Restricted numerical shadow and geometry of quantum entanglement
The restricted numerical range of an operator acting on a -dimensional Hilbert space is defined as a set of all possible expectation values of this operator among pure states which belong to a certain subset of the of set of pure quantum states of dimension . One considers for instance the set of real states, or in the case of composite spaces, the set of product states and the set of maximally entangled states. Combining the operator theory with a probabilistic approach we introduce the restricted numerical shadow of – a normalized probability distribution on the complex plane supported in . Its value at point is equal to the probability that the expectation value is equal to , where represents a random quantum state in subset distributed according to the natural measure on this set, induced by the unitarily invariant Fubini–Study measure. Studying restricted shadows of operators of a fixed size we analyse the geometry of sets of separable and maximally entangled states of the composite quantum system. Investigating trajectories formed by evolving quantum states projected into the plane of the shadow we study the dynamics of quantum entanglement. A similar analysis extended for operators on dimensional Hilbert space allows us to investigate the structure of the orbits of and quantum states of a three–qubit system.
pacs:03.67.Ac, 02.10.-v, 02.30.Tb
Recent studies on quantum entanglement, a crucial resource in the theory of quantum information processing, contributed a lot to our understanding of this deeply non-classical phenomenon (see e.g.  and references therein). In particular, some progress has been achieved in elucidating the geometry of quantum entanglement [2, 3, 4, 5, 7, 6], but several questions concerning this topic remain still open [8, 9, 10]. It was also suggested that a geometric approach is useful to classify and quantify quantum entanglement [8, 11, 12].
The phenomenon of quantum entanglement, non-classical correlations between individual subsystems, may arise in composite physical systems. Consider then the simplest composite system, which consists of two parts and can be described in a Hilbert space with a tensor product structure, . Any product pure state, , is called separable, while all other pure states are called entangled. The set of all separable pure states has the structure of the Cartesian product of the complex projective spaces , .
Among entangled pure states of a bipartite system one distinguishes the set of maximally entangled states, such that the partial trace of the corresponding projector is proportional to the identity matrix. The set contains the generalized Bell state, , and all states obtained from it by a local unitary transformation, . The set of maximally entangled states is thus equivalent to . The structure of the set and other sets of locally equivalent entangled states of a bipartite system was studied in .
In this work we propose to analyse the geometry of the set of entangled and separable quantum states using the algebraic concepts of the numerical range and numerical shadow of an operator. For any operator acting on the complex Hilbert space one defines its numerical range [14, 15] as a subset of the complex plane which contains expectation values of among arbitrary normalized pure states,
The standard notion of numerical range, often used in the theory of quantum information [19, 20, 21], can be generalized in several ways [22, 23, 24]. For instance, for an operator acting on a composite Hilbert space one defines the product numerical range  (also called local numerical range ) and a more general class of numerical ranges restricted to a specific class of states ,
Here denotes a selected subset of the set of pure quantum states of a given size . For instance, one may consider the set of all real states, or, in the case of composite spaces, the set of complex product states or the set of real maximally entangled states.
Here denotes the unique unitarily invariant measure on the set of –dimensional quantum pure states, also called Fubini-Study (FS) measure. In other words the shadow of at a given point characterizes the likelihood that the expectation of among a random pure states is equal to . Sometimes it is convenient to treat the numerical shadow as a probability measure , on a complex plane. For any measurable set it reads .
If the operator is normal, , its shadow can be interpreted as a projection of the set of classical states – the –dimensional regular simplex of probability distributions – into a two-plane . In the more general case of a non–normal its shadow can be associated with a probability distribution obtained by projecting the set of quantum pure states of size into a plane. Thus choosing a matrix to be analysed one fixes the relative position of the set and determines the direction, along which it is projection on the plane. Hence investigating all possible shadows of various operators of a given size one gathers information about the structure of the set .
where is the selected subset of . In particular, choosing the appropriate subsets of the set of pure states we define the numerical shadow restricted to real states or –coherent states. In the case of a composite Hilbert space one defines the shadow with respect to separable states or maximally entangled states. The restrictions can be combined so one can consider the shadow restricted e.g. to real separable states, or in the case of operators acting on dimensional space, which describes a three–qubit system, one may study the shadow with respect to real states or complex –states. It will be convenient to use simplified terms, so for brevity we will slightly abuse the notation and write about ’separable shadow’, ’real entangled shadow’ or ’complex GHZ shadow’ of a given operator. Using the notion of probability measure we will denote this restricted by .
On one hand, for a given matrix one may study its restricted numerical shadows determined by a given set of quantum states. Alternatively, investigating numerical shadows of various matrices of a fixed dimension with respect to a concrete subset one may analyse its geometry. In this work we study in this way the structure of the set of real, separable and maximally entangled quantum states for a two and three–qubit system.
This paper is organized as follows. In section 2 some basic properties of the numerical range and the numerical shadow are reviewed. In section 3 we discuss numerical shadow restricted to real states. In the case of operators acting on the Hilbert space with a tensor product structure, , one defines classes of separable and maximally entangled states. Numerical ranges restricted to these sets are analysed in section 4 and 5 respectively. In section 6 we show that investigations of trajectories formed by evolving quantum states projected into the plane of the shadow contributes to the understanding the dynamics of quantum entanglement. In section 7 we discuss the simplest case of composite system consisting of three parts and described in the Hilbert space . In the set of pure states acting on this space one defines two classes of entangled states called and . For any operator acting on it is then natural to introduce the shadow restricted to states locally equivalent to or , and these are investigated in section 6.
2 Standard numerical shadow and the geometry of quantum states
2.1 Classical and quantum states
Elements of a Hilbert space are used as basic objects of the quantum theory. A physical system with distinguishable states can be described by an element of the complex Hilbert space . It is assumed that such a pure quantum state is normalized, , so it belongs to the hypersphere of dimension . One identifies any two states, which differ only by a global phase, . The set of all pure quantum states acting on is therefore equivalent to the complex projective space, (see e.g. ).
Any convex combination of projectors onto pure states forms a mixed quantum state, with and . Any such state , also called a density operator, is Hermitian, positive and normalized by the trace condition, Tr. Thus the set of all density matrices of order forms a convex set of real dimensions. The projectors corresponding to the extremal points of the set of density operators form the set of pure quantum states, . In the one qubit case, , the set of pure quantum states forms the Bloch sphere, , which in this case is equivalent to the boundary of the Bloch ball . For a larger dimension the –dimensional set of pure states forms only a zero–measure subset of the dimensional boundary of the set .
Note that the definitions of the set of pure and mixed quantum states are unitarily invariant, so they can be formulated without specifying any basis in the Hilbert space. On the other hand, among quantum states, one distinguishes also the set of classical states, which are formed by the density matrices diagonal in a certain basis. Hence any classical state is represented by a normalized probability vector, such that and . The set of classical states forms thus the probability simplex . Each of corners of the simplex represents a classical pure state, and their convex hull is equivalent to . In the one qubit case, , the set of classical states forms an interval which joins two poles of the Bloch sphere and traverses across the interior of the Bloch ball of one–qubit mixed quantum states.
2.2 Numerical range and the set of quantum states
The definition (1) of the numerical range of a matrix of order specifies as the set of expectation values for normalized pure states . In fact the compact set in the complex plane can be considered as a projection of the set of quantum states onto a two plane. The following facts were recently established in .
Consider the set of classical states of size , which forms the regular simplex in . Then for each normal matrix of dimension there exists an affine rank projection of the set whose image is congruent to the numerical range of the matrix . Conversely for each rank projection there exists a normal matrix whose numerical range is congruent to image of under projection .
Let denote the set of quantum states of dimension embedded in with respect to Euclidean geometry induced by Hilbert-Schmidt distance. Then for each (arbitrary) matrix of dimension there exists an affine rank projection of the set whose image is congruent to the numerical range of the matrix . Conversely for each rank projection there exists a matrix whose numerical range is congruent to image of under projection .
Thus by fixing the dimension and selecting various operators of this order and analysing their numerical ranges we may gather information about the projections of the set of quantum states. For instance, for the projections of the Bloch sphere form ellipses (which could form circles or may reduce an intervals), and so the numerical range of any operator of size may be the result. Similarly the numerical range of operators of size can be viewed as projections of the set of single qutrit quantum states, while the numerical range for matrices of order can be associated with projections of the set [26, 28].
2.3 Measure on the space of quantum states and the numerical shadow
The Haar measure on the unitary group induces on the set of quantum pure states the unitarily invariant Fubini-Study measure . In the simplest case of this measure corresponds to the uniform distribution of points on the Bloch sphere .
For any operator of size we may compute its expectation value for random pure state chosen with respect to measure . In this way for any operator we define its numerical shadow – a probability distribution (3) on the complex plane. Note that by construction the numerical shadow is supported on the numerical range . Furthermore, the numerical shadow is unitarily invariant: .
For any normal matrix , which commutes with its adjoint, the numerical shadow covers the numerical range with the probability corresponding to a projection of a regular simplex of classical states embedded in ) onto a plane. In general, for a non–normal operator acting on , its shadow covers the numerical range with the probability corresponding to an orthogonal projection of the complex projective manifold onto a plane.
In this work we will analyse the numerical range restricted  to a certain subsets of pure quantum states and the corresponding restricted numerical shadows, with probability density determined by random states distributed on the subset according to the measure induced by the Fubini–Study measure on .
2.4 Standard numerical shadow for a diagonal matrix
Consider a diagonal matrix of size , namely . Let be an arbitrary fixed pure state in . Then the entire set of random pure states can be obtained as , where is a random unitary matrix distributed according the Haar measure. Thus the expansion coefficients of the state read .
The numerical shadow of the operator is defined as the density distribution of random numbers , where is a random state defined by the random unitary matrix . Therefore
Note that the vector of coefficients with belongs to the dimensional simplex . So the complex random variable can be treated as a scalar product, where the complex vector is given by the diagonal of the operator , while the probability vector is random. Its distribution inside the simplex depends on the distribution in the space of unitary matrices . In cases of interest for us this distribution belongs to the class of Dirichlet distributions parametrized by a real number ,
In particular, we are interested in two situations.
Real numerical shadow, generated by the Haar measure on the group of orthogonal matrices. Then the vector is distributed with respect to the statistical measure, , i.e. the Dirichlet measure with the Dirichlet parameter .
In both cases the shadow covers the entire numerical range of , equal to the convex hull of the spectrum with a density determined by the appropriate Dirichlet distribution. The case of real numerical shadow, i.e. the shadow with respect to the real pure states is treated in more detail in the subsequent section.
3 Numerical shadow with respect to real states
Quantum states belonging to a complex Hilbert space form a standard tool of the quantum theory. However, in some cases it is instructive to restrict attention to real states only. On one hand the set of the real states is easier to analyse than the full set of complex states of larger dimensionality. For instance, the phenomenon of quantum entanglement can be studied for the case of real states of a four–level systems, sometimes referred to as a pair of rebits (i.e. real bits) .
On the other hand is some physical applications it is easier to use real orthogonal rotation matrices to construct elements of the entire set of real density matrices. Therefore in this section we will study the shadow of an operator with respect to the set of quantum states with real coefficients. Such an investigation allows us to improve understanding of the structure of the set of real mixed states and its natural subsets.
3.1 Real shadow of operators of size
Consider the subset of pure quantum states of size , that can be represented by real expansion coefficients in a given basis. This set forms a dimensional real projective space .
Substituting into the definition (2) of a restricted numerical range [19, 21] the set as the subset of we arrive at the numerical range restricted to real states. In general the restricted numerical range needs not to be convex or simply connected. Similarly, restricting the integration in (4) to the set of real quantum states we obtain the shadow of the operator with respect to real states. For brevity we will also use a shorter expression, the real shadow of .
In the case of the set of real states forms the real projective space equivalent to a circle, . Thus the real shadow of a generic, non-normal matrix of size two forms a singular probability distribution supported on an elliptic curve in the complex plane, and not inside its interior.
The real shadows presented in Fig. 1 are obtained for illustrative operators of size
These probability distributions can be thus interpreted as shadows of real projective spaces , and on a plane. In the case 1(b) and 1(d) the shadow is supported on a real line, so we plot the corresponding probability distribution . In the other cases 1(a), 1(c) and 1(e) the real shadow is supported on the complex plane, so the density is encoded in the grey scale. The circle drawn by red dotted line represents the image of the sphere of dimension , in which the set of quantum states can be inscribed. The blue dotted line represents the boundary of the standard numerical range of an operator. The restricted shadow is supported in this set or its subset.
The numerical shadow carries also some information about the higher rank numerical range [30, 31] of an operator. For instance, the darker area of the shadow corresponding to a larger probability  allows one to recognize in Fig. 2 the numerical range of rank , written of selected unitary matrices. In the case of shown in Fig. 2 \subreffig:real-S3D4 it is equal to a single point at which the two diagonals of the quadrangle cross, while for the case shown in panel \subreffig:real-S3D5 the numerical range of rank two is represented by the inner pentagon located inside the numerical range in this case numerical shadow appears to be uniform in the set . In the case shown in the panel \subreffig:real-S3D7 one can see darker areas between segments connecting every other eigenvalue.
3.2 A general approach to real shadow
Given , the real shadow is taken to mean the distribution of when is a unit vector randomly chosen from the uniform distribution on .
If itself is real, we have so that the distribution of is the same as that of the symmetric . Since the uniform distribution on is invariant under orthogonal transformations, we are free to diagonalize this symmetric matrix. In other words, the real shadow of is that of , where are the (real) eigenvalues of . Thus we need the distribution of
It is known that uniform on implies that has a Dirichlet–(1/2) distribution on the simplex , ie. its density is proportional to .
As an example, let us consider the Jordan nilpotent For the real shadow of , we first note that it is the same as that of . Rescaling for convenience, we compute the real shadow of . The density of is 0 outside , satisfies , and, in view of the Dirichlet–(1/2) distribution on , is proportional to
for . In particular, . In  the case is treated in terms of hypergeometric functions and we can compute
For a general , one may write with , and by diagonalizing and we may understand the distributions of and separately, using the techniques above. Usually the joint distribution will be obscure, but we may have some hope of understanding it in certain cases. If, for example, is normal and presented in diagonal form, it may be possible to see how the horizontal and vertical distributions knit together.
Fig. 3 shows various shadows of when is a unitary (in diagonal form). The sort of 1D shadow density seen in the left panel of Fig. 3(a) seems typical; notably, the real density of a Hermitian is constant between the two middle eigenvalues. Presumably, the vertical density has a similar form. Fig. 4 presents two examples of real shadow of generic unitary matrices of dimension five. Support of the shadow does not coincide with numerical range of those matrices because a generic unitary matrix can not be diagonalized using only orthogonal matrices.
3.3 Real shadows and their moments
Since methods based on moments of the shadow distribution were so effective in the complex case, we may try to mimic them for the real shadow. It is not hard to adapt the method outlined at the beginning of the proof of Proposition 5.1 from  to show that, for
unless each component of is even, and that
Here we use the shifted factorial notation: , with the convention ; also denotes .
In principle, we can use Eq. (14) to evaluate the moments of the real shadow density of
In the complex case we were able to progress beyond this point, obtaining such effective relations for the moments as the determinant relation (see: Eq. (10) in ).
As a test case, we may try to use Eq. (15) to find the moments of , the real shadow density of , discussed in the previous example. We have
and finally we obtain that
It can be shown, that for even we have
To see this we start with equation Eq. (16). Changing the index of summation from to shows that the sum equals times itself, hence equals zero for odd . Now suppose that . The sum can be written in hypergeometric form
Kummer’s summation formula (see [38, p. 10]) implies (later we set )
Now we set and use the terminating form of Gauss’s sum (the Chu-Vandermonde formula) to obtain
Thus, setting we obtain, that the integral equals
We use the variance of a complex random variable to give quantitative insight into Figs. 2-4. The calculations leading to the following formulae are based on the values of integrals of monomials over the real or complex unit spheres. Suppose is a normal matrix, written in the form where is unitary and is diagonal with entries . Thus the columns of are eigenvectors of . Let denote the random variable and let . If is a random vector in the unit sphere in or then (where denotes the expectation). Denote the complex variance , a measure of (2-dimensional) spread of . For the complex shadow one finds
For the real shadow and when can be diagonalized by a real orthogonal matrix, that is, is orthogonal () then
which is larger than the variance of the complex shadow (see Fig. 3(b) ). In the general real case is not orthogonal, then set for . Thus is a symmetric unistochastic matrix, and its eigenvalues all lie in . By straightforward computations we find
a positive quadratic form in . The eigenvalues of lie in , and all equal when is real orthogonal.
As example consider the unitary Fourier matrix whose entries are primitive roots of unity : . Then if otherwise . The eigenvalues of are with multiplicity and with multiplicity . The matrix also contains information about approximating the eigenvectors of by points on : indeed . In particular if some is an extreme point in the convex hull of then is in the real numerical range of if and only if . This is illustrated by Fig. 4.
4 Product numerical shadow
Consider the shadow restricted to the set of pure product states. More formally, we assume that , apply the definition (4) and take , where , while and these states are normalized. The set of separable (product) pure states has the structure of the Cartesian product .
The simplest case of corresponds to the two–qubit case. The set of separable (product) pure states has then a form of the Cartesian product of two spheres . In other words this set forms a Segre embedding, .
One may also consider the shadow with respect to real separable states. In the two-qubit case this shadow corresponds to a projection of the product of real projective spaces , which forms a torus . Such a structure can be recognized on some plots shown in Fig. 5 \subreffig:real_separable_SA43– \subreffig:real_separable_S3B41.
The matrices used to obtain projections are following
One can note that the numerical ranges presented in Fig. 5 have a particular structure. Matrix is permutation equivalent to a simple sum of two matrices so its numerical range forms a convex hull of two ellipses one with focal points and the other . As these ellipses do intersect their convex hull contains four interval segments. A similar situation occurs for the matrix where two ellipses – one with focal points and the other – do not intersect and the convex hull has only two flat lines. For the matrix the numerical range is a convex hull of an ellipse with focal points at eigenvalues and the line segment between two eigenvalues . A similar situation arises for the matrix .
Consider a particular case of an operator with the tensor product structure, . Then its product numerical range is equal to the Minkowski product of numerical ranges - for more information see .
Is in this case the shadow of restricted to product states can be expressed by the numerical shadows of both operators,
Here and denote the probability measures related to the numerical shadows of and respectively, while is a measurable subset of the complex plane.
4.1 Mean and variance for the separable numerical shadow
In the case of separable shadow of matrix it is possible to obtain explicit expressions for the mean and the variance. We have
Formulas involve the partial traces and , follows from more general fact given in Appendix B. These expressions imply directly the following result.
4.2 Separable numerical shadow for a diagonal matrix
Consider a diagonal matrix defined on a composite Hilbert space, of dimension . Its diagonal elements forming the spectrum can be also represented by two indices, with and .
Let be an arbitrary fixed pure product state in , so the set of random separable pure states can be obtained as , where and are independent random unitary matrices distributed according to the Haar measure. Thus the expansion coefficients of the product state read .
The separable numerical shadow of the diagonal operator is defined as the density distribution of random numbers , where is a separable random state defined by random unitaries and . In this case one has
where is a real probability vector of size . It can be considered as a tensor product of two probability vectors and , since its components read with with and .
Thus the separable numerical shadow of a diagonal operator can be considered as a projection of the Cartesian product of classical probability simplices, . In the simplest case of the Cartesian product of two intervals (1-simplices) forms a square, which lives inside the tetrahedron of the –dimensional probability vectors.
As in the previous case we can distinguish two probability measures in the space of unitary matrices. They lead to
complex separable shadow, generated by the Haar measure on and , for which both probability vectors and are distributed uniformly with respect to the Lebesgue measure on the simplices and , respectively.
Real separable shadow, generated by the Haar measure on the orthogonal groups and , which lead to the statistical measure (Dirichlet measure with ) in both simplices.
Note that in these case the separable shadow of is supported on its product numerical range , which in general forms a proper subset of the convex hull of the spectrum. The product structure of the classical probability vector in (21), generalized for a multiple tensor product structure, is consistent with the parametrization of the product numerical range described in Prop. 12 in .
5 Maximally entangled numerical shadow
Consider an operator acting on a Hilbert space with a tensor product structure, . For simplicity let us assume that the dimensions of both subspaces are equal to so the total dimension reads . Among all pure states of the system one distinguishes the set of maximally entangled states. It contains the states equivalent with respect to a local unitary operation to the generalized Bell state, . Thus the set of maximally entangled states has the structure of , where is the discrete permutation group , [8, Ch. 15]. Choosing for the set in (4) we define the shadow of an operator with respect to the maximally entangled states. The corresponding probability measure will be denoted as .
5.1 Two qubit case:
In the simplest case of Hilbert space the set of maximally entangled states has the structure — see . Hence the numerical shadow of an operator of order four with respect to the complex maximally entangled states can be considered as a projection of the real projective space on the plane — see the shadow for some illustrative operators presented in Fig. 6 \subreffig:complex_entangled_SA44–\subreffig:complex_entangled_SA47.
If one considers a further restriction and studies the shadow with respect to real maximally entangled states, the result can be interpreted as an image of the space Observe that the illustrative shadows obtained in this case and presented in Fig. 6 \subreffig:real_entangled_SA44–\subreffig:real_entangled_SA47 show indeed projections of a circle onto the complex plane.
In the special case of a diagonal operator of size four its shadow with respect to complex maximally entangled states can be identified with a standard shadow of a reduced operator of size . This fact is formulated in the following proposition, proved in Appendix A.
Consider an diagonal matrix of order four, which acts on a composite Hilbert space