Real numerical shadow and generalized B-splines
Restricted numerical shadow of an operator of order is a probability distribution supported on the numerical range restricted to a certain subset of the set of all pure states – normalized, one–dimensional vectors in . Its value at point equals to the probability that the inner product is equal to , where stands for a random complex vector from the set distributed according to the natural measure on this set, induced by the unitarily invariant Fubini–Study measure. For a Hermitian operator of order we derive an explicit formula for its shadow restricted to real states, , show relation of this density to the Dirichlet distribution and demonstrate that it forms a generalization of the –spline. Furthermore, for operators acting on a space with tensor product structure, , we analyze the shadow restricted to the set of maximally entangled states and derive distributions for operators of order .
keywords:numerical range, probability measures, numerical shadow, B–splines
Msc:47A12, 60B05, 81P16, 33C05, 51M15
Consider a complex square matrix or order . Its standard numerical range is defined as the following subset of the complex plane,
where denotes a normalized complex vector in . Due to the Toeplitz–Hausdorff theorem this set is convex, while for a Hermitian it forms an interval belonging to the real axis – see e.g. Dav (); GR1977 (); Gu04 ().
Among numerous generalizations of this notion we will be concerned with the restricted numerical range,
where forms a certain subset of the set of normalized complex vectors of size . For instance, one can choose as the set of all real vectors, and analyze the ’real shadow’ of , denoted by . For an operator acting on a composed space, one studies also numerical range restricted to tensor product states, , and the range restricted to maximally entangled states rrange2010 (); rrange2011 (). It is worth to emphasize a crucial difference with respect to the standard notion: the resticted numerical range needs not to be convex.
In order to define a probability measure supported on numerical range of it is sufficient to consider the uniform measure on the sphere and the measure induced by the map Shd1 (); GS2012 (). Alternatively, one considers the space of quantum states – equivalence classes of normalized vectors in , which differ by a complex phase, , and works with the Haar measure invariant under the action of the unitary group shadow1 (). For any matrix one defines in this way a probability measure supported on and called numerical shadow Shd1 () or numerical measure GS2012 (). The former name is inspired by the fact that for a normal matrix this measure can be interpreted as a shadow of an uniformly covered dimensional regular simplex projected on a plane shadow1 (); gutkin2013joint (). In a similar fashion, one can consider numerical shadow of matrices over the quaternion field, defined as the pushforward measure of the uniform measure on the sphere .
Even though several papers on numerical shadow were published during the last five years Shd1 (); GS2012 (); shadow1 (), the idea to associate with the numerical range a probability measure is much older: as described in a recent review by Holbrook Hol14 () it goes back to the early papers of Davis Dav ().
Another variant of the numerical shadow of can be obtained by taking random points from the subset of the set of pure states. The corresponding probability measure , called restricted numerical shadow shadow3 (), is by definition supported in restricted numerical range . More generally, one may take an arbitrary probability measure on the set of all pure states (or on the hypersphere ) and study the measure induced in the numerical range of .
Let denote a Hermitian matrix of size , so its numerical range is an interval on the real axis. The probability distribution generated by the map , where is a random point on the unit sphere equipped with the unitary-invariant surface measure, is then equal to the shadow of .
It will be convenient to introduce the set containing density matrices of order , i.e. Hermitian positive definite operators, normalized by the trace condition, with . The set is convex as it can be considered as the convex hull of the set of projectors on the pure states of dimension – see e.g. bengtsson2006geometry (). Specifying a measure on the set of density matrices allows us to propose a more general definition of numerical shadow.
For a given matrix and a probability measure on the space of density matrices of order we define the numerical shadow of matrix with respect to as function on complex numbers
The standard numerical shadow, defined in shadow1 () and denoted by , fulfills the above definition with supported on a pure states invariant to unitary transformations. In fact all restricted numerical shadow presented in shadow3 () can be written in the above form.
The main goal of this work is to describe restricted numerical shadow for several relevant cases. For any symmetric real matrix we derive its real numerical shadow. To this end we use Dirichlet distributions, the properties of which are reviewed in Sec. 2. We demonstrate that in this case the real shadow has the same distribution as a linear combination of components of a random vector generated by the Dirichlet distribution.
In Sec. 3 we briefly discuss –splines, which correspond to complex shadows of Hermitian matrices, and show their link to generalized Dirichlet distributions Complex and real shadows of illustrative normal matrices are compared in Sec. 4, in which some results are obtained for the case of Hermitian matrices.
Main result of this work — Theorem 14, which characterizes the real shadow of real symmetric matrices, is presented in Sec. 5. Continuity of the shadow at knots is discussed in Sec. 6, while formulae for the shadow with respect of real maximally entangled states for any matrix of size are derived in Sec. 7.
2 The Dirichlet Distribution
Let in denotes the unit simplex of –point probability distributions,
The Dirichlet distribution is a measure on the simplex parameterized by a vector of real numbers ,
Note that the choice gives the flat, Lebesgue measure on the simplex, while the case corresponds to the statistical distribution – see e.g. bengtsson2006geometry ().
Set . For let and . It follows from the Dirichlet integral that
where denotes the Pochhammer product. It satisfies an important asymptotic relationship: as in the complex half-plane .
Consider the random vector corresponding to choosing a point in according to with components with . We select an arbitrary vector of real numbers ordered increasingly, , and will be concerned with the probability distribution of their weighted average,
The distribution of random variable will be denoted as
In the case some values of are repeated some formulae have to be modified. It is clear that . There is a moment generating function for .
Let denote the cumulative distribution function of , that is,
This Lemma, proof of which is provided in B , implicitly gives an expression for the moments, , because .
The mean and the variance .
In the case of it is straightforward to find the density for ,
where denotes the beta function.
Let us now return to the general case of an arbitrary dimension and consider the behavior of for . Here we require no repeated values in . This involves the intersection of the hyperplane with , which is a convex polytope whose faces are subsets of for , , and . Note that is equivalent to . The vertices of this polytope come from the intersection of hyperplanes drawn from with . Introduce the unit basis vectors () with components . There are two types of vertices:
For any given some of these vertices are in and some are not. Suppose for some with , then exactly when since the condition is , that is, . Similarly exactly when , that is, and . Thus the number of vertices is . Each vertex is an extreme point: to show this one exhibits a linear function which vanishes at the point and is positive at all other vertices. For the function accomplishes this, and for use (this applies to the vertices contained in , by inspection).
Suppose then is given by the integral of over a convex polytope with vertices lying between parallel hyperplanes. The vertices of the polytope are analytic functions of and so is analytic in and in the parameters (in broad terms, decompose the integral as a sum of iterated -fold integrals each of which has an analytic expression).
It is straightforward to find the following infinite series expression for the complementary distribution function for – see B . We assumed here that , but other repetitions are allowed.
For near (and ) behaves like and the density behaves like .
The Dirichlet distribution has a special additivity property which allows us to restrict to the situation where the ’s are mutually distinct. If two numbers ’s are equal, say then is has the same distribution as (see 6). In other words if then the distribution is the same as
When each is an integer ( there is a finite sum expression for the density in terms of piecewise polynomials (splines). This theorem is from (Shd1, , p.2070). For simplicity we state the result for the case . Let , with the convention that for and for .
Suppose , for each , then
is the partial fraction decomposition (the term with is omitted if ).
Observe that each term is itself a probability density supported on . (In the present context is the number of distinct values, differing from the statement in Shd1 () where each and some values are repeated.) The Theorem shows that the density is a piecewise polynomial of degree with discontinuities (in some order derivative) at the points . Because of this spline interpretation the quantities will henceforth be called knots.
3 B–splines and their generalization
The Dirichlet distribution is closely related to the notion of an –dimensional –spline introduced by de Boor dB1976 ().
Let be a non-trivial simplex in . On we define the B–spline of order from by
A measure version of the above definition is more useful, thus we define the normalized measure on
A non-trivial simplex can be written as where is a regular simplex and is an invertible matrix of order . The simplex is possibly translated if 0 is not a vertex of . We will use the notation
Instead of calculating the volume with respect to the flat Lebesgue measure one can use instead the Dirichlet measure with parameters instead. In this way one obtains a generalized notion of -splines.
Therefore, the distribution can be viewed as a generalized -spline. If we take any invertible matrix with the first row given by , then a generalized -spline is equal to the distribution
4 Shadows of Hermitian and real symmetric matrices
Among several probability measures defined on the set of density matrices it is convenient to distinguish a class of measures induced by the partial trace performed on a pure state on the extended system.
We say, that a density matrix of size is distributed according to the induced measure bengtsson2006geometry () if
where being a uniformly distributed, normalized random vector in and the operation of partial trace is defined for product matrices as and extended to general case by linearity. In the case of we obtain a measure on pure states and in the case of we get a Hilbert-Schmidt measure bengtsson2006geometry ().
In paper Shd1 () we showed that the (complex) shadow of a Hermitian matrix with eigenvalues (counted with multiplicity) has the distribution
From Corollary 4 the mean is and the variance is .
This follows directly from the definition of a partial trace and the additivity property of a Dirichlet distribution. As a special case we obtain, that the mixed numerical shadow with respect to flat Hilbert Schmidt distribution is given by . We can calculate mean and variance for mixed numerical shadow induced by , using Corollary 4 we have and the variance is .
Let us now return to the main subject of the paper - the shadow of a matrix of order with respect to the set of real pure states in . It is briefly called the real shadow shadow3 (), and for a real symmetric matrix it can be related to the Dirichlet distribution,
where denotes the eigenvalues of counted with multiplicity. The mean value is and the variance is .
In a close analogy to the complex case, one can also consider the shadow with respect to real mixed states obtained by an induced measure . For any real symmetric matrix this leads to the distribution , with all indices equal to . Thus the real shadow is obtained for , as required.
Henceforth we will concentrate on the distributions with pairwise distinct knots . For integer we have the interpretation as the shadow of the Hermitian matrix ( summands) where the eigenvalues of are , or the mixed numerical shadow induced by the measure . We consider the distribution as an analytic function of , for , and will find more information by extrapolating from the known formulas for integer . Start with finding explicit values of the coefficients in Theorem 8.
Suppose consists of pairwise distinct nonzero real numbers and then
The proof is provided in B.
Thus the formula in Theorem 8 is completely symmetric in , independent of the ordering. This is an ingredient in the derivation of the differential equation satisfied by the density.
Consider the case of a symmetric matrix of size . Then the density for its real shadow has an expression in terms of a -hypergeometric function which solves a certain second-order differential equation. Suppose , so formulas (14) and (10) (change to ) read for
and the series converges for any .
Let us now return to the generalized case of an arbitrary matrix order , for which condition holds. Basing on computational experiments we are in position to formulate a generalization valid for small and integers . Set and define a differential operator of order (with ) by
The differential equation has regular singular points at the knots. We will show that the density function of satisfies this equation at all , first for integer then for . The idea is to verify the equation for the interval by use of Proposition 6 and then use the symmetry property of Theorem 8 to extend the result to all intervals .
For arbitrary and
Proof. Expand the sum as
by the Chu-Vandermonde sum.
Since we intend to work with polynomials in we set . Start the verification by replacing by and apply the resulting operator to (for and generic (leaving open the possibility of being a noninteger and . At times we use the Pochhammer symbol with a negative index: for let , so that . Note that , so the result follows
In the special case we obtain since . For the result is zero. The calculations used the reversal and the Lemma with replaced by and , respectively. The upper limit of summation is because . To proceed further we introduce:
Next where denotes the elementary symmetric polynomial of degree in , . Thus
Up to a multiplicative constant, not relevant in this homogeneous equation, the density in is given by
The series terminates at . Define symmetric polynomials in (note the reversal to ) by
convergent for , then
It is required to show that the -sum vanishes for each . At there is only one term and . Replace by its definition (30) and simplify
The denominator does not vanish because . Taking out the factors depending only on the -sum becomes
There is a recurrence relation for ; the elementary symmetric function of degree in equals for . The generating function of is where
Extract the coefficient of in the following equation