Czech Technical University in Prague
Faculty of Nuclear Sciences and Physical Engineering
Břehová 7, CZ - 115 19 Prague, Czech Republic
J. Tolar
Quantum mechanics in finite-dimensional
Hilbert spaces: A classification of finite
quantum systems
Stará Lesná 2012
Outline
Abstract
Finite-dimensional quantum mechanics
Simple quantum kinematics on cyclic groups
A classification of finite quantum systems
Symmetries of simple quantum kinematics
Symmetries for composite systems
Multipartite systems
Remarks and open problems
Abstract
Quantum mechanics in Hilbert spaces of finite dimension N is
studied from the number theoretic point of view. For composite numbers N possible quantum kinematics are classified on the
basis of Mackey’s Imprimitivity Theorem for cyclic groups. This
yields also a classification of finite Heisenberg-Weyl groups and
the corresponding finite quantum systems. Simple number theory gets involved through the fundamental theorem describing
all finite discrete Abelian groups of order N as direct products of
cyclic groups, whose orders are powers of not necessarily distinct
primes contained in the prime decomposition of N.
Finite-dimensional quantum mechanics
Weyl H 1931 The Theory of Groups and Quantum Mechanics (New York: Dover) pp 272–280
J. Schwinger, Unitary operator bases, Proc. Nat. Acad.
Sci. (U.S.A.) 46, 570-579, (1960); reprinted in: J.
Schwinger, “Quantum Kinematics and Dynamics”, Benjamin, New York (1970), 63–72
Šťovı́ček P and Tolar J 1984 Quantum mechanics in a
discrete space-time Rep. Math. Phys. 20 157–170
Vourdas A 2004 Quantum systems with finite Hilbert
space Rep. Progr. Phys. 67 267–320
Kibler M R 2008 Variations on a theme of Heisenberg,
Pauli and Weyl J. Phys. A: Math. Theor. 41 375302
Simple quantum kinematics on cyclic groups
Ordinary quantum mechanics prescribes that the mathematical
quantities representing the position and momentum should be
self-adjoint operators on the Hilbert space of the system. Their
algebraic properties constitute quantum kinematics, while quantum dynamics of the system is given by the unitary group generated by the Hamiltonian which is expressed as a function of the
position and momentum operators.
According to Mackey, quantum kinematics of a system localized
on a homogeneous space M = G/H is determined by an irreducible transitive system of imprimitivity (U, E) for G based on
M in a Hilbert space H. Here U = {U (g)|g ∈ G} is a unitary
representation of G in H and E = {E(S)| S Borel subset of M }
is a projection–valued measure in H satisfying the covariance
condition
U (g)E(S)U (g)−1 = E(g −1.S).
Given a positive integer N ≥ 2, we assume that the configuration
space M is the finite set
M = ZN = {ρ|ρ = 0, 1, . . . , N − 1}
with additive group law modulo N . Since there is a natural transitive action of ZN on itself, we may consider M as a homogeneous
space of
G = ZN = {j|j = 0, 1, . . . , N − 1},
realized as an additive group modulo N with the action
G × M → M : (j, ρ) 7→ ρ + j(mod N )
and the isotropy subgroup H = {0}. For the finite set M = ZN
the covariance condition simplifies to
U (j)E(ρ)U (j)−1 = E(ρ − j),
P
where ρ ∈ M , j ∈ G, E(ρ) = E({ρ}) and E(S) = ρ∈S E(ρ).
Complete classification of transitive systems of imprimitivity up
to simultaneous unitary equivalence of both U and E is obtained
from Mackey’s Imprimitivity Theorem. Its application to our
system yields
Imprimitivity Theorem If (U , E) acts irreducibly in H, then there
is, up to unitary equivalence, only one system of imprimitivity,
where:
1. H is a Hilbert space CN with the inner product
(ϕ, ψ) =
NX
−1
ϕρ ψρ ,
ρ=0
where ϕρ, ψρ, ρ = 0, 1, . . . , N − 1, denote the components of
ϕ, ψ in the standard basis.
2. U is the induced representation U = IndG
H I called the (right)
regular representation
[U (j)ψ]ρ = ψρ+j
(j ∈ G).
Its matrix form in the standard basis is
(U (j))ρσ = δρ+j,σ .
3. E is given by
[E(ρ)ψ]σ = δρσ ψσ .
This unique system of imprimitivity has a simple physical meaning. The localization operators E(ρ) are projectors on the eigenvectors e(ρ) ∈ H corresponding to the positions ρ = 0, 1, . . . , N −1.
Since the set {e(ρ)} forms the standard basis of H in the above
matrix realization, this realization may be called position representation. Then in a normalized state ψ = (ψ0, . . . , ψN −1) the
probability to measure the position ρ is equal to
(ψ, E(ρ)ψ) = |ψρ|2.
Unitary operators U (j) act as displacement operators
U (j)e(ρ) = e(ρ−j).
In the position representation they are given by unitary matrices
equal to the powers U (j) = Aj of the one-step cyclic matrix




A ≡ U (1) = 



0
0
...
0
1
1 0 ···0
0 1 ···0
...
0 0 ....0
0 0 ....0
0
0
1
0




.



In order to describe this quantum kinematics in terms of the
equivalent discrete Weyl system we need the dual system of
unitary operators V = {V (ρ)|ρ ∈ M } acting via displacements
on the dual basis of momentum states f (j) — eigenvectors of
the commuting set {U (k) = Ak }, Af (j) = λj f (j). Eigenvalues λj
and eigenvectors f (j) are easily found using the discrete Fourier
transform F : ψ 7→ ψ̃, where
−1
1 NX
ψ̃j = √
ω −jρψρ,
N ρ=0
i.e.
1
Fjρ = √ ω −jρ,
N
and ω = exp( 2πi
N ) is the primitive N -th root of unity. Then it can
be verified that A and all its integer powers are simultaneously
diagonalized by F with the result
F AF −1 ≡ B = diag(1, ω, . . . , ω N −1) =
NX
−1
ρ=0
ω ρE(ρ).
The eigenvectors f (j) of A belonging to eigenvalues ω j are
f (j) =
NX
−1
(F −1)
(ρ) = √1
e
jρ
ρ=0
NX
−1
1
ω jρe(ρ) = √ (1, ω j , . . . , ω j(N −1))
N ρ=0
N
and the operators
V (ρ) ≡ B ρ =
NX
−1
ω ρσ E(σ)
σ=0
act on f (j) via displacements
V (ρ)f (j) = B ρf (j) = f (j+ρ).
Hence the quantum kinematics (U, E) can be equivalently replaced by the discrete Weyl system (U, V) satisfying
U (j)V (ρ) = ω jρV (ρ)U (j)
or
Aj B ρ = ω jρB ρAj .
The discrete Weyl displacement operators are defined by unitary
operators∗
W (ρ, j) = ω jρ/2V (ρ)U (j) = ω −jρ/2U (j)V (ρ).
They satisfy the composition law for a ray representation of
ZN × ZN
0 j−ρj 0 )/2
0
0
(ρ
W (ρ, j)W (ρ , j ) = ω
W (ρ + ρ, j + j 0).
According to Schwinger, the discrete Weyl system W consisting of N 2 operators W (ρ, j) provides an orthogonal operator baN
sis in the
√ space of all linear operators in C . The operators
W (ρ, j)/ N are orthogonal with respect to the inner product
Tr(W (ρ, j)W (ρ0, j 0)∗) = N δρρ0 δjj 0
∗ If
jρ/2
N is odd, the factors ωN
are well-defined on ZN × ZN .
and satisfy the completeness relation
X
W (ρ, j)W (ρ0, j 0)∗ = N 21.
ρ,j
This important result can be summarized as follows:
2 matrices S(ρ, j) =
Theorem
(J.
Schwinger
1960)
The
set
of
N
√
ρ
j
B A / N , j, ρ = 0, 1, . . . , N − 1, constitutes a complete and orthonormal basis for N × N complex matrices. Any complex
matrix in H can thus be uniquely expanded in this basis. If
N is odd, then the operator
basis can be taken in the form
√
ω jρ/2S(ρ, j) = W (ρ, j)/ N .
G.W. Mackey, “Induced Representations and Quantum
Mechanics”, Benjamin, New York (1968)
We remind our earlier notation
2 , · · · , ω N −1
B = QN = diag 1, ωN , ωN
N
and





A = PN = 




0
0
0
...
0
1
1 0 ···
0 1 ···
0 0 ···
...
0 0 ···
0 0 ···

0 0

0 0 

0 0 

.

0 1 

0 0
A classification of finite quantum systems
The cyclic group ZN = {0, 1, . . . N − 1} was a configuration space
for N -dimensional quantum kinematics of a single N -level system. However, the reasoning on the basis of the Imprimitivity
Theorem allows direct extension to any finite Abelian group as
configuration space because of a fundamental theorem describing the structure of all finitely generated Abelian groups.
Fundamental Theorem for finite Abelian groups Let G be
a finite discrete Abelian group. Then G is isomorphic with the
direct product ZN1 × · · · × ZNf of a finite number of cyclic groups
for integers N1, . . . , Nf greater than 1, each of which is a power
r
of a prime, i.e. Nk = pkk , where the primes pk need not be
mutually different.
rf
r1
In the special case of G = ZN with composite N = p1 . . . pf and
distinct primes pk > 1, the unique decomposition
r
Nk = pkk
is obtained by the Chinese Remainder Theorem.
ZN = ZN1 × · · · × ZNf ,
Returning to a general finite Abelian group as configuration
space, a transitive system of imprimitivity for G = ZN1 × · · · × ZNf
based on M = ZN1 × · · · × ZNf is equivalent to a tensor product
U = U1 ⊗ · · · ⊗ Uf , E = E1 ⊗ · · · ⊗ Ef
and acts in the Hilbert space H = H1 ⊗· · ·⊗Hf , where the dimensions are dim Hk = Nk , dim H = N1 . . . Nf . Each such system of
imprimitivity (U, E) is irreducible, if and only if each (Uk , Ek ) is
irreducible, hence if irreducible, it is unique up to unitary equivalence by the Imprimitivity Theorem.
Given a composite dimension N , then according to the Fundamental Theorem it is sufficient to take all inequivalent choices
r
of Nk = pkk with not necessarily distinct primes pk > 1 to obtain all irreducible quantum kinematics in the Hilbert space of
given finite dimension N . One can give physical interpretation
to these tensor products in the sense that the factors describe
the building blocks forming the finite quantum system.
Our results are summarized in
Theorem For a given finite Abelian group G there is a unique
class of unitarily equivalent, irreducible imprimitivity systems (U, E)
in a finite–dimensional Hilbert space H. In the special case
G = ZN the irreducible imprimitivity system is unitarily equivr
alent to the tensor product with Nk = pkk and distinct primes
p1, . . . , pf > 1.
Corollary. For a given finite Abelian group G there is a unique
class of unitarily equivalent discrete Weyl systems W in a finite–
dimensional Hilbert space H,
W = W1 ⊗ · · · ⊗ Wf .
Symmetries of simple quantum kinematics
Balian R and Itzykson C 1986 Observations sur la mécanique
quantique finie C. R. Acad. Sci. Paris 303 Série I, n.
16, 773–777
Havlı́ček M, Patera J, Pelantová E and Tolar J 2002
Automorphisms of the fine grading of sl(n, C) associated
with the generalized Pauli matrices J. Math. Phys. 43
1083-1094; arxiv: math-ph/0311015
Vourdas A and Banderier C 2010 Symplectic transformations and quantum tomography in finite quantum systems
J. Phys. A: Math. Theor. 43 042001 (9pp)
Consider those inner automorphisms acting on elements of ΠN
which induce permutations of cosets in ΠN /Z(ΠN ):
j
j
l Qi P ) = Xω l Qi P X −1 ,
AdX (ωN
N N
N N N
where X are unitary matrices from GL(N, C).
These inner automorphisms are equivalent if they define the
same transformation of cosets in ΠN /Z(ΠN ):
AdY ∼ AdX ⇔ Y QiP j Y −1 = XQiP j X −1
for all (i, j) ∈ ZN × ZN .
The group PN = ΠN /Z(ΠN ) has two generators, the cosets Q
and P (or AdQN and AdPN , respectively). Hence if AdY induces
a permutation of cosets in ΠN /Z(ΠN ), then there must exist
elements a, b, c, d ∈ ZN such that
Y QY −1 = QaP b
and
Y P Y −1 = QcP d.
It follows that to each equivalence class of inner automorphisms
AdY a quadruple (a, b, c, d) of elements in ZN is assigned.
Theorem (HPPT 2002) For integer N ≥ 2 there is an isomorphism Φ between the set of equivalence classes of inner automorphisms AdY which induce permutations of cosets and the
group SL(2, ZN ) of 2 × 2 matrices with determinant equal to 1
(mod N ),
Φ(AdY ) =
a b
c d
!
,
a, b, c, d ∈ ZN ;
the action of these automorphisms on ΠN /Z(ΠN ) is given by the
right action of SL(2, ZN ) on elements (i, j) of the phase space
PN = ZN × ZN ,
(i0, j 0) = (i, j)
a b
c d
!
.
Symmetries for composite systems
Korbelář M and Tolar J 2012 Symmetries of finite Heisenberg groups for multipartite systems J. Phys. A: Math.
Theor. 45 285305 (18pp)
Korbelář M and Tolar J 2010 Symmetries of the finite
Heisenberg group for composite systems J. Phys. A:
Math. Theor. 43 375302 (15pp); arxiv: 1006.0328
[quant-ph]
Han G 2010 The symmetries of the fine gradings of
sl(nk , C) associated with direct product of Pauli groups
J. Math. Phys. 51 092104 (15 pages)
Pelantová E, Svobodová M and Tremblay J 2006 Fine
grading of sl(p2, C) generated by tensor product of generalized Pauli matrices and its symmetries J. Math. Phys.
47 5341–5357
Multipartite systems
According to the well-known rules of quantum mechanics, finitedimensional quantum mechanics on Zn can be extended in a
straightforward way to arbitrary finite direct products
Zn1 × · · · × Znk
as configuration spaces. The cyclic groups involved describe
independent quantum degrees of freedom. The Hilbert space of
such a composite system is the tensor product
Hn1 ⊗ · · · ⊗ Hnk
of dimension N = n1 . . . nk , where n1, . . . , nk ∈ N.
For the multipartite system, quantum phase space is an Abelian
subgroup of Int(GL(N, C)) defined by
P(n1,...,nk ) = {AdM1⊗···⊗Mk | Mi ∈ Πni }.
Its generating elements are the inner automorphisms
ej := AdAj
for
j = 1, . . . , 2k,
where (for i = 1, . . . , k)
A2i−1 := In1···ni−1 ⊗Pni ⊗Ini+1···nk , A2i := In1···ni−1 ⊗Qni ⊗Ini+1···nk .
The normalizer of P(n1,...,nk ) in Int(GL(n1 · · · nk , C)) will be denoted
N (P(n1,...,nk )) := NInt(GL(n1···nk ,C))(P(n1,...,nk )),
We need also the normalizer of Pn in Int(GL(n, C)),
N (Pn) := NInt(GL(n,C))(Pn),
and
N (Pn1 )×· · ·×N (Pnk ) := {AdM1⊗···⊗Mk | Mi ∈ N (Pni )} ⊆ Int(GL(N, C)).
Further,
N (Pn1 ) × · · · × N (Pnk ) ⊆ N (P(n1,...,nk ))
.
Now the symmetry group Sp[n1,...,nk ] is defined in several steps.
First let S[n1,...,nk ] be a set consisting of k × k matrices H of 2 × 2
blocks
ni
Hij =
Aij
gcd(ni, nj )
where Aij ∈ M2(Zni ) for i, j = 1, . . . , k.
Then S[n1,...,nk ] is (with the usual matrix multiplication) a monoid.
Next, for a matrix H ∈ S[n1,...,nk ], we define its adjoint H ∗ ∈
S[n1,...,nk ] by
(H ∗)
ni
AT
.
ij =
gcd(ni, nj ) ji
Further, we need a skew-symmetric matrix
J = diag(J2, . . . , J2) ∈ S[n1,...,nk ]
where
J2 =
0 1
−1 0
!
.
Then the symmetry group is defined as
Sp[n1,...,nk ] := {H ∈ S[n1,...,nk ]| H ∗JH = J}
and is a finite subgroup of the monoid S[n1,...,nk ].
Our first theorem states the group isomorphism:
Theorem 1 (KT 2012)
∼ Sp
N (P(n1,...,nk ))/P(n1,...,nk ) =
[n1 ,...,nk ] .
Our second theorem describes the generating elements of the
normalizer:
Theorem 2 (KT 2012)
The normalizer N (P(n1,...,nk )) is generated by
N (Pn1 ) × · · · × N (Pnk )
and
{AdRij },
where (for 1 ≤ i < j ≤ k)
n −1
Rij = In1···ni−1 ⊗ diag(Ini+1···nj , Tij , . . . , Tiji
and
Tij = Ini+1···nj−1 ⊗
nj
gcd(n ,n )
Qnj i j .
) ⊗ Inj+1···nk
Corollary (KT 2012)
If n1 = · · · = nk = n, i.e.
∼ Sp (Z ).
Sp[n,...,n] =
2k n
N = nk , the symmetry group is
These cases are of particular interest, since they uncover symplectic symmetry of k-partite systems composed of subsystems
with the same dimensions. This circumstance was found, to our
knowledge, first by PST 2006 for k = 2 under additional assumption that n = p is prime, leading to Sp(4, Fp) over the field
Fp .
We have generalized this result (KT 2010) to bipartite systems
with arbitrary n (non-prime) leading to Sp(4, Zn) over the modular ring, and also to multipartite systems (KT 2012).
The corresponding result has independently been obtained by G.
Han who studied symmetries of the tensored Pauli grading of
sl(nk , C).
Remarks and open problems in FDQM
Our motivation to study symmetries of the finite Heisenberg
group not in prime or prime power dimensions but for arbitrary
dimensions stems from our previous research where we obtained
results valid for arbitrary (or odd) dimensions (e.g. our paper
TCh 2009 on Feynman’s path integral and mutually unbiased
bases). This suggests a promising open problem: using HPPT
2002 extend the results of Balian and Itzykson (BI 1986) on the
metaplectic representation of SL(2, Fp) from prime p to arbitrary
N.
A historical remark is in order: the structure of the finite–
dimensional Weyl operators was thoroughly investigated by J.
Schwinger (S 1960). He noted in particular that, if the dimension N is a composite number which can be decomposed as a
product N = N1N2, where the positive integers N1, N2 > 1 are
relatively prime, then all the Weyl operators in dimension N can
be simultaneously factorized in tensor products of the Weyl operators in the dimensions N1 and N2. J. Schwinger then states
(S 1960, pp. 578–579): “The continuation of the factorization
terminates in
N =
f
Y
νk ,
k=1
where f is the total number of prime factors in N , including
repetitions. We call this characteristic property of N the number
of degrees of freedom for a system possessing N states.” And
further, “each degree of freedom is classified by the value of the
prime integer ν = 2, 3, 5, . . . ∞.”
However, our application of the Fundamental Theorem for finite Abelian groups (ST 1984) lead to the building blocks Zpr
with r ≥ 1 as constituent configuration spaces. This fact was
independently noted by V.S. Varadarajan in a paper (VSV 1995)
devoted to the memory of J. Schwinger: “Curiously, Schwinger
missed the systems associated to the indecomposable groups ZN
where N is a prime power pr , r ≥ 2 being an integer.” And in
another paper: ”In this way he arrived at the principle that the
Weyl systems associated to Zp where p runs over all the primes
are the building blocks. Curiously this enumeration is incomplete
and one has to include the cases with Zpr where p is as before a
prime but r is any integer ≥ 1.”
It has still to be pointed out that among possible factorizations
into constituent subsystems special role is played by the factorization associated with the “best” prime decomposition of the
rf
r1
dimension N = p1 . . . pf with mutually distinct primes p1, . . . ,
pf . By to the Chinese Remainder Theorem the discrete Weyl
system of composite dimension N is equivalent to a composite
system where the constituent Weyl subsystems act in the Hilbert
r
r
spaces of relatively prime dimensions p11 ,. . . , pff .
Considering the building blocks Zpr as constituent configuration
spaces, their corresponding symmetries are (see the last Corollary) the finite symplectic groups Sp2r (Zp). Their cardinalities
can be found e.g. in O’Meara’s book. There remains, however,
an open problem presented by A. Vourdas in connection with his
study of quantum tomography — the cardinalities of the groups
Sp2r (Zn) for arbitrary positive integer n.
J. Schwinger, Proc. Nat. Acad. Sci. (U.S.A.) 46,
570-579, (1960); reprinted in: J. Schwinger, “Quantum
Kinematics and Dynamics”, Benjamin, New York (1970),
63–72
P. Šťovı́ček, J. Tolar, Quantum mechanics in a discrete
space-time, Rep. Math. Phys. 20, 157-170, (1984)
V.S. Varadarajan, Variations on a theme of Schwinger
and Weyl, Lett. Math. Phys. 34, 319-326 (1995)
Tolar J and Chadzitaskos G 2009 Feynman’s path integral and mutually unbiased bases J. Phys. A: Math.
Theor. 42 245306 (11pp)
Šulc P and Tolar J 2007 Group theoretical construction
of mutually unbiased bases in Hilbert spaces of prime
dimensions J. Phys. A: Math. Theor. 40 15099-15111
Tolar J and Chadzitaskos G 1997 Quantization on ZM
and coherent states over ZM × ZM J. Phys. A: Math.
Gen. 30 2509–2517
Chadzitaskos G and Tolar J 1993 Feynman path integral
and ordering rules on discrete finite space Int. J. Theor.
Phys. 32 517–527
Thank you for your attention