies. In the previous case we saw that the main difficulty lies in the third region, where the region of the growing sphere inside the tetrahedron is not described by a spherical triangle. The extension to higher dimensions, however, requires a thorough account of the geometrical tools required, but still it is in principle possible. So, one can find the distribution of states according to basically by computing the surface area of a growing ball of radius in dimensions (sphere) that remains inside an outer regular polytope (tetrahedron) of unit length, excluding the common regions. A dimensional sphere can be parameterized in cartesian coordinates
(31) 
with the domains for and . The definition of the polytope then is required. This problem is not trivial at all, because new geometrical situations appear in the intersection of these two bodies. In point of fact there are intermediate regimes between and appearing at integer values of (recall the previous twoqubits case), where a change in the growth of interior hypersurfaces occurs (at the values ). In any case we can always generate random states in arbitrary dimensions and compute the corresponding distributions. This is done in Fig. 2 for several cases. The relation (21) is generalized to dimensions in the form
(32) 
The distribution can be obtained analytically
(33) 
which has been numerically checked. The particular form of for arbitrary is difficult to obtain, but nevertheless one can obtain quantitative results for asymptotic values of . It may be interesting to know the position of the maximum of or the mean value , which turns to be Zyckasimp () for states generated according to (4). There is the so called Borel lemma Borel () in discrete mathematics that asserts that (translated to our problem) when you grow a ball inside , from the moment that it swallows, say, 1/2 of the volume of it, then the area outside drops very quickly with further grow. So the maximum intersection with the sphere should be approximately for the radious where the volume of the ball equals that of the polytope . The usual formulas for the volumes of ()dimensional spheres and regular unit polytopes are and , respectively.
It is then that we can assume that the position of such that is maximal. Substituting in (32), and after some algebra, we obtain the beautiful result
(34) 
In other words, for large .
We must emphasize that this type of distributions are “degenerated” in some cases, that is, different systems may present identical distributions (for instance, there is nothing different from this perspective between and systems). We do not know to what extend these distributions are physically representative of such cases, as far as entanglement is concerned. What is certain is that all states with possess a positive partial transpose. In point of fact, they are indeed separable, as shown in balls (). We merely mean by this that a state close enough to the maximally mixed state is always separable. In other words, states lying on spheres with radius are always separable.
iv.2 domain
When regarding the maximum eigenvalue as a proper degree of mixture one is able to find a geometrical picture analogue to the one of the growing sphere. In that case a nested inverted tetrahedron grows inside the outer tetrahedron representing the simplex of eigenvalues . The generalization to higher bipartite systems is similar to the case, but far much easier to implement mathematically. As in that case, we have a high degree of symmetry in the problem. The advantage is that one does not deal with curved figures but perfectly flat and sharp surfaces instead. This fact makes the general problem more approachable.
We have seen that the problem of finding how the states of a bipartite quantum mechanical system are distributed according to their degree of mixedness can be translated to the realm of discrete mathematics. If we consider our measure of mixedness to be the maximum eigenvalue of the density matrix and the dimension of our problem to be , we compute the distribution of states in arbitrary dimensions by letting an inner regular polytope to grow inside an outer unit length polytope , the vertices of the former pointing towards the centre of the faces of the latter. In fact, it can be shown that the radius of the maximum hypersphere that can be inscribed inside the inner polytope is directly related to .
By computing the surface area of strictly inside , we basically find the desired probability (density) of finding a state with maximum eigenvalue in dimensions.
To fix ideas, it will prove useful first to define the vertices of and . In fact it is essential, because we need to deal with elements of cartesian geometry in dimensions. This vectors are given as
(35) 
with being the distance from the center to any vertex of this regular polytope of unit length. One can easily check that , as required. This particular choice for the position of the vertices of this simplex is such that it simplifies going from one dimension to the next by adding a new azimuthal axis each time. This vectors comply with the relations
(36)  
(37) 
where the last equation is the general form of (26).
Once we have a well defined , to know the coordinates of is straightforward. In fact, is the reciprocication (see Sommer ()) of . This means that the coordinates of are obtained by reversing the sign of the ones of , multiplied by a suitable factor (which can be shown to be , with defined as the length between the centre of to the centre of any of its faces, which in turn points towards the vertices of ). Thus, we can relate with through a general (26)relation , such that .
Several distributions are obtained numerically by generating random states according to (4) in Fig. 3. It becomes apparent that as grows, the distributions are biased towards , in absolute agreement with the result (34).
As in the case, is distributed into regions separated at fixed values of . The general recipe for obtaining is tedious and long, but some nice general results are obtained. The distributions for the ranges a) and b) are general and read
(38)  
(39) 
respectively, where is introduced for convenience.
For the qubitqutrit system, we have . Defining and , in addition to the previous regions (38) we obtain
(40)  
(41)  
(42) 
for , and , respectively. From the previous formulas one can infer a general induction procedure. Analytical results are in excellent agreement with numerical generations.
V Nonuniqueness of a general measure for mixed states. Geometric implications
In Refs. ZHS98 (); Z99 (), a basic question regarding a natural measure for the set of mixed states was debated. As described in Secs. (7.1) and (9.1), it is know, the set of all states can be regarded as the cartesian product , where stands for the family of all complete sets of ortonormal projectors , ( being the identity matrix), and is the set of all real tuples , with and . It is universally accepted to assume the Haar measure to be the one defined over , because of its rotationallyinvariant properties. But when it turns to discuss an appropriate measure over the simplex , some controversy arises. In all previous considerations here, we have regarded the Lebesgue measure as being the “natural” one. But one must mention that Slater has argued sla1 (); sla2 () that, in analogy to the classical use of the volume element of the Fisher information metric as Jeffreys’ prior jef () in Bayesian theory, a natural measure on the quantum states would be the volume element of the Bures metric. The problem lies on the fact that there is no unique probability distribution defined over the simplex of eigenvalues of mixed states . In point of fact, the debate was motivated by the fact that the volume occupied by separable twoqubits states was found in ZHS98 () to be greater than () using the measure , something which is surprising.
One such probability distribution that is suitable for general considerations is the Dirichlet distribution Z99 ()
(43) 
with being a real parameter and the normalization constant. This is a particular case of the more general Dirichlet distribution. The concomitant probability density for variables with parameters is defined by
(44) 
with , and , and . Clearly, distribution (44) generalizes (43). This distribution admits a clear interpretation. As known, the multinomial distribution provides a probability of choosing a given collection of items out of a set of items with repetitions, the probabilities being . These probabilities are the parameters of the multinomial distribution. The Dirichlet distribution is the conjugate prior of the parameters of the multinomial distribution.
A new measure then can be defined as , where denotes the simplex of eigenvalues distributed according to (43) (The Haar measure remains untouched). Thus, one clearly recovers the Lebesgue measure for (uniform distribution), and Slater’s argumentation reduces to take in (43). For one obtains a singular distribution concentrated on the pure states only, while for , the distribution peaks on the maximally mixed state . We will see shortly that changing the continuous parameter indeed modificates the average purity (as expressed in terms of ) of the generated mixed states.
In what follows we numerically generate mixed states whose eigenvalues are distributed following (43). This is done is order to tackle the dependence of relevant quantities on the parameter . Let us consider the way mixed states are distributed according to . We focus our attention on the twoqubits instance, but similar studies can be extended to arbitrary bipartite dimensions. As shown in Fig. 4, the distributions vs. are shown for (from left to right in this order) while Fig. 5 shows analogous distributions for the maximum eigenvalue for (from right to left). Notice the different shapes. We can no longer attribute a geometrical description to except for . In Z99 () for was first derived. Here we can provide different distributions for arbitrary values.
A way to devise a certain range of reasonable values is to study the average induced for every distribution. This is performed in Fig. 6. The average value and are plotted versus . (solid line) can only be computed numerically, but luckily (dashed line) is obtained in analytical fashion for all N
(46)  
The fact that matches exact results validates all our present generations. The actual value is slightly larger than for all values of , but both of them coincide for low and high values of the parameter . It is obvious from Fig. 6 that we cannot choose distributions that depart considerably from the uniform one , because in that case we induce probability distributions that favor high or low already.
Perhaps the best way is to go straight to the question that originated the controversy on the measures: what is the dependency of the a priori probability of finding a twoqubits mixed state being separable? In Fig. 7 we depict vs. for states complying with PPT (lower curve) and those which violate the entropic inequalities (upper curve). It seems reasonable to assume that a permissible range of distributions belong to the interval , within which remains around the reference point .
However, in view of the previous outcomes we believe that the results obtained considering the uniform distribution for the simplex (the one that allow us to exploit a simple geometric analogy) remains the most natural choice possible, independent of any form that one may adopt for a generic probability distribution.
Vi Conclusions
We have introduced a geometric picture to obtain the probability of finding quantum states of twoqubits with a given degree of mixture (as measured by an appropriate function of ) is analytically found for and . In the latter case, the entropies become functions of the statistical operator’s largest eigenvalue . In point of fact, itself constitutes a legitimate measure of mixture. During the derivation of the probability (density) distributions of finding a bipartite mixed state in arbitrary dimensions with a given degree of mixture, we saw that it is more convenient to use instead of . Finally, we have derived explicitly the distribution vs. for the physical meaningful case of a qubitqutrit system ().
Acknowledgements
J. Batle acknowledges partial support from the Physics Department, UIB. J. Batle acknowledges fruitful discussions with J. Rosselló, Maria del Mar Batle and Regina Batle.
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