On a multi-dimesional generalization of the notion of orthostochastic and unistochastic matrices

0.15 MB·English

by  Eugene Gutkin

ON A MULTI-DIMESIONAL GENERALIZATION OF
THE NOTION OF ORTHOSTOCHASTIC AND
UNISTOCHASTIC MATRICES
3
1
EUGENE GUTKIN
0
2

n Abstract. Weintroducethenotionsofd-orthostochastic,d-unistochastic,
a and d-qustochastic matrices. These are the particular cases of Fd-
J bistochasticmatricesforF=R,C,H. Theconceptismotivatedby
1
mathematical physics. When d = 1, we recover the orthostochas-
1
tic, unistochastic, and qustochastic matrices respectively. This

] work exposes the basic properties of Fd-bistochastic matrices.
h
p
-
h
t
a
m Contents
[
1. Introduction 1

1 2. Definitions and basic properties 3
v
3. Relationships, symmetries, and dimension count 5
7
3 4. On (n 1)-orthostochasticity of n n matrices 7
5 − ×
5. Concluding discussion 10
2
. References 11
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3
1
:
v 1. Introduction
i
X
A square matrix P with nonnegative entries pj adding up to one
r i
a in every row and column is called a bistochastic matrix. Bistochastic
matrices come up in probability, combinatorics, mathematical physics,
geometry, optimization, etc [1, 16, 4, 12]. Let F stand for R,C or H, let
n N, and let (resp. U(F,n)) denote the set of n n bistochastic
n
ma∈trices (resp.Bthe group of linear isometries of Fn).×The matrix of
squared norms of entries of V U(F,n) is bistochastic, yielding the
squared norm mapping ν : U(F,∈n) .
n
The ranges of ν for F = R,C,→HBare the sets of orthostochastic,
unistochastic, qustochastic n n matrices, denoted by O , ,H re-
n n n
× U
spectively. The obvious inclusions O H are proper
n n n n
⊂ U ⊂ ⊂ B
Date: January 14, 2013.
1
2 EUGENE GUTKIN
if n 4 [5]. There are connections between these matrices and (gen-
≥
eralized) numerical ranges and numerical shadows [21, 15, 13, 17, 6,
9, 10, 11, 20, 14, 18]. They (especially unistochastic matrices) are of
importance in quantum physics [4, 3, 20, 8]. There is a considerable lit-
erature regarding orthostochastic, unistochastic, and qustochastic ma-
trices [19, 5, 2, 8]. However, several basic questions remain open [5, 8].
Themotivationforthepresentworkcomesfrommathematicalphysics.
The entries of a bistochastic matrix P are the probabilities of
n
∈ B
transition between the states of a classical physical system. Transi-
tions amplitudes in quantum physics are complex numbers, and the
transition probabilities are their squared absolute values. The tran-
sitions amplitudes of a quantum system with n basic states form a
unitary n n matrix U. Let P = ν(U) be the matrix of squared
×
norms of the entries in U. Then P is a bistochastic matrix describing
the corresponding classical system. Thus, a bistochastic matrix P is
unistochastic if the classical system described by P can be quantized,
yielding a quantum physical system whose transition amplitudes are
the entries of a unitary matrix U satisfying ν(U) = P.
Suppose now that the basic states of a quantum system have inter-
nal degrees of freedom. Assume, for simplicity, that the number, say
d > 1, of internal degrees of freedom is the same for all states. The
transition amplitudes become vectors vj Cd. This quantum system
i ∈
corresponds to a n n matrix V = [vj] with vector entries. The uni-
tarity condition say×s that the operatori V : Cn Cnd is an isometry.
Let Iso(C,n,d) denote the set of these isometrie→s. For V Iso(C,n,d)
∈
the corresponding bistochastic n n matrix P is given by pj = vj 2.
Thus, our quantum systems corre×spond to V Iso(C,n,d); tihe r||ediu||c-
∈
tion from a quantum to the classical system is given by the squared
norm map ν : Iso(C,n,d) . Its range is the set (C,n,d)
n n
→ B OS ⊂ B
of d-unistochastic matrices. A classical system described by P
n
∈ B
admits a quantization with d internal degrees of freedom if and only if
P (C,n,d).
T∈hOeSsameconstruction, basedonR(resp. H)leadstod-orthostochastic
matrices (R,n,d) (resp. d-qustochastic matrices (H,n,d)). To
streamlineOtShe exposition, we will refer to (F,n,dO) S as the set
n
of (F,n,d)-bistochastic or simply Fd-bistochOaSstic matric⊂esB.
We have structured the exposition as follows. In section 2 we pre-
cisely define the above notions and expose a few basic properties of
vector Fd-bistochastic matrices. Section 3 continues this exposition.
Section 4 is devoted to (n 1)-orthostochasic matrices. It contains our
−
main results, Theorem 1 and Corollary 2, and a few examples. In the
GENERALIZED UNISTOCHASTIC MATRICES 3
concluding section 5 we state a conjecture regarding d-orthostochastic
matrices.
2. Definitions and basic properties
A m n matrix is a collection of mn entries organized in m rows
×
and n columns. In general, we will denote matrices by capital letters,
and denote their entries by the corresponding low case letters with
subscripts and superscripts. For instance, let P = [pj] be a n n
i ×
matrix. The lower (resp. upper) index stands for the column (resp.
row). The entries of a matrix may be numbers, vectors, etc.
We will denote by F any of the fields R or C or H. Recall that a real
n n matrix P = [pj] is bistochastic if pj 0 and n pj = n pj =
× i i ≥ Pj=1 i Pi=1 i
1. We will denote by the set of bistochastic n n matrices.
n
For N N let FN dBenote the vector space of N×-tuples over F. Let
d,n N. ∈ThespaceFnd, decomposedasthedirect sumofncopiesofFd
will b∈e denoted either by n Fd or simply by Fnd if the decomposition
⊕i=1 i
is clear from the context.
Let (Fd,n) be the space of n n matrices with entries in Fd.
M ×
Lemma 1. A matrix V = [vj] (Fd,n) satisfies for all 1 j,k n
i ∈ M ≤ ≤
the equation
n
(1) vi,vi = δ(j,k).
Xh j ki
i=1
if and only if for all 1 j,k n we have
≤ ≤
n
(2) vj,vk = δ(j,k).
Xh i ii
i=1
Proof. Let , and , denote the scalar products in Fn and Fnd
n nd
respectivelyh. Ainy V h i (Fd,n) determines a linear operatorV : Fn
Fnd. Theadjointope∈ratMorV∗ : Fnd Fn satisfiesforallx Fn,y F→nd
→ ∈ ∈
the identity
(3) Vx,y = x,V∗y .
nd n
h i h i
Let Id denote the identity operator on Fk. Equation (1) means that
k
V : Fn Fnd is an isometry, i.e.
→
Vx,Vy = x,y .
nd n
h i h i
By equation (3), this is equivalent to V∗V = Id , i.e., equation (2). (cid:3)
n
4 EUGENE GUTKIN
Let Iso(F,n,d) denote the set of matrices V (Fd,n) satisfying
∈ M
the equivalent equations (1), (2). For V = [vj] Iso(F,n,d) we define
i ∈
the real n n matrix P by
×
(4) pj = vj,vj = vj 2.
i h i ii || i||
By equations (1) and (2), P . Thus, equation (4) defines a map-
n
∈ B
ping
ν : Iso(F,n,d) .
n
→ B
Definition1. Amatrix P is (F,n,d)-bistochastic(Fd-bistochastic
n
for brevity) if P = ν(V) fo∈r sBome V Iso(F,n,d).
∈
When d = 1 and F = R,C,H Definition 1 yields orthostochastic,
unistochastic, and qustochastic n n matrices respectively. We will
denote by (F,n,d) the set×of Fd-bistochastic matrices.
n
OS ⊂ B
Proposition 1. 1. The set (F,n,d) is closed.
n
OS ⊂ B
2. There are inclusions
(R,n,d) (C,n,d) (H,n,d), (F,n,d) (F,n,d+1).
OS ⊂ OS ⊂ OS OS ⊂ OS
3. We have
(F,n,n) = .
n
OS B
Proof. The set Iso(F,n,d) is a compact manifold and ν : Iso(F,n,d)
→
is a differentiable map, yielding claim 1. Claim 2 is immediate from
n
Bthe definitions. We will prove claim 3. Let P = [pj] and let d N
be arbitrary. Then P = ν(V),V Iso(F,n,d)i if∈tBhenre are vec∈tors
∈
vj Fd such that vj 2 = pj and the vectors
i ∈ || i|| i
w = (v1,...,vn) Fnd, 1 i n,
i i i ∈ ≤ ≤
are pairwise orthogonal. We have
(5) w ,w = v1,v1 + + vn,vn .
h i jind h i jin ··· h i jin
If for 1 k n the vectors vk,vk,...,vk are pairwise orthogonal in Fd,
≤ ≤ 1 2 n
then, by equation (5), the vectors w ,...,w arepairwise orthogonal in
1 n
Fnd. When d n, the space Fd contains n pairwise orthogonal vectors
≥ (cid:3)
with arbitrary norms.
The following is immediate from Proposition 1.
Corollary 1. Let F be any of R,C,H. Then the following holds.
1. Foranyn N there isa unique d (F,n) N suchthat (F,n,d ) =
min min
and (F∈,n,d) = for d < d . ∈ OS
n n min
B OS 6 B
2. We have
d (H,n) d (C,n) d (R,n),d (F,n) d (F,n+1),
min min min min min
≤ ≤ ≤
GENERALIZED UNISTOCHASTIC MATRICES 5
and d (F,n) n.
min
≤
Let now P . By Proposition 1, there exist d n such that
n
P (F,n,d∈).BLet d (P,F) be the minimal such d.≤Then
min
∈ OS
(6) d (F,n) = max d (P,F) : P .
min min n
{ ∈ B }
We will informally refer to d (P,F),d (F,n)) as the minimal num-
min min
ber of internal degrees of freedom for F-quantization.
There are obvious identifications:
Iso(R,n,1) = O(n), Iso(C,n,1) = U(n), Iso(H,n,1) = Sp(n).
and
(R,n,1) = O , (C,n,1) = U , (H,n,1) = H .
n n n
OS OS OS
The following is well known:
(R,2,1) = (C,2,1) = (H,2,1) = ,
2
OS OS OS B
(R,3,1) (C,3,1) = (H,3,1)
3
OS ⊂ OS OS ⊂ B
and the inclusions are proper. For n > 3 there are proper inclusions [5]
(R,n,1) (C,n,1) (H,n,1) .
n
OS ⊂ OS ⊂ OS ⊂ B
3. Relationships, symmetries, and dimension count
There are several relationships between (F,n,d)-orthostochastic ma-
trices for various values of F and d.
Proposition 2. For any n 1 and d N there are natural inclusions
≥ ∈
(C,n,d) (R,n,2d), (H,n,d) (C,n,2d)
OS ⊂ OS OS ⊂ OS
and
(H,n,d) (R,n,4d),
OS ⊂ OS
Proof. The decomposition z = x + √ 1y identifies Cd and R2d. Let
, C and , R be the complex and th−e real scalar product on Cd re-
h i h i
spectively. The relationship
( u,v C) = u,v R
ℜ h i h i
yields the proper inclusion Iso(C,n,d) Iso(R,n,2d). It is compatible
with the squared norm maps ν : Iso(R,⊂n,2d) , ν : Iso(C,n,d)
n
→ B →
, yielding the first inclusion. The second follows similarly from the
n
Bisomorphism Hd = C2d, and the third inclusion is the composition of
(cid:3)
the former two.
6 EUGENE GUTKIN
Let (R,n) be the group of permutation matrices. It acts
n
on (WR,n⊂) bMy left and by right multiplication, yielding the action
of M on (R,n) which preserves . Let Iso(Fk) be the
n n n
groWup o×f lWinear isoMmetries of Fk. The set IsoB(F,n,d) is invariant un-
der precomposition (resp. postcomposition) with elements in Iso(Fn)
(resp. Iso(Fnd)) yielding the action of Iso(Fnd) Iso(Fn) on Iso(F,n,d).
Let Diag(Fn) Iso(Fn) (resp. Diag(F,n,d) × Iso(Fnd)) be the sub-
⊂ ⊂
group of diagonal (resp. block-diagonal) isometries. By restriction,
the subgroup Diag(F,n,d) Diag(Fn) acts on Iso(F,n,d). The group
embeds naturally into×Iso(Fn) (resp. Iso(Fnd)), by permutation
n
W
(resp. block permutation) matrices yielding an action of on
n n
Iso(F,n,d). W ×W
Proposition 3. The squared norm map ν : Iso(F,n,d) (F,n,d)
is invariantunderthe action of Diag(F,n,d) Diag(Fn) a→ndOeqSuivariant
for the actions of on Iso(F,n,d) a×nd (F,n,d).
n n
W ×W OS
Proof. Let V = [vj] Iso(F,n,d). The action of Diag(Fn) multiplies
i ∈
the n vectors vj Fd, where i is fixed and 1 j n, on the right by
the same elemeint∈λ F with λ = 1. Th≤e ac≤tion of Diag(F,n,d)
i i
∈ || ||
multiplies the n vectors vj Fd, with j fixed and 1 i n, on the
left by the same isometryiU∈ Iso(Fd). These action≤s do≤not change
j
∈
the norms of vectors vj.
i
If P = ν(V) then pj = vj 2. The action of on Iso(F,n,d)
i || i|| Wn×Wn
permutes the rows and the columns of V the same way as its action on
permutes the rows and the columns of P. (cid:3)
n
B
By the preceding discussion, the group Diag(F,n,d) Diag(Fn) nat-
urallyactsonIso(F,n,d). Wedenoteby (F,n,d)the×quotient space,
i.e., (F,n,d) = Diag(F,n,d) Iso(F,Dn,Cd)/Diag(Fn). By Proposi-
tion 3D,Cthe squared norm map ν \: Iso(F,n,d) uniquely descends
n
to a mapping of (F,n,d) which we will a→lsoBdenote by ν. Thus,
(F,n,d) = ν( DC(F,n,d)) . Let dimX denote the real dimen-
n
OS DC ⊂ B
sion.
Proposition 4. The following equations hold:
1 d2 d+1
(7) dim( (R,n,d)) = (d )n2 − n,
DC − 2 − 2
(8) dim( (C,n,d)) = (2d 1)n2 (d2 +1)n+1,
DC − −
and
(9) dim( (H,n,d)) = (4d 2)n2 (d2 +d+2)n.
DC − −
GENERALIZED UNISTOCHASTIC MATRICES 7
Proof. We have
1 1
dim(Iso(R,n,d)) = (d )n2 n,dim(Iso(C,n,d)) = (2d 1)n2,
− 2 − 2 −
dim(Iso(H,n,d)) = (4d 2)n2 +n.
−
Specializing to d = 1, we recover the well known formulas
k(k 1)
dimO(k) = − ,dimU(k) = k2,dimSp(k) = 2k2 +k.
2
The groups Diag(Fn) satisfy
Diag(Rn) = 1 n,Diag(Cn) = U(1)n,Diag(Hn) = Sp(1)n,
{± }
The actions of Diag(F,n,d) and Diag(Fn) on Iso(F,n,d) are free and
commute. They are transversal, except that both Diag(C,n,d) and
Diag(Cn) contain U(1) as the group of scalar unitary matrices. This
information and the above formulas yield the claims. We leave details
(cid:3)
to the reader.
4. On (n 1)-orthostochasticity of n n matrices
− ×
Let P . By Proposition 1, d (P,R) n. We will show
n min
∈ B ≤
that if P satisfies mild non-degeneracy assumptions and n is odd, then
d (P,R) n 1. Let n N. In what follows we will use the cyclic
min
≤ − ∈
convention for indices: i+n = i.
Lemma 2. Let n N be an odd integer. Let ξ ,...,ξ be arbitrary.
1 n
∈
Then the system
(10) x +x = ξ , 1 i n,
i i+1 i+2
≤ ≤
has a unique solution
(11) 2x = ξ +ξ +ξ ξ +ξ + +ξ , 1 i n.
i i i+1 i+2 i+3 i+4 i+n−1
− − − ··· ≤ ≤
Proof. The matrix of the linear system (10) is nondegenerate, hence
the solution is unique. The reader will easily verify that equation (11)
(cid:3)
yields a solution.
Lemma 3. Let n N be an odd integer. Let ξ ,...,ξ be positive
1 n
∈
numbers satisfying the inequalities
ξ +ξ +ξ + +ξ ξ +ξ +ξ + +ξ
i i+3 i+5 i+n−2 i+1 i+2 i+4 i+n−1
··· ≤ ···
for 1 i n. Then there exists a real skew-symmetric matrix A = [aj]
≤ ≤ i
satisfying
(12) (aj)2 = ξ , 1 j n.
X i j ≤ ≤
1≤i≤n
8 EUGENE GUTKIN
Proof. Set bj = (aj)2. Then B = [bj] is a symmetric n n matrix with
i i i ×
non-negative entries. The matrix A satisfies equation (12) if and only
if B satisfies
(13) bj = ξ , 1 j n.
X i j ≤ ≤
1≤i≤n
For x ,...,x R set
1 n +
∈
b1 = b2 = x ,b2 = b3 = x ,b3 = b4 = x ,...,bn−1 = bn = x ,
2 1 n 3 2 1 4 3 2 n n−1 n−2
b1 = bn = x , and let bj = 0 for all other pairs of indices. Then B
n 1 n−1 i
satisfies equation (13) if and only if x ,...,x satisfy equation (10).
1 n
(cid:3)
The claim now follows from Lemma 2.
Theorem 1. Let n > 1 be an odd integer. Let P = [pj] be a n n
i ×
orthostochastic matrix. Suppose that P satisfies the inequalities pj = 0
i 6
for i = j and that for 1 i n we have1
6 ≤ ≤
(14) pi+pi+3+pi+5+ +pi+n−2 pi+1+pi+2+pi+4+ +pi+n−1.
i i+3 i+5 ··· i+n−2 ≤ i+1 i+2 i+4 ··· i+n−1
Then P is (n 1)-orthostochastic.
−
Proof. We will find V = [vj] Iso(R,n,n 1) such that P = ν(V). For
i ∈ −
1 j n let vj,...,vj ,vj ,...,vj Rn−1 be mutually orthogonal
≤ ≤ 1 j−1 j+1 n ∈
vectors such that pj 2 = pj. The vectors v˜j = vj(pj)−1/2 for 1 k
n,k = j, form an o||rtkh|o|normkal basis in Rn−1k. k k ≤ ≤
Fo6r 1 i n let w Rn(n−1) and the notation , be as in
i n(n−1)
≤ ≤ ∈ h i
the proof of Proposition 1. For 1 j n we set
≤ ≤
(15) vj = ajv˜j.
j X k k
k6=j
Thus, w ,...,w are the column vectors of V = [vj]. The equation
1 n i
ν(V) = P is equivalent to the two systems of quadratic equations:
(16) w ,w = 0
i j n(n−1)
h i
for i = j and
6
(17) vj 2 = pj
|| j|| j
for 1 j n.
≤ ≤
By equation (15), we have
(18) w ,w = aj +ai
h i jin(n−1) i j
1We use the cyclic convention for indices.
GENERALIZED UNISTOCHASTIC MATRICES 9
for i = j and
6
n
(19) vj 2 = (aj)2.
|| j|| X i
i=1
Thus, it suffices to find a skew-symmetric n n matrix A = [aj]
× i
satisfying
n
(20) (aj)2 = pj.
X i j
i=1
(cid:3)
Lemma 3 yields a solution.
Remark 1. The proof of Theorem 1 yields an explicit solution of the
equation ν(V) = P. Let e ,...,e Rn−1 be the standard or-
1 n−1
∈
thonormal basis. For 1 j n set vj = (pj)1/2e if k < j and
≤ ≤ k k k
vj = (pj)1/2e if k > j. The numbers aj defining the vectors
k k k−1 i
vj Rn−1 satisfy aj = 0 if i j = 1 mod n. For pairs i,j satis-
j ∈ i − 6 ±
fying i j = 1 mod n, Lemmas 2, 3 explicitly yield (aj)2 as linear
− ± i
combinations of pk where 1 k n. Set aj = (aj)2 if j < i and
k ≤ ≤ i q i
aj = (aj)2 if i < j.
i −q i
Corollary 2. Let n > 1 be an odd integer. Then
1. The set (R,n,n 1) contains all bistochastic matrices P = [pj]
OS − i
such the numbers p1,...,pn satisfy the inequalities in equation (14);
1 n
2. For an open set of P we have d (P,R) n 1.
n min
∈ B ≤ −
Proof. Let P = [pj] be a n n bistochastic matrix. By Theorem 1, P
(R,n,n 1) ifithe num×bers p1,...,pn satisfy the strict inequalitie∈s
OS − 1 n
in equation (14) and pj = 0 for i = j. This is a nonempty open set,
hence claim 2. By Propio6sition 1, 6 (R,n,n 1) contains its closure,
OS − (cid:3)
yielding claim 1.
Example 1. Let n = 3. The open set in Corollary 2 is the set of
P such that pj = 0 for i = j and
∈ B3 i 6 6
(21) p1 < p2 +p3, p2 < p3 +p1, p3 < p1 +p2.
1 2 3 2 3 1 3 1 2
Every 3 3 matrix satisfying these inequalities is 2-orthostochastic.
×
Note that the set of orthostochastic 3 3 matrices has positive codi-
×
mension in [2, 5, 19]. Inequalities (21) hold if and only if p1,p2,p3
B3 1 2 3
are the side lengths of a nondegenerate triangle. We point out that the
triangle inequalities come up as conditions of unistochasticity for 3 3
×
matrices [8].
10 EUGENE GUTKIN
Example 2. Let n = 5. The set of 4-orthostochastic 5 5 matrices in
×
Corollary 2 consists of bistochastic matrices satisfying the following:
p1 +p4 p2 +p3 +p5, p2 +p5 p3 +p4 +p1,
1 4 ≤ 2 3 5 2 5 ≤ 3 4 1
and
p3 +p1 p4 +p5 +p2, p4 +p2 p5 +p1 +p3, p5 +p3 p1 +p2 +p4.
3 1 ≤ 4 5 2 4 2 ≤ 5 1 3 5 3 ≤ 1 2 4
(These inequalities do not have an immediate geometric interpreta-
tion.) The interior of the set of 4-orthostochastic 5 5 matrices con-
×
tains [pj] such that pj = 0 for i = j and the above inequalities
i ∈ B5 i 6 6
are strict.
5. Concluding discussion
To describe precisely the sets of orthostochastic, unistochastic, and
qustochastic n n matrices for arbitrary n seems a difficult problem
ref. The concep×t of Fd-bistochastic introduced here replaces this prob-
lem with another, seemingly simpler, but still a meaningful problem:
To characterize Fd-bistochastic n n matrices for arbitrary n. Our
×
results suggest that this question is still nontrivial but less subtle.
The two questions are related. Note that, by Proposition 2, the set
of 2-orthostochastic matrices contains the set of unistochastic matri-
ces, and the set of 2-unistochastic matrices contains the set of qus-
tochastic matrices. Let n be arbitrary, and let us vary d N. By
Proposition 1, the set of Fd-bistochastic coincides with as∈d reaches
n
d (F,n) n. The above results and dimensional consBiderations sug-
min
gest that d≤ (F,n) does not indefinitely increase, as n goes to . In
min
particular, for F = R, the following conjecture is plausible. ∞
Conjecture 1. Denote by d (n) the smallest value of d such that the
min
set of d-orthostochastic matrices coincides with . Then there exists
n
B
n such that for n n we have
0 0
≥
2 d (n) 3.
min
≤ ≤
The material in section 4 suggests that value of d (n) may depend
min
on the parity of n. The author believes that the threshold dimension
should not be too large. Most likely, n = 3.
0
Acknowledgements. The work was partiallysupported by theMNiSzW
grantNN201384834andtheNCNGrantDEC-2011/03/B/ST1/00407.