# Quantum Symmetries and Cartan Decompositions in Arbitrary

## Transcript Of Quantum Symmetries and Cartan Decompositions in Arbitrary

Quantum Symmetries and Cartan Decompositions in Arbitrary Dimensions

Domenico D’Alessandro1 and Francesca Albertini2

Abstract We investigate the relation between Cartan decompositions of the unitary group and discrete quantum symmetries. To every Cartan decomposition there corresponds a quantum symmetry which is the identity when applied twice. As an application, we describe a new and general method to obtain Cartan decompositions of the unitary group of evolutions of multipartite systems from Cartan decompositions on the single subsystems. The resulting decomposition, which we call of the odd-even type, contains, as a special case, the concurrence canonical decomposition (CCD) presented in [5],[6],[7] in the context of entanglement theory. The CCD is therefore extended from the case of a multipartite system of n qubits to the case where the component subsystems have arbitrary dimension.

Keywords: Lie groups decompositions, Quantum symmetries, Quantum multipartite systems.

1 Introduction

Decompositions of Lie groups have been extensively used in control theory to design control algorithms for bilinear, right invariant, systems with state varying on a Lie group. Once it is known how to factorize a target ﬁnal state Xf as the product

Xf = X1X2 · · · Xr,

(1)

then the task of controlling to Xf can be reduced to the (simpler) task of controlling to the factors X1, ..., Xr. In quantum information theory, a factorization of the type (1) can be interpreted as the implementation of a quantum logic operation with a sequence of elementary operations. In this case, the relevant Lie group is the Lie group of unitary matrices of dimensions n, U (n). In general, a decomposition of the unitary evolution operator of the form (1) is useful to determine several aspects of the dynamics of quantum systems including the degree of entanglement (see e.g. [16]), time optimality of the evolution [11] and constructive controllability (see e.g. [8], [14]).

Most of the studies presented so far, which involve Lie group decompositions applied to the quantum systems, are concerned with low dimensional systems. For these systems, several

1Department of Mathematics, Iowa State University, Ames, IA 50011, U.S.A., Tel.:+1-515-2948130, Fax: +1515-2945454, e-mail: [email protected]

2Department of Pure and Applied Mathematics, University of Padova, Via Belzoni 7, 35100 Padova, Italy, Tel.: +39-49-8275966, Fax: +39-49-8275892, e-mail:[email protected]

1

complete and elegant results can be obtained, which also have important physical implications. Decompositions of the unitary group U (n) for large n exist and can be applied to the dynamical analysis of high dimensional quantum systems. However, the information obtained with this study is rarely as useful and of direct physical interpretation as in the low dimensional cases. For multipartite systems, this motivates the search for Lie group decompositions constructed in terms of decompositions on the single subsystems. We shall construct such type of decomposition in the present paper.

The main motivation for the study presented here was given by the recent papers [5] [6], [7]. In these papers, a decomposition of U (2n) called the Concurrence Canonical Decomposition was obtained for a quantum system of n two level systems (qubits). Such a decomposition has the above mentioned feature of being expressed in terms of elementary decompositions on the single qubit subsystems. It is related to time reversal symmetry and this raises the question of what in general the relation is between quantum mechanical symmetries and decompositions. As we shall see here, the answer to this fundamental question is instrumental in developing a general method to construct decompositions of multipartite systems from elementary decompositions of the single subsystems. We shall develop a decomposition which we call of the ‘odd-even type’ that contains the concurrence canonical decomposition as a special case.

The paper is organized as follows. In Section 2 we review the basic deﬁnitions and results concerning discrete quantum symmetries and Cartan decompositions of the Lie algebra su(n) and therefore the Lie group SU (n). We shall stress the important result that, up to conjugacies, there are only three types of Cartan decompositions which are usually labeled as AI, AII and AIII. In Section 3, we investigate the relation between Cartan decompositions and quantum symmetries and establish a one to one correspondence between Cartan decompositions and a subclass of symmetries which we call Cartan symmetries. To every Cartan decomposition of the Lie algebra u(n) and corresponding Cartan symmetry there corresponds a decomposition of the Jordan algebra of Hermitian matrices of dimension n, iu(n) equipped with the anticommutator operation. This is described in Section 4. This is also the crucial fact used to develop the general decomposition of the odd-even type for multipartite systems in arbitrary dimensions in Section 5. This decomposition is a Cartan decomposition and, in Section 6, we show how to determine its type (AI or AII). The Cartan decomposition also leads to a decomposition of the evolution of any quantum system into the product of an evolution with antisymmetric Hamiltonian and one with symmetric Hamiltonian with respect to a Cartan symmetry. This result is discussed in Remark 4.1.

2

2 Background material

2.1 Discrete Symmetries in Quantum Mechanics

Given a quantum system with underlying Hilbert space H, a quantum mechanical symmetry is deﬁned (see e.g. [9] Chapter 7, [13] Chapter 4) as a one to one and onto map Θ : H → H such that physically indistinguishable states are also mapped into physically indistinguishable states i.e. for every |ψ >∈ H and φ1 ∈ RI ,

Θ(eiφ1 |ψ >) = eiφ2 Θ(|ψ >),

(2)

for some φ2 ∈ RI . Moreover Θ preserves the inner product of two states, namely if |α˜ >:= Θ|α >, then, for any two states |α > and |β >

| < α˜|β˜ > | = | < α|β > |.

(3)

In this deﬁnition, we omit for simplicity the consideration of selection rules and assume that all the states are physically realizable. A detailed discussion of this point can be found in [9].

According to Wigner’s theorem [15], every such operation Θ can be represented as

Θ = eiφU,

(4)

where φ is a constant, physically irrelevant, real parameter, and U is either a unitary operator

or an anti-unitary one. Recall that an anti-unitary operator U , |α >→ |α˜ >:= U |α > is deﬁned

as satisfying

< β˜|α˜ >=< β|α >∗,

(5)

U (c1|α > +c2|β >) = c∗1U |α > +c∗2U |β > .

(6)

Once a basis of the Hilbert space H is chosen, an anti-unitary operator U can always be

written as

U |α >= XK|α >,

(7)

where K is the operation which conjugates all the components of the vector |α > and X is

unitary.

A symmetry Θ, whether unitary or anti-unitary, induces a transformation on the space of

Hermitian operators A as

A → ΘAΘ−1 := θ¯(A).

(8)

It is in fact easily veriﬁed that θ¯(A) is a linear and Hermitian operator. Moreover the eigenvalues of θ¯(A) are the same as those of A and a set of orthonormal eigenvectors are given by Θ|αj >,

3

where |αj > is an orthonormal basis of eigenvectors of A. It can be proved [9], [13], that, up to a phase factor, θ¯(A) := ΘAΘ−1 is the only choice that guarantees

| < α˜|θ¯(A)|β˜ >= | < α|A|β > |.

(9)

Description of the symmetry Θ is usually done by specifying how θ¯ acts on Hermitian operators rather than how Θ acts on states. This is because, Hermitian operators represent physical observables and therefore the action of θ on observables is typically suggested by physical considerations. For example, the space translation symmetry has to be such that

θ¯(xˆ) = xˆ − a,

(10)

for some constant a where xˆ is the position operator. As another example, the parity or space inversion symmetry is deﬁned such that

θ¯(xˆ) = −xˆ.

(11)

On the other hand, speciﬁcation of θ¯ on an irreducible set of observables uniquely determines Θ up to a phase factor [9]. Recall that an irreducible set of observables {Aj} is deﬁned such that if an observable B commutes with all of the {Aj}, then B is a multiple of the identity.

An observable H is said to satisfy a symmetry Θ or to be symmetric with respect to Θ if

θ¯(H) = H,

(12)

or equivalently

ΘH = HΘ.

(13)

It is said to be antisymmetric with respect to Θ if

θ¯(H) = −H ↔ ΘH = −HΘ.

(14)

A special type of symmetry is the time reversal symmetry. In classical mechanics a time reversal symmetry changes a system into one which evolves with time reversal trajectories. This suggests to deﬁne a time reversal symmetry in quantum mechanics so that θ¯ acts on the position xˆ and momentum operator pˆ according to

θ¯(xˆ) = xˆ,

(15)

θ¯(pˆ) = −pˆ.

(16)

This implies that the corresponding Θ transforms momentum eigenvectors |p > as

Θ|p >= | − p > .

(17)

4

If the system under consideration has no spin degree of freedom, then xˆ and pˆ form an irreducible set of observables and therefore (15) and (16) uniquely specify the transformation Θ on the state. Moreover, from the deﬁnition of angular momentum Lˆ := xˆ × pˆ, we obtain

θ¯(Lˆ) = θ¯(xˆ) × θ¯(pˆ) = −Lˆ.

(18)

For a system with spin angular momentum Sˆ, we impose by deﬁnition, according to (18)

θ¯(Sˆ) = −Sˆ,

(19)

and xˆ, pˆ, Sˆ form an irreducible set of observables. If |m > is an eigenvector of the (spin) angular momentum corresponding to eigenvalue m, we have

Θ|m >= | − m > .

(20)

From these speciﬁcations, it is possible to obtain an explicit expression of the time reversal symmetry for a system of N particles with spin operators Sˆ1,...,SˆN . It is given (in a basis of tensor products of the eigenstates of the z− component of the spin operators) by (see [9])

Θ

=

e−

iπ h¯

(

Sˆ1

,y

+

Sˆ2

,y

+

...

+

SˆN

,y

)

K

,

(21)

where Sˆj,y is the y component of the spin operator corresponding to the j−th particle, j = 1, ..., N , and K is the conjugation operator (same as in (7)).

2.2 Cartan involutions and decompositions of su(n)

We discuss next the Cartan decompositions for the Lie algebra su(n). What we say could be generalized to general semisimple Lie algebras. We refer to [10] for more details.

A Cartan decomposition of su(n) is a vector space decomposition

su(n) = K ⊕ P,

(22)

where the subspaces K and P satisfy the commutation relations

[K, K] ⊆ K,

(23)

[K, P] ⊆ P,

(24)

[P, P] ⊆ K.

(25)

In particular, notice that K is a subalgebra of su(n). A Cartan decomposition of su(n) induces a factorization of the elements of the Lie groups SU (n). Let us denote by eL the connected Lie

5

group associated to a generic Lie algebra L. Then, given a Cartan decomposition (22), every element X in SU (n) can be written as

X = KP,

(26)

where K ∈ eK and P is the exponential of an element of P. Moreover if A is a maximal Abelian subalgebra of su(n), with A ⊆ P, then one can prove that

∪K∈eK KAK∗ = P.

(27)

This implies that one can write P in (26) as P = K1AK1∗ with K1 ∈ eK and A ∈ eA. Therefore every element X in SU (n) can be written as

X = K1AK2,

(28)

with K1, K2 ∈ eK and A ∈ eA. This is often referred to as KAK decomposition. A Cartan involution of su(n) is a homomorphism θ : su(n) → su(n) such that θ2 is equal to

the identity on su(n). Associated to a Cartan decomposition (22) is a Cartan involution which is equal to the identity on K and multiplies by −1 the elements of P, i.e.

θ(K) = K, ∀K ∈ K,

(29)

θ(P ) = −P, ∀P ∈ P.

(30)

Therefore, given a Cartan decomposition, relations (29) and (30) determine a Cartan involution

θ. Viceversa given a Cartan involution θ, the +1 and −1 eigenspaces of θ determine a Cartan

decomposition.

According to a theorem of Cartan [10], there exist only three types of Cartan decompositions

for su(n) up to conjugacy. More speciﬁcally, given a Cartan decomposition (22) there exists a

matrix H ∈ SU (n) such that K := HKH∗, P := HPH∗, where K and P fall in one of the

following cases labeled AI, AII and AIII3 .

AI

K = so(n), P = so(n)⊥,

(31)

where so(n) is the Lie algebra of real skew-Hermitian matrices of dimension n, so(n)⊥ is the vector space over the reals of purely imaginary skew-Hermitian matrices. The corresponding Cartan involution, which we denote by θI , returns the complex conjugate of a matrix, i.e.

θI (A) := A¯.

(32)

3In the following deﬁnitions and in the rest of the paper the inner product < A, B > in su(n) is deﬁned as < A, B >:= T r(AB∗) and it is proportional to the Killing form (see e.g. [10])

6

AII

n K = sp( ),

P = sp( n )⊥,

(33)

2

2

where, we are assuming n even, and sp( n2 ) is the Lie algebra of symplectic n × n matrices i.e.

the Lie algebra of skew-Hermitian matrices A satisfying

AJ + JAT = 0.

(34)

The matrix J is deﬁned as

0 In

J :=

2.

(35)

−I n 0

2

The corresponding Cartan involution θII is given by

θII (A) := J A¯J −1 = −J A¯J.

(36)

AIII. In this case K is the set of all the skew-Hermitian matrices A of the form

R0

A=

,

(37)

0S

where R ∈ u(p), S ∈ u(q), p, q > 0, p + q = n and T r(R) + T r(S) = 0. P is equal to K ⊥. The corresponding Cartan involution is given by

θIII (A) := Ip,qAIp,q,

(38)

Ip×p

0

where the matrix Ip,q is deﬁned as the block matrix Ip,q :=

.

0 −Iq×q

Several authors have proposed Lie algebra decompositions for su(n) that, although special

cases of the general Cartan decomposition, are of particular signiﬁcance in some contexts. For

example, Khaneja and Glaser [12] (see also [4] for the relation of this decomposition with Cartan

decomposition) have factorized unitary evolutions in SU (2n), namely unitary evolution of n two

level quantum systems (qubits), into local operations i.e. operations on only one qubit and two-

qubits operations. This result has consequences both in the study of universality of quantum

logic gates and in control theory. In the latter context, one would like to decompose the task

of steering the evolution operator to a prescribed target into a sequence of steering problems to

intermediate targets with a determined structure.

Another decomposition which is of particular interest to us is the Concurrence Canonical

Decomposition (CCD) of su(2n) which was studied in [5], [6], [7] in the context of entanglement

and entanglement dynamics. In this decomposition, K and P are real span of tensor products,

multiplied by i, of n 2 × 2 matrices chosen in the set {I2×2, σx, σy, σz}, where σx,y,z are the

x, y, z Pauli matrices. In particular, K is spanned by tensor products with an odd number of

7

Pauli matrices and P is spanned by tensor products with an even number of Pauli matrices. It was shown in [5], [6] that for n even this decomposition is a Cartan AI decomposition and for n odd is a Cartan AII decomposition. One of the primary goals of the present paper is to extend the CCD to the case of multipartite systems of arbitrary dimensions. The CCD was also used in [1], [2] to characterize the input-output equivalent models of networks spin 12 , in a problem motivated by parameter identiﬁcation for spin Hamiltonians. Generalizations of these results for networks of spins of any value, in view of the results presented here, will be given in a forthcoming paper [3].

3 Relation between Cartan decompositions and symmetries

The results of [5], [6] associate to the Concurrence Canonical Decomposition a time reversal symmetry. In particular, there is a relation between the involution θ corresponding to the CCD and the time reversal symmetry Θ in (21) (with N = n the number of spin assumed all equal to 21 ). This relation is given by

θ(A) = ΘAΘ−1, ∀A ∈ su(2n),

(39)

where the right hand side needs to be interpreted as composition of operators. It is also easily seen, using only the fact that the time reversal symmetry is antiunitary and the general formula (7), that, if θ¯ is the time reversal symmetry on observables iA, we have

θ¯(iA) := ΘiAΘ−1 = −iΘAΘ−1 = −iθ(A).

(40)

This rises the question of whether there is in general a one to one correspondence between symmetries Θ, θ¯ (8), and Cartan involutions θ and therefore Cartan decompositions. Also the question arises on whether formula (see (40))

θ¯(iA) = −iθ(A), ∀A ∈ u(n)

(41)

is always valid. We shall investigate these issues in this section. We shall see that only a particular class of symmetries, which we call Cartan symmetries give rise to Cartan involutions.

Deﬁnition 3.1 A symmetry Θ is called a Cartan symmetry if and only if Θ2 is equal to the identity up to a phase factor.

Cartan symmetries have the property that applied two times to any state return the physical state unchanged. For example the time reversal symmetry and the parity (11) are Cartan symmetries while the space translation symmetry (10) is not a Cartan symmetry.

8

Whether or not a symmetry is a Cartan symmetry can be veriﬁed once we have its representation in a given basis i.e. (cf. (7))

Θ|α >= XK|α >,

(42)

where X is unitary and K is the identity if Θ is a unitary symmetry and is the conjugation of all the components of |α > if Θ is antiunitary. Θ is a Cartan symmetry if and only if

XX¯ = eiφIn×n,

(43)

for some φ ∈ RI in the antiunitary case and X2 = eiφIn×n for some φ ∈ RI in the unitary case. This is clearly of the particular orthonormal basis chosen. If Θ is antiunitary and T is a unitary transformation which transforms one orthonormal basis into another and XK describes the action of the symmetry in one basis then T XT¯∗K describes the action of the symmetry in the new basis. It is easily seen that if X satisﬁes (43) so does T XT¯∗ and an analogous fact holds for unitary symmetries.

Generalizing the approach in [5] [6] we now give the following deﬁnition.

Deﬁnition 3.2 The transformation induced by a symmetry Θ on su(n) is deﬁned as

θ(A) := ΘAΘ−1.

(44)

Notice this deﬁnition is analogous to the one of symmetries θ¯ on observables (8) which we repeat here with diﬀerent notations:

θ¯(iA) := ΘiAΘ−1, ∀A ∈ su(n).

(45)

In order to give an expression of the induced transformation in a given basis, we consider the antiunitary and the unitary case separately. In the antiunitary case, if K is the conjugation Θ = XK, Θ−1 = X¯ ∗K = KX∗, θ which gives

θ(A) = XA¯X∗.

(46)

Analogously, one obtains

θ(A) = XAX∗,

(47)

in the unitary case.

Theorem 1 The transformation θ on su(n) induced by a symmetry (Θ, θ¯) is a Cartan involution if and only if (Θ, θ¯) is a Cartan symmetry. Moreover, if (Θ, θ¯) is antiunitary, we have

∀A ∈ su(n),

θ¯(iA) := −iθ(A).

(48)

9

Moreover, if (Θ, θ¯) is unitary, we have ∀A ∈ su(n),

θ¯(iA) := iθ(A).

(49)

Proof. It is easily veriﬁed that θ deﬁned in (46) or (47) is a homomorphism. Moreover, assume

Θ is a Cartan symmetry. Then we calculate (in the antiunitary case and analogously in the

unitary case)

θ2(A) = X(XA¯X∗)X∗ = XX¯ AX¯ ∗X∗ = A,

(50)

where in the last equality we have used the fact that Θ is a Cartan symmetry. Therefore the associated θ is a Cartan involution.

Conversely consider a Cartan involution θ on su(n), induced by a symmetry Θ. Then we want to show that Θ is a Cartan symmetry.

Since θ must be of the type AI, AII or AIII, we must be able to write it as (32), (36) or (38) up to conjugacy. In particular there exists a unitary T such that (case AI)

θ(B) = T T¯∗B¯T¯T ∗, ∀B ∈ su(n),

(51)

or such that (case AII)

θ(B) = T J T¯∗B¯T¯J −1T ∗, ∀B ∈ su(n),

(52)

or such that (case AIII)

θ(B) = T Ip,qT ∗BT Ip,qT ∗,

(53)

in the AIII case 4 We take Θ in the cases AI, AII and AIII given by (cf. (46) and (47))

Θ = T T¯∗K,

(54)

Θ = T JT¯∗K,

(55)

and

Θ = T Ip,qT ∗,

(56)

respectively. It is easily veriﬁed that these are all Cartan symmetries, i.e. XX¯ = I with X = T T¯∗, X = T J T¯∗ and X2 = In×n with X = T Ip,qT ∗. Moreover the choice is unique, up to a

phase factor which does not change the property of the symmetry of being a Cartan symmetry,

as the set of matrices su(n) is an irreducible set of skew-Hermitian operators. This concludes

the proof of the theorem.

2

4In the case AI in appropriate coordinates the involution is equal to conjugation. If T is the matrix that makes the change of coordinates, every B ∈ su(n) can be written as B = T AT ∗ for a unique A in su(n) and therefore A = T ∗BT . Now θI (B) = T A¯T ∗ and replacing A = T ∗BT , one obtains (51). The other cases are analogous.

10

Domenico D’Alessandro1 and Francesca Albertini2

Abstract We investigate the relation between Cartan decompositions of the unitary group and discrete quantum symmetries. To every Cartan decomposition there corresponds a quantum symmetry which is the identity when applied twice. As an application, we describe a new and general method to obtain Cartan decompositions of the unitary group of evolutions of multipartite systems from Cartan decompositions on the single subsystems. The resulting decomposition, which we call of the odd-even type, contains, as a special case, the concurrence canonical decomposition (CCD) presented in [5],[6],[7] in the context of entanglement theory. The CCD is therefore extended from the case of a multipartite system of n qubits to the case where the component subsystems have arbitrary dimension.

Keywords: Lie groups decompositions, Quantum symmetries, Quantum multipartite systems.

1 Introduction

Decompositions of Lie groups have been extensively used in control theory to design control algorithms for bilinear, right invariant, systems with state varying on a Lie group. Once it is known how to factorize a target ﬁnal state Xf as the product

Xf = X1X2 · · · Xr,

(1)

then the task of controlling to Xf can be reduced to the (simpler) task of controlling to the factors X1, ..., Xr. In quantum information theory, a factorization of the type (1) can be interpreted as the implementation of a quantum logic operation with a sequence of elementary operations. In this case, the relevant Lie group is the Lie group of unitary matrices of dimensions n, U (n). In general, a decomposition of the unitary evolution operator of the form (1) is useful to determine several aspects of the dynamics of quantum systems including the degree of entanglement (see e.g. [16]), time optimality of the evolution [11] and constructive controllability (see e.g. [8], [14]).

Most of the studies presented so far, which involve Lie group decompositions applied to the quantum systems, are concerned with low dimensional systems. For these systems, several

1Department of Mathematics, Iowa State University, Ames, IA 50011, U.S.A., Tel.:+1-515-2948130, Fax: +1515-2945454, e-mail: [email protected]

2Department of Pure and Applied Mathematics, University of Padova, Via Belzoni 7, 35100 Padova, Italy, Tel.: +39-49-8275966, Fax: +39-49-8275892, e-mail:[email protected]

1

complete and elegant results can be obtained, which also have important physical implications. Decompositions of the unitary group U (n) for large n exist and can be applied to the dynamical analysis of high dimensional quantum systems. However, the information obtained with this study is rarely as useful and of direct physical interpretation as in the low dimensional cases. For multipartite systems, this motivates the search for Lie group decompositions constructed in terms of decompositions on the single subsystems. We shall construct such type of decomposition in the present paper.

The main motivation for the study presented here was given by the recent papers [5] [6], [7]. In these papers, a decomposition of U (2n) called the Concurrence Canonical Decomposition was obtained for a quantum system of n two level systems (qubits). Such a decomposition has the above mentioned feature of being expressed in terms of elementary decompositions on the single qubit subsystems. It is related to time reversal symmetry and this raises the question of what in general the relation is between quantum mechanical symmetries and decompositions. As we shall see here, the answer to this fundamental question is instrumental in developing a general method to construct decompositions of multipartite systems from elementary decompositions of the single subsystems. We shall develop a decomposition which we call of the ‘odd-even type’ that contains the concurrence canonical decomposition as a special case.

The paper is organized as follows. In Section 2 we review the basic deﬁnitions and results concerning discrete quantum symmetries and Cartan decompositions of the Lie algebra su(n) and therefore the Lie group SU (n). We shall stress the important result that, up to conjugacies, there are only three types of Cartan decompositions which are usually labeled as AI, AII and AIII. In Section 3, we investigate the relation between Cartan decompositions and quantum symmetries and establish a one to one correspondence between Cartan decompositions and a subclass of symmetries which we call Cartan symmetries. To every Cartan decomposition of the Lie algebra u(n) and corresponding Cartan symmetry there corresponds a decomposition of the Jordan algebra of Hermitian matrices of dimension n, iu(n) equipped with the anticommutator operation. This is described in Section 4. This is also the crucial fact used to develop the general decomposition of the odd-even type for multipartite systems in arbitrary dimensions in Section 5. This decomposition is a Cartan decomposition and, in Section 6, we show how to determine its type (AI or AII). The Cartan decomposition also leads to a decomposition of the evolution of any quantum system into the product of an evolution with antisymmetric Hamiltonian and one with symmetric Hamiltonian with respect to a Cartan symmetry. This result is discussed in Remark 4.1.

2

2 Background material

2.1 Discrete Symmetries in Quantum Mechanics

Given a quantum system with underlying Hilbert space H, a quantum mechanical symmetry is deﬁned (see e.g. [9] Chapter 7, [13] Chapter 4) as a one to one and onto map Θ : H → H such that physically indistinguishable states are also mapped into physically indistinguishable states i.e. for every |ψ >∈ H and φ1 ∈ RI ,

Θ(eiφ1 |ψ >) = eiφ2 Θ(|ψ >),

(2)

for some φ2 ∈ RI . Moreover Θ preserves the inner product of two states, namely if |α˜ >:= Θ|α >, then, for any two states |α > and |β >

| < α˜|β˜ > | = | < α|β > |.

(3)

In this deﬁnition, we omit for simplicity the consideration of selection rules and assume that all the states are physically realizable. A detailed discussion of this point can be found in [9].

According to Wigner’s theorem [15], every such operation Θ can be represented as

Θ = eiφU,

(4)

where φ is a constant, physically irrelevant, real parameter, and U is either a unitary operator

or an anti-unitary one. Recall that an anti-unitary operator U , |α >→ |α˜ >:= U |α > is deﬁned

as satisfying

< β˜|α˜ >=< β|α >∗,

(5)

U (c1|α > +c2|β >) = c∗1U |α > +c∗2U |β > .

(6)

Once a basis of the Hilbert space H is chosen, an anti-unitary operator U can always be

written as

U |α >= XK|α >,

(7)

where K is the operation which conjugates all the components of the vector |α > and X is

unitary.

A symmetry Θ, whether unitary or anti-unitary, induces a transformation on the space of

Hermitian operators A as

A → ΘAΘ−1 := θ¯(A).

(8)

It is in fact easily veriﬁed that θ¯(A) is a linear and Hermitian operator. Moreover the eigenvalues of θ¯(A) are the same as those of A and a set of orthonormal eigenvectors are given by Θ|αj >,

3

where |αj > is an orthonormal basis of eigenvectors of A. It can be proved [9], [13], that, up to a phase factor, θ¯(A) := ΘAΘ−1 is the only choice that guarantees

| < α˜|θ¯(A)|β˜ >= | < α|A|β > |.

(9)

Description of the symmetry Θ is usually done by specifying how θ¯ acts on Hermitian operators rather than how Θ acts on states. This is because, Hermitian operators represent physical observables and therefore the action of θ on observables is typically suggested by physical considerations. For example, the space translation symmetry has to be such that

θ¯(xˆ) = xˆ − a,

(10)

for some constant a where xˆ is the position operator. As another example, the parity or space inversion symmetry is deﬁned such that

θ¯(xˆ) = −xˆ.

(11)

On the other hand, speciﬁcation of θ¯ on an irreducible set of observables uniquely determines Θ up to a phase factor [9]. Recall that an irreducible set of observables {Aj} is deﬁned such that if an observable B commutes with all of the {Aj}, then B is a multiple of the identity.

An observable H is said to satisfy a symmetry Θ or to be symmetric with respect to Θ if

θ¯(H) = H,

(12)

or equivalently

ΘH = HΘ.

(13)

It is said to be antisymmetric with respect to Θ if

θ¯(H) = −H ↔ ΘH = −HΘ.

(14)

A special type of symmetry is the time reversal symmetry. In classical mechanics a time reversal symmetry changes a system into one which evolves with time reversal trajectories. This suggests to deﬁne a time reversal symmetry in quantum mechanics so that θ¯ acts on the position xˆ and momentum operator pˆ according to

θ¯(xˆ) = xˆ,

(15)

θ¯(pˆ) = −pˆ.

(16)

This implies that the corresponding Θ transforms momentum eigenvectors |p > as

Θ|p >= | − p > .

(17)

4

If the system under consideration has no spin degree of freedom, then xˆ and pˆ form an irreducible set of observables and therefore (15) and (16) uniquely specify the transformation Θ on the state. Moreover, from the deﬁnition of angular momentum Lˆ := xˆ × pˆ, we obtain

θ¯(Lˆ) = θ¯(xˆ) × θ¯(pˆ) = −Lˆ.

(18)

For a system with spin angular momentum Sˆ, we impose by deﬁnition, according to (18)

θ¯(Sˆ) = −Sˆ,

(19)

and xˆ, pˆ, Sˆ form an irreducible set of observables. If |m > is an eigenvector of the (spin) angular momentum corresponding to eigenvalue m, we have

Θ|m >= | − m > .

(20)

From these speciﬁcations, it is possible to obtain an explicit expression of the time reversal symmetry for a system of N particles with spin operators Sˆ1,...,SˆN . It is given (in a basis of tensor products of the eigenstates of the z− component of the spin operators) by (see [9])

Θ

=

e−

iπ h¯

(

Sˆ1

,y

+

Sˆ2

,y

+

...

+

SˆN

,y

)

K

,

(21)

where Sˆj,y is the y component of the spin operator corresponding to the j−th particle, j = 1, ..., N , and K is the conjugation operator (same as in (7)).

2.2 Cartan involutions and decompositions of su(n)

We discuss next the Cartan decompositions for the Lie algebra su(n). What we say could be generalized to general semisimple Lie algebras. We refer to [10] for more details.

A Cartan decomposition of su(n) is a vector space decomposition

su(n) = K ⊕ P,

(22)

where the subspaces K and P satisfy the commutation relations

[K, K] ⊆ K,

(23)

[K, P] ⊆ P,

(24)

[P, P] ⊆ K.

(25)

In particular, notice that K is a subalgebra of su(n). A Cartan decomposition of su(n) induces a factorization of the elements of the Lie groups SU (n). Let us denote by eL the connected Lie

5

group associated to a generic Lie algebra L. Then, given a Cartan decomposition (22), every element X in SU (n) can be written as

X = KP,

(26)

where K ∈ eK and P is the exponential of an element of P. Moreover if A is a maximal Abelian subalgebra of su(n), with A ⊆ P, then one can prove that

∪K∈eK KAK∗ = P.

(27)

This implies that one can write P in (26) as P = K1AK1∗ with K1 ∈ eK and A ∈ eA. Therefore every element X in SU (n) can be written as

X = K1AK2,

(28)

with K1, K2 ∈ eK and A ∈ eA. This is often referred to as KAK decomposition. A Cartan involution of su(n) is a homomorphism θ : su(n) → su(n) such that θ2 is equal to

the identity on su(n). Associated to a Cartan decomposition (22) is a Cartan involution which is equal to the identity on K and multiplies by −1 the elements of P, i.e.

θ(K) = K, ∀K ∈ K,

(29)

θ(P ) = −P, ∀P ∈ P.

(30)

Therefore, given a Cartan decomposition, relations (29) and (30) determine a Cartan involution

θ. Viceversa given a Cartan involution θ, the +1 and −1 eigenspaces of θ determine a Cartan

decomposition.

According to a theorem of Cartan [10], there exist only three types of Cartan decompositions

for su(n) up to conjugacy. More speciﬁcally, given a Cartan decomposition (22) there exists a

matrix H ∈ SU (n) such that K := HKH∗, P := HPH∗, where K and P fall in one of the

following cases labeled AI, AII and AIII3 .

AI

K = so(n), P = so(n)⊥,

(31)

where so(n) is the Lie algebra of real skew-Hermitian matrices of dimension n, so(n)⊥ is the vector space over the reals of purely imaginary skew-Hermitian matrices. The corresponding Cartan involution, which we denote by θI , returns the complex conjugate of a matrix, i.e.

θI (A) := A¯.

(32)

3In the following deﬁnitions and in the rest of the paper the inner product < A, B > in su(n) is deﬁned as < A, B >:= T r(AB∗) and it is proportional to the Killing form (see e.g. [10])

6

AII

n K = sp( ),

P = sp( n )⊥,

(33)

2

2

where, we are assuming n even, and sp( n2 ) is the Lie algebra of symplectic n × n matrices i.e.

the Lie algebra of skew-Hermitian matrices A satisfying

AJ + JAT = 0.

(34)

The matrix J is deﬁned as

0 In

J :=

2.

(35)

−I n 0

2

The corresponding Cartan involution θII is given by

θII (A) := J A¯J −1 = −J A¯J.

(36)

AIII. In this case K is the set of all the skew-Hermitian matrices A of the form

R0

A=

,

(37)

0S

where R ∈ u(p), S ∈ u(q), p, q > 0, p + q = n and T r(R) + T r(S) = 0. P is equal to K ⊥. The corresponding Cartan involution is given by

θIII (A) := Ip,qAIp,q,

(38)

Ip×p

0

where the matrix Ip,q is deﬁned as the block matrix Ip,q :=

.

0 −Iq×q

Several authors have proposed Lie algebra decompositions for su(n) that, although special

cases of the general Cartan decomposition, are of particular signiﬁcance in some contexts. For

example, Khaneja and Glaser [12] (see also [4] for the relation of this decomposition with Cartan

decomposition) have factorized unitary evolutions in SU (2n), namely unitary evolution of n two

level quantum systems (qubits), into local operations i.e. operations on only one qubit and two-

qubits operations. This result has consequences both in the study of universality of quantum

logic gates and in control theory. In the latter context, one would like to decompose the task

of steering the evolution operator to a prescribed target into a sequence of steering problems to

intermediate targets with a determined structure.

Another decomposition which is of particular interest to us is the Concurrence Canonical

Decomposition (CCD) of su(2n) which was studied in [5], [6], [7] in the context of entanglement

and entanglement dynamics. In this decomposition, K and P are real span of tensor products,

multiplied by i, of n 2 × 2 matrices chosen in the set {I2×2, σx, σy, σz}, where σx,y,z are the

x, y, z Pauli matrices. In particular, K is spanned by tensor products with an odd number of

7

Pauli matrices and P is spanned by tensor products with an even number of Pauli matrices. It was shown in [5], [6] that for n even this decomposition is a Cartan AI decomposition and for n odd is a Cartan AII decomposition. One of the primary goals of the present paper is to extend the CCD to the case of multipartite systems of arbitrary dimensions. The CCD was also used in [1], [2] to characterize the input-output equivalent models of networks spin 12 , in a problem motivated by parameter identiﬁcation for spin Hamiltonians. Generalizations of these results for networks of spins of any value, in view of the results presented here, will be given in a forthcoming paper [3].

3 Relation between Cartan decompositions and symmetries

The results of [5], [6] associate to the Concurrence Canonical Decomposition a time reversal symmetry. In particular, there is a relation between the involution θ corresponding to the CCD and the time reversal symmetry Θ in (21) (with N = n the number of spin assumed all equal to 21 ). This relation is given by

θ(A) = ΘAΘ−1, ∀A ∈ su(2n),

(39)

where the right hand side needs to be interpreted as composition of operators. It is also easily seen, using only the fact that the time reversal symmetry is antiunitary and the general formula (7), that, if θ¯ is the time reversal symmetry on observables iA, we have

θ¯(iA) := ΘiAΘ−1 = −iΘAΘ−1 = −iθ(A).

(40)

This rises the question of whether there is in general a one to one correspondence between symmetries Θ, θ¯ (8), and Cartan involutions θ and therefore Cartan decompositions. Also the question arises on whether formula (see (40))

θ¯(iA) = −iθ(A), ∀A ∈ u(n)

(41)

is always valid. We shall investigate these issues in this section. We shall see that only a particular class of symmetries, which we call Cartan symmetries give rise to Cartan involutions.

Deﬁnition 3.1 A symmetry Θ is called a Cartan symmetry if and only if Θ2 is equal to the identity up to a phase factor.

Cartan symmetries have the property that applied two times to any state return the physical state unchanged. For example the time reversal symmetry and the parity (11) are Cartan symmetries while the space translation symmetry (10) is not a Cartan symmetry.

8

Whether or not a symmetry is a Cartan symmetry can be veriﬁed once we have its representation in a given basis i.e. (cf. (7))

Θ|α >= XK|α >,

(42)

where X is unitary and K is the identity if Θ is a unitary symmetry and is the conjugation of all the components of |α > if Θ is antiunitary. Θ is a Cartan symmetry if and only if

XX¯ = eiφIn×n,

(43)

for some φ ∈ RI in the antiunitary case and X2 = eiφIn×n for some φ ∈ RI in the unitary case. This is clearly of the particular orthonormal basis chosen. If Θ is antiunitary and T is a unitary transformation which transforms one orthonormal basis into another and XK describes the action of the symmetry in one basis then T XT¯∗K describes the action of the symmetry in the new basis. It is easily seen that if X satisﬁes (43) so does T XT¯∗ and an analogous fact holds for unitary symmetries.

Generalizing the approach in [5] [6] we now give the following deﬁnition.

Deﬁnition 3.2 The transformation induced by a symmetry Θ on su(n) is deﬁned as

θ(A) := ΘAΘ−1.

(44)

Notice this deﬁnition is analogous to the one of symmetries θ¯ on observables (8) which we repeat here with diﬀerent notations:

θ¯(iA) := ΘiAΘ−1, ∀A ∈ su(n).

(45)

In order to give an expression of the induced transformation in a given basis, we consider the antiunitary and the unitary case separately. In the antiunitary case, if K is the conjugation Θ = XK, Θ−1 = X¯ ∗K = KX∗, θ which gives

θ(A) = XA¯X∗.

(46)

Analogously, one obtains

θ(A) = XAX∗,

(47)

in the unitary case.

Theorem 1 The transformation θ on su(n) induced by a symmetry (Θ, θ¯) is a Cartan involution if and only if (Θ, θ¯) is a Cartan symmetry. Moreover, if (Θ, θ¯) is antiunitary, we have

∀A ∈ su(n),

θ¯(iA) := −iθ(A).

(48)

9

Moreover, if (Θ, θ¯) is unitary, we have ∀A ∈ su(n),

θ¯(iA) := iθ(A).

(49)

Proof. It is easily veriﬁed that θ deﬁned in (46) or (47) is a homomorphism. Moreover, assume

Θ is a Cartan symmetry. Then we calculate (in the antiunitary case and analogously in the

unitary case)

θ2(A) = X(XA¯X∗)X∗ = XX¯ AX¯ ∗X∗ = A,

(50)

where in the last equality we have used the fact that Θ is a Cartan symmetry. Therefore the associated θ is a Cartan involution.

Conversely consider a Cartan involution θ on su(n), induced by a symmetry Θ. Then we want to show that Θ is a Cartan symmetry.

Since θ must be of the type AI, AII or AIII, we must be able to write it as (32), (36) or (38) up to conjugacy. In particular there exists a unitary T such that (case AI)

θ(B) = T T¯∗B¯T¯T ∗, ∀B ∈ su(n),

(51)

or such that (case AII)

θ(B) = T J T¯∗B¯T¯J −1T ∗, ∀B ∈ su(n),

(52)

or such that (case AIII)

θ(B) = T Ip,qT ∗BT Ip,qT ∗,

(53)

in the AIII case 4 We take Θ in the cases AI, AII and AIII given by (cf. (46) and (47))

Θ = T T¯∗K,

(54)

Θ = T JT¯∗K,

(55)

and

Θ = T Ip,qT ∗,

(56)

respectively. It is easily veriﬁed that these are all Cartan symmetries, i.e. XX¯ = I with X = T T¯∗, X = T J T¯∗ and X2 = In×n with X = T Ip,qT ∗. Moreover the choice is unique, up to a

phase factor which does not change the property of the symmetry of being a Cartan symmetry,

as the set of matrices su(n) is an irreducible set of skew-Hermitian operators. This concludes

the proof of the theorem.

2

4In the case AI in appropriate coordinates the involution is equal to conjugation. If T is the matrix that makes the change of coordinates, every B ∈ su(n) can be written as B = T AT ∗ for a unique A in su(n) and therefore A = T ∗BT . Now θI (B) = T A¯T ∗ and replacing A = T ∗BT , one obtains (51). The other cases are analogous.

10