SYMMETRY REDUCTIONS OF LINDBLADEQUATIONS - SIMPLE EXAMPLESAND APPLICATIONS Текст научной статьи по специальности «Физика»

УДК 530.145, 512.81

DOI 10.19110/1994-5655-2021-6-49-52

А.А. КАРАБАНОВ

РЕДУКЦИИ СИММЕТРИИ УРАВНЕНИЙ ЛИНДБЛАДА - ПРОСТЫЕ ПРИМЕРЫ

И ПРИЛОЖЕНИЯ

Школа физики и астрономии Ноттингемского университета, Великобритания

[email protected]

A.A. KARABANOV

SYMMETRY REDUCTIONS OF LINDBLAD EQUATIONS - SIMPLE EXAMPLES AND APPLICATIONS

School of Physics and Astronomy, University of Nottingham,

UK

Аннотация

Открытая квантовая динамика в марковском приближении описывается основным уравнением Линдблада. Динамика Линдблада замкнута в алгебре Ли Л = su(n), т.е. имеет su(n) симметрию. Мы говорим, что уравнение Линдблада допускает редукцию симметрии, если оно имеет инвариантное векторное подпространство Ло С Л с Ли-алгебраической структурой. Редукции симметрии ограничивают динамику на меньшие подпространства, которые дополнительно являются алгебрами Ли.

В заметке описаны тривиальные редукции, основанные на приводимости гамильтониана и операторов Линдблада. Представлены примеры нетривиальных редукций в пределе бесконечной температуры и редукций Майораны с сохранением четности. Обсуждаются приложения к открытой спиновой динамике.

Ключевые слова:

открытые квантовые системы, уравнение Линдблада, редукция симметрии

Abstract

Open quantum dynamics in the Markovian approximation is described by the Lindblad master equation. The Lindbladian dynamics is closed in the Lie algebra Л = su(n), i.e. it has su(n) symmetry. We say that the Lindblad equation admits a symmetry reduction if it has an invariant vector subspace Ло С Л with the Lie algebraic structure. Symmetry reductions restrict dynamics to smaller subspaces that additionally are Lie algebras.

In these notes, trivial reductions relying on the reducibility of the Hamiltonian and Lindblad operators are described. Examples of nontrivial reductions in the infinite temperature limit and the parity preserving Majorana reductions are presented. Applications to open spin dynamics are discussed.

Keywords:

open quantum systems, Lindblad master equation, symmetry reduction

Introduction

The open quantum dynamics in terms of the positive density operator in the Markovian approximation is described by the Lindblad master equation [1]

p = Mp =-i[H,p] + Dp, D

1

m

E

k=i

Yk L(Vk),

(1)

L(V)p = VpVf - 2 (V]Vp + pVW)

Here H is the Hamiltonian, D is the dissipator built with the traceless Lindblad operators Vk, Tr Vk = 0, and the non-negative rates Yk ^ 0.

The density operator has trace 1 and at any time t is written in the form

p(t) = n-1I + po(t), Tr po(t) = 0, po(t) = po(t)t

where n is the dimension of the Hilbert space, I is the unit operator, p0(t) is the traceless Hermitian operator. The first term does not change in time. The vector space A of all possible traceless parts generate the Lie algebra (with the usual commutation of operators) that is isomorphic to su(n), the algebra of traceless anti-Hermitian n x n operators. Indeed, multiplying traceless Hermitian

operators by the complex unit, we come to traceless anti-Hermitian operators. Thus, the Lindbladian dynamics of Eq. (1) is closed in the Lie algebra A = su(n), i.e., it has the su(n) symmetry.

We say that Eq. (1) admits a symmetry reduction if it has a smaller invariant vector space A0 C A with a Lie algebraic structure. In other words, A0 is a Lie algebra (with the usual commutator of operators) and any trajectory that starts in A0 remains there for any times,

[Ло, Ло] С Ло, ро(0) е Лс

Vt po(t) e Ao.

Obviously, A0 is a subalgebra of the total symmetry algebra su(n). Symmetry reductions restrict the dynamics into smaller invariant subspaces that additionally have a Lie algebraic structure.

Since all initial conditions within A0 generate trajectories that stay within A0 for all times, it is necessary that the action of the superoperator M to the unit operator belongs to A0 and A0 is invariant under the action of M,

MI e A0, MA0 C A0. (2)

Eq. (2) gives the criterion for the subalgebra A0 to be a symmetry reduction.

In particular, the full Krylov subspace generated by the powers MkI is within any symmetry reduction algebra A0. The subspace Ki and so all symmetry reductions contain also the 1-dimensional subspace spanned by the equilibrium state (perhaps not unique),

span {pq} e A0, n-1 MI + Mp*0 = 0. (3)

Thus, for existence of symmetry reductions it is necessary that the subspace Ki is a proper subspace of the total algebra su(n),

dim Ki < n2 — 1, K! = span[MkI, k =1, 2, ...}.

(4)

Indeed, the Krylov subspace Ki is an invariant subspace of Eq. (1), i.e., trajectories starting in Ki remain there all the time. In general, dim Ki = n2 — 1, the Krylov subspace coincides with the total symmetry algebra su(n), Eq. (1) does not have proper invariant subspaces and hence does not admit symmetry reductions.

Eqs. (2), (3), (4) show that the Hamiltonian and Lindblad operators should satisfy special conditions for Eq. (1) to have symmetry reductions. In these notes, we discuss first trivial symmetry reductions relying on splitting the Hilbert space by reducibility of the Hamiltonian and Lindblad operators. Then we present two examples of nontrivial symmetry reductions: the reduction to the infinite temperature limit MI = 0 and the reduction by the parity Z2-grading of the total algebra su(n) realised as a Majorana reduction. In the first example, the symmetry reduction is due to a constraint to the dissipation rates 7k. The second example is valid for any dissipation rates. The relevant applications to open spin dynamics are pointed out and briefly discussed.

1. Trivial reduction

The Hamiltonian is an Hermitian operator. The set of Lindblad operators (even in useful physical models) is typically not very large and is subdivided into a set

of Hermitian operators and a set of Hermitian-conjugate pairs,

V5

2q

Vl i = V2q_!

q

1,

mi,

H = HV2m1+p = V2mi+p, P =1,...,m2.

We have then

D = — 2^2 Y2m1+p[V2m1+p, [V2mi+p, ']] +

p=i

mi

(5)

+ £ |72q L(V2q )+ Y2q_iL(Vl )} .

q=i

In particular,

mi

MI = J>2q — Y2q-l)[^2q, ^]. (6)

q=1

Let the set of the Hermitian-conjugate pairs of Lindblad operators V2q, V2iq be reducible, i.e., possesses a common invariant vector subspace X c h of the Hilbert space of a lower dimension,

V2qX С X, V2qX С X, q =1, 0 < dim X = m < n.

, mi ,

Then the Hermitian-conjugate part of the dissipator (given by the second term in Eq. (5)) is closed in the set A0 of all traceless operators p0 e su(n) that preserve the reduced subspace X,

p е Лс

poX С X.

The set A0 is closed with respect to commutation of operators and so forms a Lie algebra. This algebra is isomorphic to the su(n)-normalizer of the algebra su(m) spanned by traceless operators on the reduced subspace X. If additionally the Hamiltonian and the Her-mitian Lindblad operators belong to A0,

V2mi+P, H e A0 = N(su(m)),

then the subalgebra A0 is a symmetry reduction of Eq. (1). Since the complementary subspace Xc = h\X, dim Xc = n — m is also invariant for V2q, , the i construction is applicable to Xc. Hence,

same

Ло

N (su(m)) = N (su(n — m))

where the subalgebra su(n-m) is spanned by traceless operators on the complementary subspace Xc.

We call such symmetry reduction a trivial reduction, as it describes the situation where the initial Hilbert space, being formally n-dimensional, is in fact split into two independent subspaces of lower dimensions m, n— m < n that are not dynamically connected. We have (after a suitable permutation of Hilbert states)

H, Vk e A0 =( A I

dim A = m x m, 0 < m < n.

In the special case where the Lindblad operators are all Hermitian, V2q = 0, q = 1, ..., mi, the symmetry reductions are any subalgebras which normalizers contain the operators V2mi+p, H.

2. Infinite temperature reduction

Let the unit operator annihilate the right-hand side of Eq. (1),

MI =0 (7)

and the operators [V2q,v2iq] are linearly independent. Then it follows from Eq. (6) that

Y2q = Y2q-1 = Yq ■

As a result, according to Eq. (5),

(8)

1

D = " 2 ^ 72m1+p[V2m1+p, [V2mi+p, •]]-

p=1

mi

([Vîq, [V2q, •]] + [^q, [V*q, •]]) ■

(9)

q=i

Let Ao be the subalgebra generated by the Hermitian-conjugate pairs of Lindblad operators and let the Hermitian Lindblad operators and the Hamiltonian belong to the normalizer of Ao,

[V2m1+pi Ao], [H, Ao] C Ao {V2q, V^q}. (10)

It follows then from Eq. (9) that A0 is a symmetry reduction of Eq. (1).

According to Eq. (7), the density operator p = n-1I is an incoherent equilibrium solution to Eq. (1), featuring the case where all the pure states of the system have equal probabilities. This case corresponds to the infinite temperature limit of both Fermi-Dirac and Bose-Einstein statistics that in this case coincide with the corresponding limit of the Boltzmann distribution. For this reason, we call the symmetry reduction given by Eqs. (7), (8), (10) an infinite temperature reduction.

As an example, consider the Markovian Lindblad open dynamics of a single spin in a magnetic field. The Hamiltonian contains the Zeeman splitting and the coherent driving parts,

H = nsz + ^ (S+ + S-).

The Lindblad operators are the usual raising and lowering operators characterizing the longitudinal relaxation plus the z-operator that describes the transverse relaxation,

Vi,

S± = Sx ± iSy, V3 = Sz

We assume that the initial value for the dynamics belongs to the spin algebra so(3) (treated as an isomorphic copy of su(2)),

P(0)

i

I + Pz Sz + P+S+ + p-S-,

(11)

[Sz ,S±] = ±S±, [S+,S-] = 2Sz

where n = 2s+1 is the dimension ofthe Hilbert space, s is the spin quantum number (any half-integer or integer), pz±± are some constants.

Let 71 = 72 = 7, i.e., the Lindblad operators S± have the same dissipation rates. Then it can be easily verified that we are under conditions of the infinite temperature reduction described earlier in this section. For any time t the dynamics remains closed in so(3), i.e., is reduced to a 3-dimensional dynamics of the constants Pz,±,

p(t)

n

i

I + Pz (t)Sz + P+(t)S+ + p- (t)S- ■

The latter is described by the well-known Bloch equations. This result is valid for any spin quantum number s.

Let now s > 1/2 and 71 = 72. Then Eq. (7) no longer holds and the dynamics, even starting in so(3) as in Eq. (11), at t > 0 comes out ofthe spin algebra so(3). Indeed, the (non-driven) thermal equilibrium state is represented now by a diagonal matrix that cannot be represented by a combination of the unit operator and the operator Sz,

Pth = Z 1

exp

hi Sz ~kT

diag{pi, ■ ■ ■, pn} = n I + /3Sz■

As a matter of fact, higher orders Srz, r > 1, of the operator Sz occur that do not belong to so(3). This corresponds to the case of finite temperatures.

It can be shown that in the presence ofthe coherent terms in the Hamiltonian, w1 = 0, and for y1 = y2, the above spin dynamics does not admit symmetry reductions: the dimension of any (non-equilibrium) trajectory equals n2 — 1, the dimension ofthe total symmetry algebra su(n). For w1 = 0 and any y1>2 the dynamics admits the symmetry reduction represented by the algebra of traceless diagonal nxn matrices that is the Cartan subalgebra of su(n).

3. Parity preserving reduction

The total symmetry algebra A = su(n) is a formal superalgebra, i.e., it admits a Z2-grading into even and odd subspaces respected by the commutation,

A = Ao + A1, [Ai, Aj] c A(i+i)(

mod 2) •

Here the even subspace A0 is a subalgebra, while the odd subspace A1 (being not a subalgebra) complements the even subalgebra to the total algebra A.

It can be assumed that the Z2-grading of su(n) is inherited from a Z2-grading ofthe associative algebra of n x n matrices. An example valid for any n is the representation of traceless (anti-)Hermitian operators by matrix elements plj, 1 < l, j < n, that belong to even and odd collateral diagonals,

As = span {plj : l — j = s (mod 2)}, s = 0, 1. (12)

The following result is a simple subsequence of the above construction. If the Hamiltonian belongs to the even subspace, while each Lindblad operator belongs entirely to one of the two grading subspaces,

H e Ao, Vk e Ao or A1,

2

then the even subalgebra A0 is a symmetry reduction of Eq. (1). We call such symmetry reduction a parity preserving reduction.

For large n, the Z2-grading given by Eq. (12) may be not unique. Below we consider an example of parity preserving reduction based on the so-called Majorana fermions.

The Majorana fermions form a set of 2m nx n operators al, l = 1, ..., 2m, that anti-commute with each other and square to the unit operator,

af = I, aiaj + aj ai = 0, l = j.

Consider the even and odd parts of the Clifford algebra generated by the Majorana fermions, i.e., the vector spaces spanned by even and odd order products of the operators al,

2k

Ло = span j JJ^ air, lr = 1, ..., 2m,

r=i

k = 1, ..., m|,

2k i

Л1 = span j

r=i

air, lr = 1, ..., 2m,

k = 1,

, m .

Denote A0, A 1 the subspaces of the above subspaces that contain only traceless operators. The even subspace A0 is closed with respect to the operator commutation and so forms a Lie algebra. It follows from the anti-commutation of the Majorana fermions that the algebra A0 contains the algebra of anti-symmetric quadratic forms of the operators al that is isomorphic to the algebra so(2m) of (generally complex) anti-symmetric operators. Hence, A0 is an extension of so(2m) by all even order products of al.

The above construction suggests that the subdivision A = A0 + A1 is a Z2-grading of the total symmetry algebra. Thus, if the Hamiltonian belongs to A0 and each Lindblad operator belongs to either A0 or A1, then the algebra A0 is a symmetry reduction of Eq. (1). In other words, for any time t the traceless part of the density operator is represented by a combination of even order products of the operators al. We call this a Majorana reduction.

As an example, consider the system of m > 1 interacting spin-1/2 particles (for example, qubits). This system is described by the tensor product of m 4-dimen-sional operator spaces spanned by the set of 2 x 2 operators 1, Sx, Sy, Sz where 1 is the unit operator and Sa,

a = x,y, z, are the standard spin-1/2 operators (of the 2-representation of so(3)). As each spin has two states and the trace of the density operator equals 1, the whole symmetry algebra is Л = su(2m) of dimension 4m — 1.

Denote Skx, Sky, Skz the spin operators generated by the kth spin - that placed in position k of the tensor product: Ska = 1 x ... x 1 x Sa x 1... x 1. These individual operators for different spins k = k' commute and generate the algebra so(3) within their subspaces with the same k. It can be easily verified that the set of 2m operators

a2k-1 = 2k Skx Ssz, s<k

a2k = 2 Sky Ssz, s<k

k = 1, . . . , m,

are Majorana fermions.

Here, the Majorana chain

(13)

2m_i

m_ 1

H = hi [ai, ai+i] = (^lSlz + DiSixSi+i,x)

i=i

i=i

leads to the Hamiltonian of the coherently driven 1-di-mensional Ising model (with respect to the x axis). Another combination

H

2m_i m_i

= hi [ai, ai+i] +

i=i k=i m_ 1

J2 (QiSiz1 nx

i=i

hi [ai, ai+i] + /2 hk [a2k_i, a2k+2] =

+ nf sixsi+i,x + ny Siy Si+i,

leads to the Hamiltonian of the driven anisotropic Heisenberg XY 1-dimensional chain. Adding higher even order Majorana products, we can get interacting spin systems of higher dimensions. In all cases, choosing various combinations of odd or even order Majorana products as the Lindblad operators, we generate various dissipation models.

Other physically important examples of symmetry reductions relying on parity can be suggested.

References

1. Breuer H.P., Petruccione F. The theory of open quantum systems. Oxford University Press, 2010. 636 p.

Статья поступила в редакцию 28.10.2021.