Criteria for dynamically stable decoherence-free subspaces and

PHYSICAL REVIEW A 77, 052301 共2008兲

Criteria for dynamically stable decoherence-free subspaces and incoherently generated coherences Raisa I. Karasik,1,2,3 Karl-Peter Marzlin,3 Barry C. Sanders,3,4 and K. Birgitta Whaley1,2 1

Applied Science and Technology, University of California, Berkeley, California 94720, USA Berkeley Quantum Information Center and Department of Chemistry, University of California, Berkeley, California 94720, USA 3 Institute for Quantum Information Science, University of Calgary, Calgary, Alberta T2N 1N4, Canada 4 Centre for Quantum Computer Technology, Macquarie University, Sydney, New South Wales 2109, Australia 共Received 16 November 2007; published 1 May 2008兲

2

We present a detailed analysis of decoherence free subspaces and develop a rigorous theory that provides necessary and sufficient conditions for dynamically stable decoherence free subspaces. This allows us to identify a special class of decoherence free states which rely on incoherent generation of coherences. We provide examples of physical systems that support such states. Our approach employs Markovian master equations and applies primarily to finite-dimensional quantum systems. DOI: 10.1103/PhysRevA.77.052301

PACS number共s兲: 03.67.Pp, 42.50.Ct

I. INTRODUCTION

The theory of decoherence-free 共or noiseless兲 subspaces 共DFSs兲 provides an important strategy for the passive preservation of quantum information 关1–4兴 and has been subjected to successful but restricted experimental tests 关5–10兴. DFS theory has been developed in the context of algebraic symmetries in the interaction Hamiltonian by Zanardi and Rasetti 共ZR兲 关2,11兴 and by Lidar, Bacon, and Whaley 关12兴, and in the context of semigroup dynamics via quantum master equations by Lidar, Chuang, and Whaley 共LCW兲 关3兴 and by Shabani and Lidar 关13兴. The latter analysis has been extended to the theory of DFS in the non-Markovian regime 关13,14兴 and to DFS with imperfect initialization 关13兴. Analogs of DFS have also been proposed in the quantum channel formulation 关15兴. Potential advantages of DFS include quantum circuit simplifications by reducing the demands for quantum error correction and for quantum memory and transport. Given the significance of DFS for realization of quantum information processing, it is important that the foundations of the theory are rigorous and unambiguous, especially in preparation of future experiments to test and exploit the DFS. DFS have been defined as collections of states that undergo unitary evolution in the presence of couplings to the environment 共i.e., decoherence effects兲. However, unitary evolution of a quantum state can arise in a number of ways and this fact has resulted in the development of related, but different definitions of DFS in the literature. In the context of Markovian master equations, DFS have frequently been defined as a collection of states for which dissipative 共decoherence兲 part of the Markovian master equation is zero. In this paper, we give a detailed analysis of DFS based on Markovian semigroup master equations defined for finitedimensional Hilbert spaces. We provide a definition for dynamically stable DFS that identifies a new subclass of DFS states. This definition allows us to develop a theorem that gives necessary and sufficient conditions for the existence of dynamically stable DFSs and that provides a constructive protocol for determining the composition of all DFSs. The existence of this class of DFS states is based on the rather counterintuitive phenomenon that the decoherence 1050-2947/2008/77共5兲/052301共12兲

term, which is of Lindblad type, can generate coherences between these states and other states which are then exactly cancelled by the system Hamiltonian. This is in striking contradiction to the usual role of the decoherence term, which includes damping of coherences rather than cancellation of these. This phenomenon is different from techniques to suppress decoherence like dynamical decoupling 关16–19兴 and continuous error correction 关20,21兴 because it does not rely on the relative time scale for the system evolution and decoherence processes. We show that the existence of these incoherently generated coherences results from an invariance of the Markovian master equation under a specific transformation and provide two examples of physical systems that support these special decoherence-free 共DF兲 states. We also indicate that this phenomenon is rare and does not exist for finite-dimensional systems subject to common quantum noise processes such as spontaneous emission, thermal excitation, and dephasing. II. BACKGROUND

Our analysis of the DFS is predicated on treating the open system as Markovian and thus using Markovian master equations. A system S with state vectors in finite-dimensional Hilbert space HS is coupled to a reservoir R with state vectors in Hilbert space HR. The system-plus-reservoir 共S + R兲 dynamics are fully described by the Hamiltonian ˆ +H ˆ ˆ =H ˆ 丢 1ˆ + 1ˆ 丢 H H S R S R int

共2.1兲

ˆ , and H ˆ the system, reservoir, and interaction ˆ ,H with H S R int Hamiltonians, respectively, and 1ˆ the identity operator. We are interested in the evolution of the system’s reduced state ␳, which is obtained by tracing over the reservoir’s degrees of freedom 共subject to appropriate assumptions: no initial correlations between S and a Markovian R and weak S-R coupling兲. Dynamics of the reduced density matrix ␳ deˆ + ⌬ˆ , with ˆ =H pends on the effective-system Hamiltonian H S eff ˆ ⌬ the Hermitian contribution that stems from the interaction between the quantum system and the reservoir. The evolution

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KARASIK et al.

of the reduced density matrix ␳ also depends on the decoherence superoperator N2 −1

1 S LD关␳兴 = 兺 akl共关Fˆk, ␳Fˆ†l 兴 + 关Fˆk␳,Fˆ†l 兴兲, 2 k,l=1

共2.2兲

where A = 共akl兲 is a positive semidefinite and timeindependent matrix and NS = dim共HS兲. The set of operators F = 兵Fˆk ;k = 0, . . . ,N2S − 1其,

Fˆ0 ⬅ 1ˆ ,

共2.3兲

is a basis for the space of all linear transformations defined on HS. The evolution of ␳ is given by the master equation ˆ , ␳兴 + L 关␳兴. ␳˙ = − i关H eff D

共2.4兲

Henceforth, we refer to Eq. 共2.4兲 as Markovian open system dynamics. Equation 共2.4兲 is of Lindblad form 关22,23兴. This means that Eq. 共2.4兲 generates a completely positive trace preserving map and thereby ensures that the solution to the master equation has all the required properties of a physical density matrix at all times. ˆ and L 关␳兴 in Eq. 共2.4兲 We note here that the terms ⌬ D originate from the system-reservoir coupling and induce a modified unitary evolution and decoherence in system S. In this paper we are interested in characterizing states that undergo purely unitary time evolution.

Remark 1. We require states to be pure in Definition 1, i.e., ␳ = 兩␺典具␺兩 and ␳2共t兲 = ␳共t兲, in order to ensure that all states satisfying this definition do indeed undergo unitary evolution. For mixed states, it is possible to find states that are formally decoherence-free 共i.e., LD关␳共t兲兴 = 0 for all t兲, yet do not undergo a proper unitary dynamics. Examples of such states include the completely mixed state ␳共t兲 = 1ˆ S / NS in a system that decoheres through dephasing only, as well as many thermal states. The reason that these states are not decohering according to definition 1 is that the incoherent transition rates between any two states in the mixture are equal, without the dynamics of each being unitary. Such states contain minimal information and are not useful for the purpose of quantum information processing. Hence we focus in this paper on pure states. Our results can be generalized to specific mixtures of pure decoherence-free states 关3兴 but not to a thermal state. This definition of a restricted DFS implies that a pure state ␳共t兲 would remain pure during its evolution and thus be an ideal tool to process quantum information. It is therefore important to find criteria for the existence of a restricted DFS in a given quantum system. We provide such criteria in Sec. III B below. Before deriving these criteria, we first discuss the conditions for an instantaneous DFS for which LD关␳共t兲兴 = 0 at a fixed time t, tightening existing conditions that have been derived previously in the Markovian setting 关3兴. A. Instantaneous decoherence-free states

III. RESTRICTED DECOHERENCE-FREE SUBSPACES

DFSs have been defined as collections of states that evolve in a unitary fashion 关2–4,11–13,25兴. We shall refer to the complete set of such states as a dynamically stable DFS. From now on we will always assume that the state of the system ␳共t兲 satisfies Eq. 共2.4兲. In this section, we characterize the restricted subset of DFS states ␳ fulfilling LD关␳共t兲兴 = 0 for all t. Conditions for the more general definition of all states evolving in a unitary fashion will be considered in Sec. IV. The traditional approach to DFSs in the context of Markovian master equations requires that LD关␳共t兲兴 = 0 关3兴. This constitutes a condition for an instantaneous DFS, i.e., a DFS at a specific time t. However, satisfying the condition LD关␳共t兲兴 = 0 at one specific time alone, i.e., having an instantaneous DFS, does not guarantee unitary evolution. The acˆ modifies the tion of the effective system Hamiltonian H eff density matrix ␳共t兲 and thus at a later time t⬘ ⬎ t, ␳共t⬘兲 might no longer describe a pure decoherence-free state. The condition LD关␳共t兲兴 = 0 for all t further ensures that state ␳共t兲 undergoes pure unitary evolution and thus extends the notion of an instantaneous DFS. Nevertheless, as we will demonstrate explicitly in Sec. V, this condition is sufficient, but not necessary, for the existence of a DFS. Therefore we shall refer to the set of states satisfying this sufficiency condition as a restricted DFS. Definition 1. Let the time evolution of a state ␳共t兲 be given by Markovian open system dynamics. Then a state ␳共t兲 is a pure restricted decoherence-free state if ␳2共t兲 = ␳共t兲 and LD关␳共t兲兴 = 0 , ∀ t.

The condition for existence of instantaneous DFS states has been previously studied by LCW, who formulated the following theorem which we restate here and then tighten. Theorem 1 (LCW theorem 关3兴). A necessary and sufficient condition for generic decoherence-free dynamics 共i.e., ˜ = span关兵兩i典其兴其 of the register HilLD关˜␳兴 = 0兲 in a subspace 兵H bert space is that all basis states 兩i典 are degenerate eigenstates of all the error generators F: Fˆk兩i典 = ck兩i典 ∀ k,

i.

共3.1兲

If the error generators 兵Fˆk其 can be closed as a semisimple Lie ˜ are annihilated by all error genalgebra, then all states in H ˆ erators F, i.e., Fk兩i典 = 0兩i典 ∀ k , i. We first identify and remedy shortcomings corresponding to the following two examples concerning: 共i兲 zero eigenvalues for the matrix 共akl兲 in Eq. 共2.2兲 and 共ii兲 nonzero eigenvalues ck for the error generators in Eq. 共3.1兲. These two examples demonstrate the associated difficulties with this description. Example 1. Consider a three-level system with basis 兵兩0典 , 兩1典 , 兩2典其 with vanishing Hamiltonian Hef f and LD关␳兴 of the form 共2.2兲 with NS = 3. A is an 8 ⫻ 8 matrix because there are N2S − 1 = 8 error generators in this example. We choose the components of A as akl = 1; for k , l = 1 , 2 and zero otherwise, so that only the error generators Fˆ1 = 兩0典具1兩 and Fˆ2 = 兩0典具2兩 are relevant. We can rewrite A as

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†

A = B⌳B ,

B=

1

冑2

冢

1 1 1 1 O

O

冑21

冣 冢 冣 2 0

,

Jˆ1 = 兩1典具1兩,

O

, ⌳= 0 0 O O 共3.2兲

where O denotes submatrices with all components equal to zero and 1 is a 6 ⫻ 6 identity matrix. Obviously A possesses zero eigenvalues. The state 兩␺典 = 2−1/2共兩1典 − 兩2典兲 is in a DFS because LD关兩␺典具␺兩兴 = 关Heff,兩␺典具␺兩兴 = 0.

NS2−1

blkFˆk 兺 k=1

共3.4兲

that transform Eq. 共2.2兲 to the “diagonalized” decoherence superoperator

LD关兩0典具0兩兴 = −

LD关␳兴 = 2共关Jˆ1, ␳Jˆ†1兴 + 关Jˆ1␳,Jˆ†1兴兲

=

兩0典具1兩 + 兩0典具2兩

冑2

.

⌫ˆ ⬅ 兺 ␭lJˆ†l Jˆl .

共3.5兲

共3.6兲

with

冑2

共3.9兲

M

where we have relabeled the eigenvalues 兵␭l其 and operators 兵Jˆl其 so that ␭l ⬎ 0 for l = 1 , . . . , M ⱕ NS2 − 1 and ␭l = 0 otherwise. In this form the zero eigenvalues of A do not contribute to LD关␳兴. In the above example we then have

Fˆ1 + Fˆ2

兩0典具1兩 + 兩1典具0兩 ⫽ 0. 2

Hence 兩0典 does not exhibit decoherence-free dynamics, yet it is an eigenstate of all the error generators. This example shows that, in addition to excluding zero eigenvalues of A, a second condition for the eigenstates of error generators Jˆl corresponding to nonzero eigenvalues is required in order to tighten the LCW theorem. Here we establish a simple and complete criterion for identifying whether a given state is an instantaneous DFS for a general decoherence process described by a Markovian master equation. We begin with a definition of the decoherence operator ⌫ˆ for a system having decoherence superoperator LD关␳兴, Eq. 共3.5兲, of the form

M

1 LD关␳兴 = 兺 ␭l共关Jˆl, ␳Jˆ†l 兴 + 关Jˆl␳,Jˆ†l 兴兲, 2 l=1

J1 =

共3.8兲

In order to close the Lie algebra L generated by operators Jˆ1 and Jˆ2, we need to introduce the operator Jˆ3 = 兩0典具1兩. The subalgebra spanned by Jˆ3 forms an Abelian ideal so that L is not a semisimple Lie algebra. The state 兩0典 is an eigenstate of all decoherence operators Jˆ1兩0典 = 0兩0典, Jˆ2兩0典 = 1兩0典, Jˆ3兩0典 = 0兩0典. Therefore, according to the LCW theorem, 兩0典 should be decoherence-free. However,

共3.3兲

On the other hand, we have Fˆ1,2兩␺典 = ⫾ 2−1/2兩0典 ⫽ c1,2兩␺典, so that 兩␺典 is not an eigenstate of the error generators Fˆk, and the LCW theorem would thus not identify 兩␺典 as a DFS state. This problem originates from the inclusion of zero eigenvalues of A 关26兴 and can be resolved as follows. Let B = 共bkl兲 be a unitary matrix that diagonalizes the matrix A and let ␭k ⱖ 0 be the positive eigenvalues of matrix A. Then we can define Lindblad operators Jˆl =

Jˆ2 = 兩0典具0兩 + 兩0典具1兩.

共3.7兲

Then Jˆ1兩␺典 = 0兩␺典. Thus the LCW theorem holds if it is modified to require that 兩␺典 is an eigenstate of the operators 兵Jˆl其 rather than of 兵Fˆk其. Example 1 illustrates an aspect of master equations that is important for the study of DFS, namely, that to determine a DFS, it is essential to use the “diagonalized” form of the decoherence superoperator. This has been implicitly recognized in some recent literature 关13,27兴, but not all. Example 1 shows that using a nondiagonalized form of the decoherence superoperator can lead to errors. Example 2. Consider a two-level system with basis 兵兩0典 , 兩1典其 and LD关␳兴 of the form 共3.5兲 with M = 2, ␭1 = ␭2 = 1 and the decoherence operators

共3.10兲

l=1

The eigenvalues of ⌫ˆ correspond to the inverse lifetimes of the corresponding eigenstates. Proposition 2. LD关兩␺典具␺兩兴 = 0

共3.11兲

if and only if Jˆl兩␺典 = cl兩␺典 for all l = 1 , . . . , M and ⌫ˆ 兩␺典 = g兩␺典, M ␭l兩cl兩2. where g = 兺l=1 Proof. First we prove that the conditions in proposition 2 are sufficient. Suppose there exists a state 兩␺典 such that Jˆl兩␺典 = cl兩␺典 and ⌫ˆ 兩␺典 = g兩␺典. Then M

兺 ␭lJˆl兩␺典具␺兩Jˆ†l = l=1

冉兺 冊 M

␭l兩cl兩2 兩␺典具␺兩

共3.12兲

l=1

and for ␳ = 兩␺典具␺兩 L D关 ␳ 兴 =

1 2

冉兺 M

l=1

冊

2␭lJˆl␳Jˆ†l − ␳⌫ˆ − ⌫ˆ ␳ = 0,

共3.13兲

M because g = 兺l=1 ␭l兩cl兩2. To prove that the conditions in the proposition are necessary, we now assume LD关␳兴 = 0. First we show that Jˆl兩␺典 = cl兩␺典. Using Eq. 共3.5兲 we evaluate the expression

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0 = 具␺兩LD关␳兴兩␺典 = 兺 ␭l共兩具␺兩Jˆl兩␺典兩2 − 储Jˆl兩␺典储2兲. 共3.14兲 l=1

⬜ Jˆl兩␺典 can generally be written as Jˆl兩␺典 = cl兩␺典 + 兩␺⬜ l 典 with 兩␺l 典 some 共non-normalized兲 state that is orthogonal to the state 兩␺典. Substituting this into Eq. 共3.14兲 yields M

⬜ ␭ l具 ␺ ⬜ 兺 l 兩␺l 典 = 0. l=1

共3.15兲

Because ␭l ⬎ 0 for l = 1 , . . . , M we find 储兩␺⬜典l储 = 0, i.e., Jˆl兩␺典 = cl兩␺典 l = 1 , . . . , M. We also need to show that ⌫ˆ 兩␺典 = g兩␺典. Consider a subspace V⬜ = 兵兩兩␺⬜典兩具␺⬜兩␺典 = 0其.

共3.16兲

Then 1 0 = 具␺⬜兩LD关␳兴兩␺典 = − 具␺⬜兩⌫ˆ 兩␺典 ∀ 兩␺⬜典 僆 V⬜ . 2 共3.17兲 This is only possible if ⌫ˆ 兩␺典 = g兩␺典. The value of g can be derived as follows. Consider the operator

loses energy. For decay processes the corresponding operators Jˆl can be represented by upper triangular matrix with zeros along the diagonal and thus can have only zeros for its eigenvalues. Excitation processes are correspondingly represented by lower triangular matrices with zeros along the diagonal and therefore also have only zero eigenvalues. Consequently this kind of noise automatically satisfies corollary 3 and hence the second condition in proposition 2 共i.e., ⌫ˆ 兩␺典 = g兩␺典兲 is redundant for quantum systems with noise comprised of independent decay and excitation processes. The second example where the second condition in proposition 2 becomes redundant is pure dephasing. During dephasing 关28兴, a quantum system remains in the same state, but acquires a phase. Thus for quantum noise due to dephasing, there exists a representation such that all Jˆl are diagonal. In this case, Jˆl commutes with Jˆ†l , which means that Jˆl is a normal operator 关29兴. As a result, we know that eigenstates of Jˆl are also eigenstates of Jˆ†l 关29兴. Consequently the action of each term contributing to ⌫ˆ in Eq. 共3.10兲 is of the form Jˆ†l Jˆl兩␺典 = cⴱl cl兩␺典 with 兩␺典 a simultaneous eigenstate of all Jˆl and Jˆ†l , so that we automatically have ⌫ˆ 兩␺典 = g兩␺典 with g a constant. Thus in this situation the second condition of proposition 2 also becomes redundant.

M

ˆ = 兺 ␭ cⴱJˆ . G l l l

ˆ and G ˆ † with The state 兩␺典 is an eigenstate of operators G M ␭l兩cl兩2 and g, respectively. Then eigenvalues g⬘ = 兺l=1 ˆ †兩␺典储2 = 具␺兩G ˆG ˆ †兩␺典 = gg⬘ , 共g⬘兲2 = 储G

B. Existence criteria for dynamically stable restricted DFS

共3.18兲

l=1

共3.19兲

assuming that the state 兩␺典 is normalized. Therefore, g M ␭l兩cl兩2. 䊏 = 兺l=1 Remark 2. Our proposition 2 immediately removes example 2 because the example does not satisfy the condition ⌫ˆ 兩␺典 = g兩␺典. Proposition 2 provides the required tightened version of the LCW theorem with the two additional conditions and makes correct predictions with regard to examples 1 and 2. Example 2 explicitly illustrates the shortcomings of the LCW theorem and elucidates the exact nature of the problem that we have remedied by introducing proposition 2. Corollary 3. Suppose that all eigenvalues of operators Jˆl for l = 1 , . . . , M are equal to zero; then the state 兩␺典 is an instantaneous pure decoherence-free state if and only if it is an eigenvector of the decoherence operator 共3.10兲 with an eigenvalue of zero. We now discuss two examples for which the second condition in proposition 2, ⌫ˆ 兩␺典 = g兩␺典, is rendered redundant by virtue of the intrinsic physics, i.e., the physical system automatically satisfies this condition. These examples are drawn from the most common types of noise in quantum systems, namely, excitation, decay, and dephasing. The first example is noise due to excitation or decay. During excitation, the system transits to a different state and gains energy. For decay, the system changes its state and

In Sec. III A, we determined when pure states ␳共t兲 are decoherence-free at a fixed time t. However, as mentioned before, this is not sufficient for unitary dynamics because the ˆ can drive ␳共t兲 out of the effective system Hamiltonian H eff decoherence-free subspace at a later time t. In this section we will incorporate time evolution in our considerations in order to establish more precise criteria for the existence of DFS at all times. This will characterize restricted DFS that satisfy LD关␳共t兲兴 = 0 ∀ t, i.e., dynamically stable DFS. The following theorem establishes the composition of HDFS in terms of all initial states that undergo decoherence-free evolution. Theorem 4. Let the time evolution be given by Markovian open system dynamics Eq. 共2.4兲. The space span兵兩␺1典,兩␺2典, . . . ,兩␺K典其 = Ejump

共3.20兲

is a DFS for all time t if and only if Ejump is a subspace that ˆ and that has basis vectors satisfying is invariant under H eff Jˆl兩␺k典 = cl兩␺k典,

⌫ˆ 兩␺k典 = g兩␺k典

共3.21兲

M ␭l兩cl兩2. for all l = 1 , . . . , M and for all k = 1 , . . . , K with g = 兺l=1 Proof. 关Note: we have used the notation Ejump to emphasize the relation with the operators Jˆ which correspond to the jump operators in a quantum trajectory description 共see below兲.兴 To prove that the conditions of theorem 4 are necessary, we suppose Ejump is a DFS. Then any state

兩␺共0兲典 僆 HDFS can be written as

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兩␺共0兲典 = 兺 ␣k兩␺k典,

共3.23兲

k=1

which evolves in time according to 兩␺共t兲典 = U共t兲兩␺共0兲典

共3.24兲

ˆ 兲 because L 关兩␺共t兲典具␺共t兲兩兴 = 0 ∀ t. As with U共t兲 = exp共−itH eff D 兩␺共t兲典 僆 HDFS = Ejump ,

共3.25兲

we can write the evolving state as K

兩␺共t兲典 = 兺 ␣k共t兲兩␺k典.

共3.26兲

k=1

As the state 兩␺共0兲典 僆 HDFS is arbitrary, this implies that K

K

k=1

k=1

ˆ 兺 ␣ 兩␺ 典 = 兺 ␤ 兩␺ 典, H eff k k k k

共3.27兲

ˆ leaves H i.e., the Hamiltonian H eff DFS invariant. Since LD关兩␺共0兲典具␺共0兲兩兴 = 0 for all 兩␺共0兲典 僆 Ejump, by proposition 2 we know that basis states 兩␺k典 must be eigenstates of Jˆl and ⌫ˆ ∀ l = 1 , . . . , M. To complete the proof that the conditions of theorem 4 are necessary, we have to show that the eigenvalues cl and g are equal for all basis states 兩␺k典. We do this by establishing a contradiction. Suppose two arbitrary basis states 兩␺k典 and 兩␺k⬘典 have different eigenvalues Jˆl兩␺k典 = cl,k兩␺k典,

Jˆl兩␺k⬘典 = cl,k⬘兩␺k⬘典.

共3.28兲

ence. Repeated application of L and the linearity of LD allows us to infer that the action of LD will not contribute to any power Ln关␳兴. Consequently, we find for the time evolution of an arbitrary density matrix ␳共0兲 formed of states 僆Ejump, etL关␳共0兲兴 = e−i关Heff,. . .兴␳共0兲 = e−itHeff␳共0兲eitHeff . 共3.33兲 Hence, ␳共t兲 僆 HDFS, since it shows unitary evolution under Heff. 䊏 Remark 3. By adopting definition 1 for DFS, we were able ˆ as a to derive that Ejump is invariant under action of H eff consequence of the definition, obviating the need to impose this condition as part of the definition itself, as was done in Ref. 关11兴. Next we provide a proposition with conditions for Ejump that are equivalent to requiring Ejump to be invariant under ˆ , i.e., to one of the conditions in theorem 4 above. The H eff advantage of these alternative conditions is that it might be easier to verify them for systems that support DFS with large dimensions. Proposition 5. Let Ejump be a space spanned by some or all of a collection of simultaneous degenerate eigenstates of Jˆl with eigenvalues cl for l = 1 , . . . , M that are also eigenstates ˆ if of ⌫ˆ with eigenvalue g. Then Ejump is invariant under H eff and only if ˆ ,Jˆ 兴兩␺典 = 0兩␺典 关H eff l

for all l = 1, . . . ,M

共3.34兲

and

Then the state 兩␺典 =

兩 ␺ k典 + 兩 ␺ k⬘典

冑2

共3.29兲

is not an eigenstate of Jˆl. However, because 兩␺典 僆 HDFS, proposition 2 implies that 兩␺典 must be an eigenstate. Hence the eigenvalues must be equal for all basis states. To prove that the conditions of theorem 4 are sufficient, we consider the action of the Liouvillean ˆ , . . .兴 + L 关¯兴 L关¯兴 ⬅ − i关H eff D

共3.30兲

ˆ ,⌫ˆ 兴兩␺典 = 0兩␺典 关H eff

for all 兩␺典 僆 Ejump. Proof. Let 兩␺典 be an eigenstate of Jˆl 共l = 1 , . . . , M兲 and ⌫ˆ , ˆ leaves with respective eigenvalues cl and g. Suppose H eff ˆ 兩␺典 is an eigenstate of operators Jˆ Ejump invariant. Then H eff l and ⌫ˆ with respective eigenvalues cl and g. Consequently, Eqs. 共3.34兲 and 共3.35兲 hold. Suppose now that Eqs. 共3.34兲 and 共3.35兲 are true for any 兩␺典 僆 Ejump. Then

on a coherence 兩␺0典具␺0⬘兩 between two arbitrary states 兩␺0典 , 兩␺0⬘典 僆 Ejump. We then have

再

⌫ˆ ⌫ˆ LD关兩␺0典具␺0⬘兩兴 = 兺 ␭l Jˆl兩␺0典具␺0⬘兩Jˆ†l − 兩␺0典具␺0⬘兩 − 兩␺0典具␺0⬘兩 2 2 l = 0,

冎

共3.31兲

because the states are eigenstates of Jˆl and ⌫ˆ . Hence the action of L can be reduced to

ˆ 兩␺典兲 = c 共H ˆ 兩␺典兲, Jˆl共H eff l eff

ˆ leaves E ˆ 兩␺ 典 ⬅ 兩␺ 典. Because H with H eff jump invariant we eff 0 1 know that 兩␺1典 , 兩␺1⬘典 僆 Ejump. This is a superposition of two coherences that are of the same type as the original coher-

ˆ 兩␺典兲 = g共H ˆ 兩␺典兲, ⌫ˆ 共H eff eff 共3.36兲

ˆ 兩␺典 is in E 䊏 i.e., H eff jump for all 兩␺典 僆 Ejump. Remark 4. For the specific examples of decoherence processes describing either dephasing, decay, or excitation in a finite-dimensional spinlike systems, it was already shown in Ref. 关4兴 that the set of states

ˆ ,兩␺ 典具␺⬘兩兴 = − i兩␺ 典具␺⬘兩 + i兩␺ 典具␺⬘兩, L关兩␺0典具␺0⬘兩兴 = − i关H eff 0 1 0 0 0 1 共3.32兲

共3.35兲

兵兩␺1典,兩␺2典, . . . ,兩␺K典其

共3.37兲

LD关兩␺i典具␺i兩兴 = 0

共3.38兲

that fulfills

ˆ . spans a DFS if it is invariant under H eff

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Jˆl兩␺k典 = cl兩␺k典

In the previous section we have studied pure, restricted DFS that fulfill the sufficiency criterion of definition 1, namely, LD关␳共t兲兴 = 0 for all t. However, for unitary dynamics, it is only necessary that the purity Tr关␳2共t兲兴 of a state is preserved during the evolution 共given an initially pure state, i.e., with Tr关␳2共0兲兴 = 1兲. This motivates us to provide a more general definition for the DFS as a collection of states that evolves in a unitary fashion. This is a larger set than the restricted DFS studied in Sec. III. Let D共H兲 be the set of all density matrices that describe pure states and that are defined for a quantum system associated with the Hilbert space H. Definition 2. Let the time evolution of an open quantum system with Hilbert space HS be given by Markovian open system dynamics Eq. 共2.4兲. Then a decoherence free subspace HDFS is a subspace of HS such that all pure states ␳共t兲 僆 D共HDFS兲 fulfill

⳵t Tr关␳2共t兲兴 = 0 ∀ t ⱖ 0,

with

Tr关␳2共0兲兴 = 1. 共4.1兲

We show in the Appendix that all states statisfying definition 2 undergo unitary evolution and therefore constitute a true DFS. Remark 5. In definition 2, it is absolutely crucial that the purity of the state ␳共t兲 is preserved for all times t, since nonunitary evolution is possible at intermediate times even when the initial state and final states are pure. For example, consider a two-level atom subject to decay in the upper level by spontaneous emission. For this system, an initially pure excited state will eventually evolve into a pure ground state. Thus, initial and final states are both pure. However, for any intermediate time between t = 0 and t → ⬁, the state of the system will be an incoherent mixture of ground and excited states. Therefore, despite starting out as a pure excited state, this system will not remain pure at all times and will therefore not be classified as DFS by definition 2. Using definition 2 and Eq. 共2.4兲, it immediately follows that 关30,31兴

⳵t Tr关␳2共t兲兴 = 2具LD关␳共t兲兴典.

共4.2兲

Therefore, DFS exist when 具LD关␳兴典 = 0, which is less restrictive than LD关␳兴 = 0. We will analyze the difference between these two conditions in Sec. V. In the remainder of this section we present a theorem that gives necessary and sufficient conditions for the existence of DFS according to definition 2 and also provide a constructive protocol for determining both when DFS exist and the exact specification of all states within the DFS. Theorem 6. Let the time evolution of an open quantum system described in a finite-dimensional Hilbert space be ˆ . The space governed by Eq. 共2.4兲 with time-independent H eff span兵兩␺1典,兩␺2典, . . . ,兩␺K典其 = Ejump

for all l = 1 , . . . , M and for all k = 1 , . . . , K and Ejump is invariant under M

ˆ + i 兺 ␭ 共cⴱJˆ − c Jˆ†兲. ˆ =H H eff l l l l l ev 2 l=1

共4.5兲

This guarantees identification of all DFS as well as explicit construction of all DF states. Proof. Before proceeding with the proof, we point out that Ejump contains only those eigenvectors of Jˆl that span an ˆ , i.e., the dimension of invariant subspace with respect to H ev EJump may be equal to or less than the number of simultaneous linearly independent eigenvectors of the jump operators Jˆl 共l = 1 , . . . , M兲 and there is no requirement in the theorem that these be the same. We begin the proof by observing that for any M-dimensional vector bជ with complex coefficients b1 , . . . , b M 关32兴

冋

册

M

i 共bជ 兲 关␳兴 − i 兺 ␭l共bⴱl Jˆl − blJˆ†l 兲, ␳ , LD关␳兴 = ˜LD 2 l=1

共4.6兲

where M

˜L共bជ 兲关␳兴 = 1 兺 ␭ 兵关J ˜ 共b 兲, ␳˜J†共b 兲兴 + 关J ˜ 共b 兲␳,J ˜ †共b 兲兴其 l l l l l D l l l l 2 l=1 共4.7兲 with ˜Jl共bl兲 = Jˆl − blˆI. Now we prove that the conditions of theorem 6 are sufficient. Any state 兩␺共t兲典 僆 Ejump

共4.8兲

can be written as K

兩␺共t兲典 = 兺 ␣k共t兲兩␺k典

and

␳共t兲 = 兩␺共t兲典具␺共t兲兩. 共4.9兲

k=1

Then for cជ = 共c1 , . . . , c M 兲 共cជ 兲 关␳兴 − i LD关␳共t兲兴 = ˜LD

冋

冋

M

i 兺 ␭l共cⴱl Jˆl − clJˆ†l 兲, ␳ 2 l=1

M

i = − i 兺 ␭l共cⴱl Jˆl − clJˆ†l 兲, ␳ 2 l=1

册

册 共4.10兲

because ˜Jl共cl兲兩␺共t兲典 = 共Jˆl − clˆI兲兩␺共t兲典 = 0兩␺共t兲典. Hence the time evolution for ␳共t兲 is given by ˆ , ␳共t兲兴 ␳˙ 共t兲 = − i关H ev

共4.11兲

so that ˆ , ␳共t兲兴␳共t兲其 = 0. ⳵t Tr关␳2共t兲兴 = 2 Tr关␳˙ 共t兲␳共t兲兴 = 2 Tr兵− i关H ev

共4.3兲

is a DFS satisfying definition 2 if and only if the basis vectors fulfill

共4.4兲

共4.12兲 Thus Ejump is a DFS.

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To prove that the conditions are necessary we suppose Ejump is a DFS. For fixed t0,

simultaneous eigenstates of a collection of operators 共Jˆ1 , . . . , Jˆ M in this case兲, is invariant under the action of a ˆ if and only if all states in the space are eigenHamiltonian H states of the corresponding commutator operators Bl ˆ , Jˆ 兴 with eigenvalues 0, for all l = 1 , . . . , M. Thus, in or= 关H l der to construct DFS from EJ, we 共i兲 consider all basis elements 兩␺1典 , . . . , 兩␺N典 for EJ, 共ii兲 form linear combinations of these as ␣1兩␺1典 + ¯ + ␣N兩␺N典 with ␣1 , . . . , ␣N free parameters, and 共iii兲 then determine all possible sets 兵␣i , i = 1 , . . . N其 such that the resulting states are eigenstates of Bˆl ˆ , Jˆ 兴 with eigenvalue 0 for all i = 1 , . . . , M. The resulting = 关H l states form a complete basis for the DFS specified by eigenvalues c1 , . . . , c M of the jump operators. Different choices of eigenvalues for the jump operators can give rise to different DFS and are constructed similarly. 䊏 Remark 6. Theorem 6 applies only to finite-dimensional spaces. To demonstrate this we consider a damped harmonic oscillator described by the master equation

K

兩␺共t0兲典 = 兺 ␣k共t0兲兩␺k典,

␳共t0兲 = 兩␺共t0兲典具␺共t0兲兩,

k=1

共4.13兲 which implies 0 = ⳵t Tr关␳2共t0兲兴 = 2 Tr关␳共t0兲LD关␳共t0兲兴兴 = 具␺共t0兲兩LD关␳共t0兲兴兩␺共t0兲典.

共4.14兲

Suppressing the dependence on t0 we see that Eq. 共4.14兲 has the same form as Eq. 共3.14兲. Repeating the same argument we are led to the condition Jˆl兩␺典 = cl兩␺典. Furthermore, the line of reasoning presented in Eqs. 共3.28兲 and 共3.29兲 proves that the eigenvalues cl are equal for all basis states 兩␺k典. Thus we can conclude that Jˆl兩␺共t0兲典 = cl共t0兲兩␺共t0兲典, for any t0. At this point, it appears that the eigenvalue cl共t兲 may depend on time t, but we argue now that cl共t兲 must be time independent for finite-dimensional systems, i.e., cl共t兲 = cl. Equation 共2.4兲 is a system of linear, first-order differential equations with constant coefficients; the components of ␳共t兲 and 兩␺共t兲典 are therefore continuous functions in time. Jˆl兩␺共t兲典 = cl共t兲兩␺共t兲典 then implies that cl共t兲 must be continuous, too. On the other hand, the time independent operator Jˆl acts on a finite-dimensional Hilbert space so that its spectrum is discrete. Thus, cl共t兲 must be constant. According to Eq. 共4.10兲 for ␳共t兲 = 兩␺共t兲典具␺共t兲兩 with Jˆl兩␺共t兲典 = cl兩␺共t兲典, we then have ˆ , ␳共t兲兴 共4.15兲 ˆ , ␳共t兲兴 + L 关␳共t兲兴 = − i关H ␳˙ 共t兲 = − i关H eff D ev and ˆ t兲␳共0兲exp共iH ˆ t兲 ␳共t兲 = exp共− iH ev ev

共4.16兲

ˆ t兲兩␺共0兲典 because H ˆ is time indepenwith 兩␺共t兲典 = exp共−iH ev ev ˆ . The dent. Thus 兩␺共t兲典 僆 Ejump if Ejump is invariant under H ev Hilbert space HS can in principle contain multiple DFS. Each DFS is characterized by an M-tuple 共c1 , . . . , c M 兲 of eigenvalues of all the jump operators Jˆ1 , . . . , Jˆ M for states in Ejump. Now we demonstrate that theorem 6 is constructive, i.e., we explain how to use theorem 6 to determine all basis states in all DFS. Condition 1 of theorem 6, i.e., Eq. 共4.4兲, states that DFS are subsets of all common degenerate eigenstates of the jump operators Jˆ1 , . . . , Jˆ M . We label the space of all common degenerate eigenstates of the jump operators as EJ, with dimension N. The space EJ is determined by a collection of corresponding eigenvalues of the jump operators c1 , . . . , c M . A different collection of eigenvalues might give rise to a different nonempty space EJ and constitute potentially a different DFS. In order to satisfy condition 2 of theorem 6, i.e., invariance under Eq. 共4.5兲, we need to restrict the space EJ to ˆ . We accomplish a subspace Ejump that is invariant under H ev this task using proposition 5. Proposition 5 says that a space that is spanned by some or all of the collection of degenerate

␥ ␳˙ 共t兲 = − i关␻0aˆ†aˆ, ␳共t兲兴 + 关2aˆ␳共t兲aˆ† − aˆ†aˆ␳共t兲 − ␳共t兲aˆ†aˆ兴, 2 共4.17兲 with aˆ the annihilation operator. A coherent state ␳共t兲 = 兩␣共t兲典具␣共t兲兩, with ␣共t兲 = ␣e−共i␻0−␥/2兲t and ␣ a complex constant, is a solution to this master equation. 兩␣共t兲典 is an eigenstate of aˆ with unit purity, but it is not invariant under the ˆ 共t兲 = ␻ aˆ†aˆ + i␥ 关␣ⴱ共t兲aˆ − ␣共t兲aˆ†兴. evolution Hamiltonian H ev 0 2 Theorem 6 does not apply to this example because a coherent state 兩␣共t兲典 is an eigenstate of aˆ with a time-dependent eigenvalue ␣共t兲. Thus, the analysis following Eq. 共4.14兲 in the proof of theorem 6 does not apply. We now provide a constructive protocol for DFS, i.e., we present a procedure that explicitly determines the existence and composition of all DFS. Protocol 1. 1. We consider all Lindblad operators Jˆl l = 1 , . . . , M, from Eq. 共3.5兲 and calculate common eigenstates for these operators, i.e., eigenstates 兩␺i典 satisfying Jˆl兩␺i典 = cl兩␺i典, for l = 1 , . . . , M. Note for each l, all 兩␺i典 are eigenstates of Jˆl with the same eigenvalue. However, eigenvalues for different index l need not be the same. The number of such common eigenstates is N ⱖ 0. For N = 0, there is no DFS. 2. When N ⬎ 0 we construct a space EJ = span兵兩␺i典 , i = 1 , . . . , N其. This space is characterized by a set of eigenvalues c1 , . . . , c M of the jump operators. According to theorem 6, any DFS must be a subspace of EJ. Note that according to theorem 6, dim共HDFS兲 ⱕ N. 3. Now we apply proposition 5 to the conditions determined by theorem 6 to select out the states in EJ that belong to a DFS 共note that there is no restriction to a single DFS兲. The conditions of theorem 6 require that the DFS be invariant under Hev. According to proposition 5, a state 兩␺i典 in EJ ˆ is a zero eigenthat is invariant under some Hamiltonian H ˆ , Jˆ 兴. Thus, the DFS is a collection of state of the operator 关H l simultaneous eigenstates of Jˆl that are also eigenstates of all

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ˆ , Jˆ 兴, with eigenvalue 0, i.e., HDFS Bl = 关H ev l = 兵兩␺i典 : 兩␺i典 僆 EJ and Bˆl兩␺i典 = 0兩␺i典其. We can therefore simˆ , Jˆ 兴 and use these to seply construct the operators Bˆl = 关H ev l lect out the states 兩␺i典 in Ejump that also fulfill Bˆl兩␺i典 = 0兩␺i典. Doing this for all l gives the basis for the DFS that is specified by eigenvalues c1 , . . . , c M , of total dimension dim共Ejump兲 ⱕ N. This protocol must be repeated for all possible combinations of common eigenvalues of Jˆ1 , Jˆ2 , . . . , Jˆ M . This will generate all existing DFS. These results provide simple constructive criteria for the existence of DFS and for exact specification of all DFS states for any given Markovian master equation 关33兴.

V. INCOHERENTLY GENERATED COHERENCES

In this section we analyze the difference between the two definitions of decoherence free subspaces introduced in this paper, i.e., definitions 1 and 2. The following example demonstrates that there are states that satisfy definition 2 but do not satisfy the criteria of definition 1. Example 3. Consider a two-level system with basis states 兩1典 and 兩0典 in presence of a single decoherence operator Jˆ = ␴+ + ␴z, ␭ = 2, and Hamiltonian H = ␴y. Here ␴y = i共兩0典具1兩 − 兩1典具0兩兲 and ␴x = 兩0典具1兩 + 兩1典具0兩. Then the state 兩␺典 = 兩1典 has nonzero action of LD on it, with LD关兩␺典具␺兩兴 = i关␴y , 兩␺典具␺兩兴 = −␴x ⫽ 0, and yet it corresponds to a stationary solution of the master equation that fulfills 具LD关␳兴典 = 0. In absence of the Hamiltonian H, the superoperator LD关兩␺典具␺兩兴 would therefore evolve the system into a state that differs from 兩␺典 and that would, in general, therefore be subsequently affected by decoherence. However, in this example, H exactly cancels the effect of the decoherence superoperator at t = 0 and hence for all times, ensuring that 兩␺典 remains stationary and decoherence-free. The reason why 兩␺典 is stable in this example is that LD关␳兴 initially acts coherently on 兩␺典 and the coherences generated by this are canceled by the action of the unitary term −i关H , 兩␺典具␺兩兴. It is easy to see that the initial coherence is generated between the state 兩␺典 = 兩1典 and the orthogonal state 兩0典. However, this cancellation is specific to these two states and does not hold for an arbitrary state of the two-level system, for which additional terms of the form of Eq. 共4.7兲 will be present in the action of LD关␳兴. Consequently, if the Hamiltonian term were absent, LD关␳兴 would generate an incoherent evolution of the quantum state 兩␺典 because it only acts coherently on this particular state and its orthogonal complement, so that once the initial state has been infinitesimally coherently transformed to some superposition of these states, LD关␳兴 will gain some contribution of incoherent terms of the form of Eq. 共4.7兲. The existence of states that satisfy definition 2 but not definition 1 is surprising because it disagrees with the following, physically intuitive argument. Inserting the decoherence operator ⌫ˆ into Eq. 共3.5兲 we can rewrite the master equation Eq. 共2.4兲 as

M

ˆ ␳ + i␳H ˆ † + 兺 ␭ Jˆ ␳Jˆ† ␳˙ = − iH NH l l l NH

共5.1兲

l=1

ˆ − i⌫ˆ / 2. In ˆ ⬅H with the non-Hermitian Hamiltonian H NH eff ˆ the context of quantum trajectory methods 关34兴, H NH generM ates a continuous evolution in time, while 兺l=1␭lJˆl␳Jˆ†l is interpreted as generating sudden quantum jumps at random times. The dynamics in absence of quantum jumps is given by ˆ

ˆ†

␳共t兲 = e−itHNH␳共0兲eitHNH .

共5.2兲

ˆ are both Hermitian operators, they correSince ⌫ˆ and H eff ˆ , respond to the anti-Hermitian and Hermitian parts of H NH ˆ cannot eliminate the effect of the despectively. Hence, H eff coherence operator ⌫ˆ . Naively, one also would expect that M ␭lJˆl␳Jˆ†l cannot cancel the effect the quantum jump term 兺l=1 ˆ of ⌫. This argument leads to the conclusion that setting LD关␳共t兲兴 = 0 ∀ t is the only way to achieve unitary evolution. Example 3 demonstrates that this argument is not correct in general. The existence of these states can be quite generally explained with reference to the properties of Markovian master equations. Equation 共4.6兲 shows that the Markovian master equation 共2.4兲, is invariant under the following transformation 关32兴: Jˆl → Jˆl − blˆI

l = 1, . . . ,M ,

共5.3兲

ˆ + i 兺 ␭ 共bⴱJˆ − b Jˆ†兲. ˆ →H H l l l l l eff eff 2 l=1

共5.4兲

for M

Here bl ranges over all complex numbers. This transformation implies that LD关␳兴 can always rewritten as a sum of two terms 关see Eq. 共4.6兲兴, namely, 共i兲 a decoherence super-operator with respect to the transformed jump operators Jˆl − blIˆ and 共ii兲 a commutator between ␳ and ˆ ⬅ i 兺 M ␭ 共bⴱJˆ − b Jˆ†兲. Thus, the the Hermitian operator H l l D 2 l=1 l l l general decoherence superoperator LD关␳兴 always contains a commutator with a Hermitian operator and, therefore, can in principle generate unitary evolution under suitable circumˆ , ␳兴 stances. Consequently, we cannot simply interpret i关H eff and LD关␳兴 in the Markovian master equation 共2.4兲, as terms that always generate unitary and decohering dynamics, respectively. Under special circumstances, in particular, when the state ␳ is generated by a common eigenstate of all jump operators Jˆl, Eq. 共4.7兲 vanishes and the decoherence superoperator conˆ : sists then of just the commutator with Hermitian H D ˆ , ␳兴. LD关␳兴 = − i关H D

共5.5兲

In this situation, a state ␳ decoheres because the Hermitian ˆ generated by L 关␳兴 drives state ␳ out of the Hamiltonian H D D subspace where its purity is preserved, and not because LD关␳兴 causes a genuine decay in a quantum system. It is then

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ˆ might possible that the effective system Hamiltonian H eff compensate for such unitary leakage, resulting in preservation of the subspace. This is the situation described in example 3 above. For a given system, it may therefore be possible to construct or to modify the effective Hamiltonian ˆ and to specifically to ensure negation of the effects of H D thereby stabilize states ␳ in the subspace corresponding to one or more common eigenstates of the jump operators Jˆl. Very recently Ticozzi and Viola 关35兴 have discussed ideas that are similar to this observation. Here we will derive general criteria for when this phenomenon can occur and provide several physical examples. Example 3 thus illustrates an intriguing general phenomenon, i.e., that even though decoherence in an open system S is caused by the interaction between system S and its reservoir, decoherence can be completely eliminated by a particular system Hamiltonian under special circumstances. We would like to emphasize here that this effect is very different from well-studied decoherence suppression phenomena such as dynamical decoupling 关16–19兴. In dynamical decoupling, decoherence effects are suppressed because the time scale for the system Hamiltonian is much faster than the time scale for the decay processes. The system Hamiltonian changes the state of the system so rapidly that decoherence processes are adiabatically eliminated. In contrast, in example 3 the decoherence effects simply do not take place, as in any DFS. To understand more generally under which conditions the incoherent part LD关␳兴 of the master equation can support a DFS through the generation of coherences, we introduce now the following special class of DFS states. Definition 3. States with incoherent generation of coherences 共IGC兲 are pure states ␳共t兲 that evolve unitarily, ⳵t Tr关␳2共t兲兴 = 0 ∀ t ⱖ 0, but that have LD关␳共t兲兴 ⫽ 0 at some time共s兲 t. Theorem 6 allows us to investigate the difference between restricted DFS states in the sense of definition 1 and subspaces composed of IGC states, which we shall refer to as IGC subspaces. Assume state ␳ = 兩␺典具␺兩 fulfills Jˆl兩␺典 = cl兩␺典 so that 具LD关␳兴典 = ⳵t Tr关␳2共t兲兴 = 0. Then

states do not exist in systems with decoherence due to independent decay, excitation or dephasing processes. Hence, while the intuitive physical idea that LD关␳兴 = 0 should be fulfilled for DFS states is mathematically wrong, it nevertheless is valid for many physical applications.

LD关␳兴 = 兺 共2兩cl兩2␳ − clJ†l ␳ − cⴱl ␳Jl兲.

共5.6兲

l

The following result can be inferred from this equation. Corollary 7. An IGC state 兩␺典 can only exist if, for at least one l, Jˆl兩␺典 = cl兩␺典 with cl ⫽ 0, and 兩␺典 is not an eigenstate of 兺lclJˆ†l . Remark 7. This result implies that IGC states and IGC subspaces are rare in nature. In particular, IGC states do not exist when the operators Jˆl are normal 共关Jˆl , Jˆ†l 兴 = 0 关29兴兲 since in this case any 兩␺典 僆 HDFS would necessarily also be an eigenstate of Jˆ†l . This implies that IGC states do not exist in systems subject to decoherence due to dephasing 共see also the discussion below corollary 3兲. In decoherence processes that involve population exchange, such as spontaneous emission or thermal excitation, a DFS state typically corresponds to an eigenstate of Jˆl with cl = 0, as discussed earlier, and so corollary 7 precludes any IGC in this situation. Thus IGC

Examples of IGC states

While IGC states do not exist for the most common physical decoherence models, they nevertheless can occur under special circumstances as the following examples illustrate. Example 4. Driven two-level atom in squeezed vacuum reservoir. We consider a two-level atom that interacts with a radiation field. We also assume that the radiation field is prepared in a squeezed vacuum state. Then this system is described by the master equation 关32兴. ˆ , ␳兴 − ␥0 c2共␴ ␴ ␳ + ␳␴ ␴ − 2␴ ␳␴ 兲 ␳˙ = − i关H S + − + − − + 2 −

␥0 2 s 共␴−␴+␳ + ␳␴−␴+ − 2␴+␳␴−兲 2

+ ␥0共sc␴−␳␴− + sc␴+␳␴+兲,

共5.7兲

with s ⬅ sinh共r兲, c ⬅ cosh共r兲, and r the 共real兲 squeezing parameter. Equation 共5.7兲 assumes Lindblad form when we introduce the operator Jˆ = c␴− + s␴+ .

共5.8兲

For r ⫽ 0 the operator Jˆ has two nonzero eigenvalues ⫾冑sc with eigenstates 兩␺⫾典 = ⫾ 冑s兩0典 + 冑c兩1典. Each corresponds to a ˆ stationary IGC state if the Hamiltonian is given by H S ␥0 = ⫾ 2 冑sc共s − c兲␴y, i.e., the DFS 共IGC subspaces兲 are at most one dimensional. However, one state is not sufficient to encode a qubit. Next we will explore how one can generate an IGC subspace large enough to be used for quantum information processing. Example 5. N two-level atoms in the Dicke limit in squeezed vacuum reservoir. We consider N two-level atoms in a squeezed vacuum reservoir in the Dicke limit 关36兴. We model the dynamics of this system by the master equation ˆ , ␳兴 + ␥ 关2Jˆ␳Jˆ† − Jˆ†Jˆ␳ − ␳Jˆ†Jˆ兴 ␳˙ = − i关H D 2

共5.9兲

with N

Jˆ = 兺 Jˆn

and

Jˆn = c␴ˆ n− + s␴n+ ,

共5.10兲

n=1

where the index n refers to the nth atom. To find IGC states, we need to construct all eigenstates of the operator Jˆ. They are given by all possible tensor products of 兩␺⫾典. For a given eigenstate, let n+ be number of 兩␺+典 components and n− = N − n+ the number of 兩␺−典 components. Then the corresponding eigenvalue is 共n+ − n−兲冑sc. For n+ = n− the eigenvalue is zero, indicating that the state is not of

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IGC type. All other eigenstates of Jˆ become IGC states if the ˆ = ␥ 共n − n 兲冑sc共s driving Hamiltonian assumes the form H D 2 + − N ␴ny. − c兲Sˆy with Sˆy = 兺n=1 States with the same n+ have the same eigenvalue and form an IGC subspace. The dimension of the IGC subspace is given by the binomial coefficient 共 nN+ 兲. For N ⱖ 3, there exist IGC subspaces with dimension greater than 1 and can be used to encode quantum information. For example, for n = 3 the IGC subspace corresponding to the eigenvalue 冑sc can be spanned from the nonorthogonal basis states

duced in Eq. 共4.10兲, for instance, in the case ␻0共t兲 = ␻0 and g共t兲 = 2i ␥␣e−i␻0t, then purity and mean photon number 具aˆ†aˆ␳共t兲典 are both preserved, i.e., such a state does not experience decoherence or attenuation. Example 7. Two-photon absorber. A two-photon absorber pumped by a two-photon parametric process constitutes another example of IGC states. The master equation for this system is

兩␺1典 = − 冑c共兩101典 + 兩011典兲 + 冑s共兩100典 + 兩010典兲,

ˆ = i␥ 关共␣ⴱ兲2aˆ2 − ␣2aˆ†2兴. The two coherent states 兩⫾␣典 with H D 2 are solutions to this master equation 关37–40兴 and are both eigenstates of the two-photon annihilation operator aˆ2 with the same eigenvalue ␣2. Hence they form a two-dimensional IGC subspace.

兩␺2典 = − c冑c兩111典 + s冑c兩100典 − c冑s兩011典 + s冑s兩000典, 兩␺3典 = 冑c共兩011典 − 兩110典兲 − 冑s共兩001典 − 兩100典兲. 共5.11兲 The encoding efficiency of the IGC subspace 共the number of logical qubits log2兵dim关IGC共N兲兴其兲 approaches unity just as in the case of the usual DFS 关2,3兴. These examples seem to suggest that in finite-dimensional systems IGC states only appear in relatively exotic situations. However, it is quite simple to find realistic examples in infinite systems. We give two examples below, noting that since theorem 6 does not apply to infinite dimensional systems, these states may not be invariant under time evolution with H, despite the preservation of the state purity. Example 6. Driven damped harmonic oscillator. Consider a driven harmonic oscillator with annihilation operator aˆ and

␥ LD关␳兴 = 关2aˆ␳aˆ† − aˆ†aˆ␳ − ␳aˆ†aˆ兴, 2

共5.12兲

ˆ = ␻ 共t兲aˆ†aˆ + gⴱ共t兲aˆ + g共t兲aˆ† . H eff 0

共5.13兲

This model is frequently used to describe photons in a lossy single-mode cavity that is coherently pumped by a driving field of amplitude g共t兲. A coherent state 兩␣典 is an eigenstate of aˆ with eigenvalue ␣ and fulfills LD关兩␣典具␣兩兴 ⫽ 0 for ␣ ⫽ 0. In absence of a driving field 关g共t兲 = 0兴 and for constant frequency ␻0, this state remains a coherent state 关32兴. This also holds in the general case; for a given function g共t兲 and initial condition ␣共0兲, a coherent state evolves as the pure state ␳共t兲 = 兩␣共t兲典具␣共t兲兩 with

␣共t兲 = ␣共0兲eF共t兲 − i

冕

t

dt⬘eF共t兲−F共t⬘兲g共t⬘兲,

共5.14兲

0

and F共t兲 ⬅ 兰t0dt⬘关−i␻0共t⬘兲 − ␥ / 2兴. Hence, purity is preserved and a coherent state corresponds to an IGC state. As we noted before, theorem 6 does not apply to infinitedimensional systems. Thus, purity is preserved for coherent states, although they are not invariant under the evolution Hamiltonian. For a coherent state, the effect of the decoherence term is reduced to an attenuation of the oscillator: it describes loss of energy 共or photons兲 but not loss of coherence. However, if the choice of g共t兲 is consistent with the transformation intro-

ˆ , ␳兴 + ␥ 共2aˆ2␳aˆ†2 − aˆ†2aˆ2␳ − ␳aˆ†2aˆ2兲 共5.15兲 ␳˙ = − i关H D 2

VI. SUMMARY AND DISCUSSION

We have examined two different definitions of states ␳共t兲 of a quantum system that undergo unitary evolution in the presence of decoherence effects, both of which are related to the preservation of state purity. For the restricted DFS introduced in definition 1, the decoherence term in the master Eq. 共2.4兲 vanishes, LD关␳共t兲兴 = 0. This is a sufficient but not a necessary criterion for unitary evolution. General DFS states can be characterized by the less stringent condition ⳵t Tr关␳2共t兲兴 = 0 for all t ⱖ 0 or, equivalently, 具LD关␳共t兲兴典 = 0, that was introduced in definition 2. With theorems 4 and 6 we have established rigorous conditions for the existence of dynamically stable DFS of both types in finite-dimensional systems. We also provided a simple constructive protocol for explicit computation of states in all such DFS that exist. DFS states that fulfill 具LD关␳共t兲兴典 = 0 but for which LD关␳共t兲兴 ⫽ 0 are especially interesting. These states satisfy Definition 2 but not 1. We have shown that these states rely on incoherent generation of coherences 共IGC兲. This means that they correspond to a set of states on which the decoherence term LD关␳共t兲兴 acts similar to a Hamiltonian term, generating coherences with other states that are cancelled out by the system Hamiltonian. We showed that existence of these IGC states constitutes a general phenomenon that can be understood in terms of basic properties of the Markovian master equation. In particular, their existence was shown to result from an invariance of the master equation with respect to a continuous transformation, implying a dynamical symmetry that links the coherent and decoherent terms and that results in cancellation of the decoherent terms in certain circumstances. IGC states form a subset of DFS states and as such may be useful for quantum information processing. A general strategy to generate such states is to first identify states on which the decoherence superoperator acts coherently for infinitesimal times and then to stabilize these states to make IGC states by adding a Hamiltonian to the master equation that cancels the action of LD关␳共t兲兴 on the states. We have studied under which decoherence conditions a system admits IGC states. For finite-dimensional systems the most common

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decoherence models, including independent dephasing and incoherent excitation or deexcitation processes, do not allow for IGC states. In other words, the set of restricted DFS is then identical with the full DFS. IGC states in finitedimensional systems can only appear in relatively exotic situations, for instance, when the system interacts with a reservoir that is prepared in a squeezed vaccum state. On the other hand, in infinite-dimensional systems it is easy to find examples of IGC states, e.g., coherent states in a lossy single-mode optical cavity.

To prove that the condition is necessary, suppose that Tr关Et共␳0兲2兴 = 1 for a given pure state ␳0. Et共␳0兲 can be written in the operator sum representation as 关42兴 Et共␳0兲 = 兺 Eˆk␳0Eˆ†k , k

兺k Eˆ†k Eˆk = 1ˆ .

共A6兲

For a state of purity one, the CP map on the pure state ␳0 = 兩␺典具␺兩 can then be written as Et共兩␺典具␺兩兲 = 兺 兩␺k典具␺k兩,

兩␺k典 ⬅ Eˆk兩␺典.

共A7兲

k

ACKNOWLEDGMENTS

We thank D. Lidar, A. Shabani, and P. Zanardi for valuable critical comments on a previous version of this work and D. Lidar for helpful comments on this manuscript. We also thank H. Wiseman for a valuable discussion and L. Viola for bringing Ref. 关35兴 to our attention. Financial support by iCORE, NSERC, and CIFAR is gratefully acknowledged. R.I.K. and K.B.W. thank the NSF for financial support under ITR Grant No. EIA-0205641, and the Defense Advanced Research Projects Agency 共DARPA兲 and the Air Force Laboratory, Air Force Material Command, USAF, under Contract No. F30602-01-2-0524.

This is a uniform mixture of states 兩␺k典. Such a mixture can only be of purity one if all 兩␺k典 are proportional to each other, i.e., if 兩␺k典 = ␭k兩˜␺典,

where 兩˜␺典 is a normalized state defined as 兩˜␺典 = E1兩␺典 / 储E1兩␺典储, provided that 储E1兩␺典储 ⫽ 0. If this is not the case, the normalized state can be defined in terms of any other Eˆk for which 储Eˆk兩␺典储 ⫽ 0. There must be at least one such operator Eˆk, since if there is no such operator, then Et共␳0兲 = 0, which would contradict the assumption that purity is preserved. Using Eq. 共A8兲 in Eq. 共A7兲, we then obtain Et共兩␺典具␺兩兲 = 兺 Eˆk兩␺典具␺兩Eˆ†k ,

APPENDIX: EQUIVALENCE BETWEEN UNITARY DYNAMICS AND PURITY PRESERVATION

E t共 ␳ 0兲 =

共A1兲

if and only if Et共␳0兲 preserves purity Tr关Et共␳0兲2兴 = 1. Proof. We first show that the condition is sufficient. Assume that Et共␳0兲 = Ut␳0U†t . Then Tr关Et共␳0兲2兴 = Tr共Ut␳0U†t Ut␳0U†t 兲,

共A9兲

k

Here we show that definition 2, according to which a state is DF if its purity does not change from initial unit value, is equivalent to requiring unitary dynamics for Markovian master equations. A very similar result has been derived by Chefles 关41兴. In this appendix we present an alternative derivation and provide a different formulation that can be directly applied to the main part of this work. We start with the fact that the evolution of any quantum state ␳0 is always described by a completely positive tracepreserving linear map 共CP map兲 Et共␳0兲. For such a map the following proposition holds. Proposition 8. Let ␳0 be a density matrix with purity one, i.e., Tr共␳20兲 = 1. Then the action of a linear CP map Et共␳0兲 is unitary Ut␳0U†t

共A8兲

共A2兲

=Tr共Ut␳0␳0U†t 兲,

共A3兲

=Tr共␳0␳0兲,

共A4兲

=1.

共A5兲

= 兺 兩␭k兩2兩˜␺典具˜␺兩.

共A10兲

k

Now we use the unit normalized state 兩˜␺典 to write

兺k 兩␭k兩2 = 兺k 具˜␺兩␭ⴱk ␭k兩˜␺典,

共A11兲

= 兺 具␺兩Eˆ†k Eˆk兩␺典,

共A12兲

=具␺兩1ˆ 兩␺典,

共A13兲

=1,

共A14兲

k

so that Eq. 共A10兲 becomes Et共兩␺典具␺兩兲 = 兩˜␺典具˜␺兩.

共A15兲

Hence Et maps 兩␺典 to 兩˜␺典. Such a map between unit normalized pure states can always be written as a unitary transformation Ut and explicit construction of Ut can be made, e.g., by using a Householder reflection 关43兴. Hence Et共兩␺典具␺兩兲 = Ut兩␺典具␺兩U†t .

共A16兲

䊏 Under the usual assumption of and initially factorizable system and environment state, the time evolution of a system state ␳共t兲 can always be described by a time-dependent CP map Et共␳0兲 acting on the initial state ␳0 关42兴. By proposition

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8, requiring unitary dynamics for all t is then equivalent to the condition Tr关␳2共t兲兴 = 1 ∀ t ⱖ 0. For dynamics that are governed by a Markovian master Eq. 共2.4兲, ␳共t兲 is a differentiable operator-valued function. Hence the time derivative

of ␳共t兲 is well defined and purity preservation can be expressed as ⳵t Tr关␳2共t兲兴 = 0 ∀ t ⱖ 0 with Tr共␳20兲 = 1. Therefore, for Markovian dynamics, definition 2 is equivalent to requiring a DF state to undergo unitary dynamics.

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