DECOHERENCE-FREE SUBSPACES FOR OPEN QUANTUM RANDOM WALKS ON GRAPHS

We study decoherence-free subspaces in a type of Quantum Markov Semigroups called continuous-time open quantum random walks on graphs. We measure the temporary changes of quantum correlations using geometric quantum discord with bures distance under some assumptions about the semigroup. In particular, we characterize the decay of correlations to zero, showing that turns out to be closely related with the structure of decoherence-free subspace. AMS Subject Classification: 47D06, 46L55, 81S22


Introduction
Continuous-time open quantum random walks (COQRW) on graphs were introduced using the continuous-time limits of open quantum random walks in [19].It was shown that the limit processes are represented by Quantum Markov Semigroups (QMSs) T = (T t ) t≥0 , i.e., T is a weakly * -continuous semigroup of completely positive, identity preserving, normal linear maps on the von Neumann algebra B(h) of all linear bounded operators on a given complex separable Hilbert space h.QMSs are a non-commutative extension of Markov semigroups defined in classical probability, they represent an evolution without memory of a microscopic system in accordance with the laws of quantum physics and fit into the framework of open quantum systems (see [12], [13]).The semigroup T corresponds to the Heisenberg picture in the sense that given any observable x, T t (x) describes its evolution at time t.In this way, given a density matrix ρ, its dynamics (Schrodinger picture or predual semigroup) is given by the semigroup T * t (ρ), where tr(ρT t (x)) = tr(T * t (ρ)x).
Several aspects of temporal evolutions described by COQRWs have been investigated.By example in [2], and [19], relationship with non-Markovian generalized Lindblad master equations is studied.In [6] (in a particular class of CO-QRWs) convergence to a steady state regardless of the initial state when a graph is connected is showed.In this work we are interested in study decoherence-free subspaces in continuous-time open quantum random walks on graphs.
Decoherence occurs when a quantum system interacts with its environment in an irreversible way.Decoherence and noise ([4, 14, 15, 20, 21, 25] and references therein) typically affect quantum features of a state over its time evolution, however it may be possible to find states with a unitary evolution in some "good" portion of a system.
Two main approaches to decoherence of open quantum systems have been proposed in the literature; both are based on quantum Markov semigroups.
Blanchard and Olkiewicz [3], starting from an algebraic setting, defined environment induced decoherence and found many physical models where the system algebra decomposes as the direct sum of two pieces: a subalgebra, called the decoherence-free algebra, where the semigroup acts homomorphically, a Banach subspace where the semigroup action is purely dissipative (see e.g.papers [4,5] and the references therein) and vanishing as time tends to infinity.The decoherence-free subalgebra was later characterised in [11,13] as the commutant (or generalised commutant for unbounded operators) of certain families of operators arising form the GKSL (Gorini-Kossakowski-Sudarshan-Lindblad) representation of the generator.In particular, decoherence-free subalgebras of COQRWs were studied in [10].
In the approach to decoherence proposed by Lidar et al. [15,8] registers of a quantum computer are modeled by a quantum open system on a finitedimensional Hilbert space h.The time evolution of states is described by a predual semigroup T * on the Banach space of trace class operators on h.Given that, for a quantum computer to execute a quantum algorithm efficiently, it needs to evolve by unitary quantum dynamics, in this aprroach, is defined a particular sector where a quantum algorithm is executable: the decoherence-free subspaces.
A subspace h f of h is decoherence-free if the time evolution of states ω supported in h f is given by ω → e −itK ωe itK for some self-adjoint operator K on h f .Decoherence-free subspaces were identified in [8] (see also [24]) under some physical (somewhat implicit) assumptions, we refer to [15] for an introduction to the theory of decoherence-free subspaces with a lot of examples and applications to protection of quantum information.
Lidar et al. papers, however, are concerned only with finite-dimensional systems and focus on explicit physical models.Moreover, his method essentially depends on the choice of an orthonormal basis at the outset.This basis is determined by the spectral analysis of the coefficients of the GKSL generator of the quantum Markov semigroup.As a result, this method cannot be extended to infinite dimensions, or to the case of continuous spectra and unbounded coefficients of the GKSL-generator.In [1] they look at the decoherence-free subspace issue from a mathematical point of view and study the following problem: given a quantum Markov semigroup on the algebra B(h) with generator represented in a generalised GKSL form, characterising its decoherence-free subspaces for a possibly infinite dimensional Hilbert space h.We use this approach by study decoherence-free subspaces and temporary changes of quantum correlations in COQRWs.
The structure of the paper is as follows.Section 2 contains the definition of decoherence-free subspaces for any QMSs and some preliminary remarks.In Section 3 we introduce COQRW on graphs and we study his decoherencefree subspace.The temporary change of quantum correlations using geometric quantum discord with bures distance under some assumptions about the semigroup is presented in Section 4.

Decoherence-Free Subspaces
Fron now on, we denote the von Neumann algebra of all bounded operators on the Hilbert space h by B(h).We recall that an state ω on B(h) is a positive, trace-one, operator on h, in particular it is a clase trace operator on h.The support supp(ω) of ω is the closed subspace of h generated by eigenvectors with strictly positive eigenvalues.Definition 1.A subspace h f of h is called decoherence-free (DF) if there exists a self-adjoint operator K on h f such that for all state ω with support in for all t ≥ 0. From now on, we will call that K is associated with h f .
Note that a self-adjoint operator K on h f can always be extended to the whole Hilbert space h, therefore DF subspaces could be defined in an equivalent way with a self-adjoint operator K on h leaving the subspace h f invariant.In a more precise way, for an unbounded K, this means that e −itK (h f ) ⊆ h f for all t ∈ R.Moreover, note that a DF subspace is necessarily closed with respect to the norm topology of h.
Remark.COQRW on graphs are a particular case of predual semigroup of a QMS, where the generator L * of predual semigroup has a well-known representation.In general , any generator of a QMS is described by the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) form for all state ρ on B(h), where s≥1 is a finite or infinite sequence and the series s≥1 L * s L s converge strongly .(See[18], theorem 3.16, pag 271.)Remark.Recall that the operators L ℓ , H ∈ B(h) in a GKSL representation of L are not unique, we may, for instance, translate each L ℓ by adding multiples z ℓ ½ of the identity operator ½, with ℓ |z ℓ | 2 < ∞.In this way we obtain another . We refer to [18] p.272-273 for a detailed discussion on this subject.Using the structure in the generator of semigroup is possible to give necessary and sufficient conditions to find DFs of an arbitrary QMS.This sentence is true when the coefficients of L * are bounded operator and its also true in the not bounded case.We present the result in the bounded case.(General result is founded in [1], proposition 7 ).

Theorem 2.
A subspace h f is a DF subspace with associated self-adjoint operator K, if and only if in any GKSL representation of L by means of operators L ℓ , G there exist complex numbers λ ℓ (ℓ ≥ 1) and a real number r such that ℓ≥1 |λ ℓ | 2 < ∞ and Proof.Consider the GKSL representation of the generator L and h f is a DF subspace with associated operator K.
By the well-known polarisation identity (1) also holds for ω = |u v| with u, v ∈ h f .Differentiating we have (3) it follows that u is an eigenvector of all L ℓ , i.e.
The identity (4) for w ∈ h f also yields We now prove that eigenvalues λ ℓ (u) do not depend on the choice of the vector u ∈ h f .Note, first of all, that eigenvectors u, v in Since, the Hilbert space h being separable, the spectrum of p f L ℓ p f is at most countable, hence totally disconnected.The function on the unit sphere of is continuous and so its range must be connected.It follows that the function u → λ ℓ (u) is constant.Now, rewriting (3) as we see that u and v are also eigenvectors for G + iK.The eigenvalues z(u) and z(v) fulfill the identity hence for some r(u) ∈ R. Finally, replacing this in ( 6), we see that r(u) must be independent of u ∈ h f .Conversely, suppose that 1. and 2. hold, then we compute immediately the above relationship also holds for e −i(t−s)K u e −i(t−s)K v and we have, and h f is decoherence-free.
Remark.The above result shows that, translating the operators L ℓ by −λ ℓ , we find another GKSL representation of L with In this way, since ℓ≥1 (L ′ ℓ ) * L ′ ℓ vanishes on h f , we find as self-adjoint operator K associated with h f the generator of the oneparameter group originating from the action of the semigroup in the new GKSL representation of L.
This theorem provides a recipe for finding DF subspaces.First of all look for common eigenspaces for all the operators L ℓ , then, translate L ℓ to L ℓ − λ ℓ ½ with eigenvalues λ ℓ finding a new GKSL representation of the generator L. The intersection of common eigenspaces of all the operators L ℓ is now the common kernel of all the operators L ℓ − λ ℓ ½.Finally check that the operator G founded in the new GSKL representation of L leaves the common kernel invariant and is anti self-adjoint on this subspace.
Remark.A K-invariant subspace of a DF subspace is itself a DF subspace, therefore we will be interested in maximal DF subspaces.

COQRWs: Some Properties and Associated Decoherence-Free Subspaces
In this section we describe the model, some properties and DFs of COQRWs.
Let G be a graph such that the set of vertexes is given by V = {1, . . ., d}, with d ≥ 2. To each vertex j ∈ V we associate an element e j in the canonical basis of C d , moreover, all vertex in V have degrees of freedom given, in this article, by the simplest space C 2 equipped with an orthonormal basis (f j ) j=1,2 , therefore the dynamic is describe by operators in h = C 2 ⊗ C d in the following way: for all j, m ∈ V such that j = m, we fix B mj ∈ M 2 (C) standing for the effect of passing from the vertex m to the vertex j.We also define the sum of all the effects when one leaves the vertex m B * mm B mm :=

Ker(B mj ).
The continuous-time open quantum random walk (COQRW) is the predual semigroup T * = (T * t ) t≥0 acting on the set of class-trace operator I(h) defined in h = C 2 ⊗ C d , its Lindblad generator is given by for all ρ ∈ I(h), where with H = H * bounded operator in h.
Remark.Let B : C 2 → Rang(B), P : C d → Rang(P ) be linear operators then (see theorem 2.19 in [9]).We denote {e m } ⊥ to the orthogonal subespace of subspace generated by e m .

Proposition 4.
j∈V Proof.We take A description of decoherence-free subspace for COQRW is obtained by applying Theorem 2, Proposition 3, Proposition 4, and equality (9).Theorem 5. Let h f be DF subspace of COQRW and with K = H associated self-adjoint operator.
Proof.First note that the only eigenvalue of an operator with m = j and B mj = 0 is 0. Indeed, if we suppose that exist u = k,s u ks f k ⊗ e s eigenvector of L mj with eigenvalue associated λ = 0 then 0 for all m = j, i.e., λu rm = 0 for all r and m = j.
then u = 0 (we suppose λ = 0 ) and by other hand, if u is eigenvector then u = 0, this is a contradiction.Therefore λ = 0. Second,by (9) note that m,j∈V m =j Finally, we see that if u ∈ m,j∈V m =j Ker(L mj ) then L * mj L mj u = 0 for all m = j.Taking K = H we otain Applying Theorem 2 with K = H we obtain that .
Using Propositions 3 and 4 it follows that Remark.Theorem 5 establish the sectors, where a quantum algorithm describe by COQRWs on graphs is executable.
Hereinafter, we study some properties of COQRWs when H = 0, in this case, we obtain that Writing any ρ in h = C 2 ⊗C V as a block matrix with respect to the canonical basis of B(C V ) then we obtain We call diagonal subspace, and denote it by D the subspace generated by the elements X kk ⊗|e k e k | with X kk ∈ M 2 (C).Let E : B(h) → D be the conditional expectation with range D defined by and let E * be the predual map on trace class operators with range i.e., E * : I(h) → D * where By equality (10) and positivity of σ, it follows the positivity of σ pp ∈ M 2 (C) for all p ∈ V.Moreover, tr(σ) = 1 if and only if with c p := tr( σ pp ) ≥ 0 for all p ∈ V, Additionally, we obtain that if COQRWs has null-Hamiltonian, that is, his generator is given by ( 8) with H = 0 then hence for all t ≥ 0, i.e., D * and D * of f are T * -invariants, equivalently for all t ≥ 0.

Temporary Change in Quantum Correlations
In this section we measure temporary changes in quantum correlations using geometric quantum discord with Bures distance when COQRWs has null-Hamiltonian, that is, his generator is given by ( 8) with H = 0.In particular, we characterize the decay of correlations to zero, showing that turns out to be closely related with the structure of decoherence-free subspaces.
The Bures distance can be used to bound from below and above the trace distance d 1 (ρ, σ) = tr(|ρ − σ|) (see [7]) as follows: Theorem 6.Let T be a COQRW with null-Hamiltonian then the following assertions are equivalent: with at most a unique m ∈ V − V 0 such that Ker(B mm ) = 0.  14) we obtain that In particular, taking ̺ = U m ZU * j for an arbitrary operator Z = (Z rs ) r,s=1,2 ∈ M 2 (C), we obtain and recursively, we get where δ jm (Z) := − 1 2 (D m Z + ZD j ) and δ By ( 4) and ( 20), we obtain that with d p m ≥ 0, d q j ≥ 0 for all m, j ∈ V, m = j and p, q ∈ {1, 2}, for any ̺ ∈ M 2 (C) and t ≥ 0.

If we suppose that
with at most a unique m ∈ V − V 0 such that Ker(B mm ) = 0, then Ker(B * mm B mm ) = 0 (by equation ( 7) and proposition 3) with B * mm B mm = 0 (m ∈ V − V 0 ) therefore exists at most a unique m ∈ V such that B * mm B mm can have at most one zero eigenvalue, i.e.,exists at most a unique m ∈ V and exists at most a unique p ∈ {1, 2} such that d p m = 0 and d q j > 0 for all (j, q) = (m, p).By By (23), we obtain that (U * m ̺U j ) sr = (U * m ̺U j ) rs = 0.This is a contradiction with (24) .Therefore exist at most a unique m ∈ V −V 0 such that Ker(B mm ) = 0, then (c) ⇒ (a).