A Protocol For Cooling and Controlling Composite Systems by Local Interactions
aa r X i v : . [ qu a n t - ph ] N ov A Protocol For Cooling and Controlling Composite Systems by Local Interactions
Daniel Burgarth and Vittorio Giovannetti Computer Science Department, ETH Z¨urich, CH-8092 Z¨urich, Switzerland NEST-CNR-INFM & Scuola Normale Superiore, piazza dei Cavalieri 7, I-56126 Pisa, Italy
We discuss an explicit protocol which allows one to externally cool and control a composite systemby operating on a small subset of it. The scheme permits to transfer arbitrary and unknown quantumstates from a memory on the network (“upload access”) as well as the inverse (“download access”).In particular it yields a method for cooling the system.
I. INTRODUCTION
Repetitive applications of the same quantum trans-formation have been exploited to achieve noise protec-tion [1], cooling, state preparation [2, 3, 4], and quantumstate transfer [5]. Motivated by the above results, inRef. [6] we developed a scheme for controlling larger sys-tems when control is only assumed to be available on asubsystem. Once this is achieved, apart from cooling andstate preparation, it is also possible to perform arbitraryquantum data processing (e.g. measurements, unitaryrotations). This is similar in spirit to universal quantuminterfaces of Ref. [7], but our approach allows us to spec-ify explicit protocols and to give lower bounds for fideli-ties. These techniques are also related with the “asymp-totic completeness” property introduced by K¨ummererand Maassen [4, 8] which allows one to control a systemby coupling it with quantum mediators.In the present paper we review the scheme of Ref. [6]by showing how arbitrary quantum states can be writteninto (i.e. prepared on) a large system, and read from it,by local control only. This implies that arbitrary quan-tum operations on the system state can be performed.An important specific task is the cooling of the systemto its ground state. Using some heuristic argument, wewill provide an estimate of the convergence time of thecooling and we will test it with some numerical examples.We develop the protocol in several steps. First, we showthat the system of interest can be actively brought to itsground state by replacing its controlled part with fresh“cold” qubits from a memory. We then find that cool-ing implies that the information about the initial systemstate is transferred into the memory, and design a lin-ear map that decodes this information. Since this mapis generally not unitary, we use the polar decompositionto find its best unitary approximation. The fidelity ofinformation decoding can then be lower bounded by theoverlap of the system state with its ground state. Finallywe design the reverse operation allowing us to transfer in-formation from the memory to the system.The material is organized as follows. In Sec. II theprotocol is presented in its general lines. In Sec. IIIwe give a detailed derivation of the coding and decod-ing transformations and derive bounds for the fidelities.Numerical estimations of the protocol performances aregiven in Sec. IV focusing on the case of locally controlledHeisenberg-like coupled spin networks. Conclusion and remarks are in Sec. V while technical material is pre-sented in the Appendices.
II. THE PROTOCOL
Consider a composed system described by the Hilbertspace H = H C ⊗ H ¯ C ⊗ H M . We assume that full control(the ability to prepare states and apply unitary transfor-mations) is possible on system C and M, but no controlis available on system ¯ C. Moreover, we assume that C and ¯ C are coupled by a time-independent Hamiltonian H. We show here that under certain assumptions, if thesystem C ¯ C is initialized in some arbitrary state we cantransfer (“download”) this state into the system M byapplying some operations between M and C only. Like-wise, by initializing the system M in the correct state,we can transfer (“upload”) arbitrary states on the sys-tem C ¯ C. The system M functions as a quantum memory and must be at least as large as the system C ¯ C . Assketched in Fig. 1 we can imagine the memory to be splitinto sectors M ℓ , having the same dimension of C , i.e. H M = N Lℓ =1 H M ℓ with dim H M ℓ = dim H C . A. Downloading info from C ¯ C to M The downloading protocol we present here is composedby two stages: a swapping stage, in which at regular timeintervals we couple the subsystem C to the first L memo-ries M ; and a decoding stage in which we apply a unitarytransformation to the first L memories in order to re-cover the initial state of C ¯ C . As we will see for any finite L our analysis does not guarantee that the fidelity be-tween the recovered state and the initial state of C ¯ C isperfect. However, in Sec. III A it will be shown that byaugmenting L one can make the fidelity arbitrarily closeto one.We assume that the memory is initialized in a factor-ized state of the form | i M ≡ L O ℓ =1 | i M ℓ , (1)where | i is a state whose properties will be specifiedin the following. To download a generic state, we let the FIG. 1: Example of the model discussed in the text. Here thesystem C ¯ C is formed by 7 spins characterized by some time-independent Hamiltonian H (the coupling are represented byred lines connecting the spins). The system C ¯ C can onlybe controlled by acting on a (small) subsystem C (in thiscase represented by the uppermost spin of the network). Thecoupling H can - in some cases - mediate the local control on C to the full system C ¯ C. In our case, system C is controlledby performing regular swap operations S ℓ between it and a2-dim quantum memory M ℓ . system C ¯ C to evolve for a while according to its Hamilto-nian H , perform a unitary gate which couples C to oneof the sectors of M , let C ¯ C evolve again and so forth.More specifically, at step ℓ of the protocol we perform aunitary swap S ℓ ≡ S CM ℓ between system C and system M ℓ [9]. After the L th swap operation the protocol stops.This is the swapping stage and it is characterized by theunitary operator W ≡ S L U S L − U · · · S ℓ U · · · S U , (2)where U = exp [ − iHt ] is the time-evolution of C ¯ C forsome fixed time interval t. As discussed in Ref. [6], thereduced evolution of the system ¯ C under the transfor-mation (2) can be expressed in terms of the followingcompletely positive CP map [10] τ ( ρ ¯ C ) ≡ Tr C (cid:2) U ( ρ ¯ C ⊗ | i C h | ) U † (cid:3) , (3)where | i C is the state that is swapped in from the mem-ory and Tr C [ · · · ] indicate the partial trace over the sub-system C . Indeed, after L swaps the state of ¯ C is ob-tained by taking the partial trace with respect to C and M of the vector W ( | ψ i C ¯ C | i M ) where | ψ i C ¯ C is the initial state of C ¯ C , i.e. ρ ( L )¯ C = Tr CM (cid:2) W ( | Ψ i C ¯ C h Ψ | ⊗ | i M h | ) W † (cid:3) = τ ◦ τ ◦ · · · ◦ τ | {z } L − ρ ′ ¯ C ) ≡ τ L − ( ρ ′ ¯ C ) , (4)where “ ◦ ” represents the composition of super-operators [10] and ρ ′ ¯ C ≡ Tr C (cid:2) U | ψ i C ¯ C h ψ | U † (cid:3) , (5)(for an explicit derivation of this expression see Ap-pendix A).Our main assumption is that the map τ is ergodic withpure fixed point which we denote as | i ¯ C . Explicitly thismeans that the only state which is left invariant by τ isthe vector | i ¯ C , i.e. τ ( ρ ¯ C ) = ρ ¯ C ⇐⇒ ρ ¯ C = | i ¯ C h | . (6)As shown in Refs. [11, 12] this implies that the channel τ is relaxing (mixing), that islim n →∞ τ n ( σ ¯ C ) = | i ¯ C h | , (7)for all σ ¯ C . This condition gives rise to the controllabilityof the system. Indeed from Eq. (7) it follows that forsufficiently large L , an initial state of the form | ψ i C ¯ C ⊗| i M can be approximated as W (cid:0) | ψ i C ¯ C ⊗ | i M (cid:1) ≈ | i C ¯ C ⊗ | Φ( ψ ) i M . (8)The right hand side of this equation factorizes into purestates because the transformation W is unitary, and be-cause both the initial state of C ¯ C and M and the finalstate of C ¯ C are pure. This implies that, in the asymp-totic limit of infinitely many protocol steps (i.e. L ≫ C ¯ C has been “cooled” into the state | i C ¯ C while all the information regarding the initial state | ψ i C ¯ C must be contained in the vector | Φ( ψ ) i M [13]. Further-more, it is at least intuitively clear that such informationcan be recovered by the application of a proper unitary“decoding” operation V † on M which does not dependon the input state of the system ( decoding stage), i.e. [14] V † | Φ( ψ ) i M ≈ | ψ i M . (9)At a mathematical level, the convergence of the down-loading protocol described above only depends upon theinvariant property (6) — see Ref. [6]. In Sec. III we willbriefly review such a proof and provide a characterizationof the unitary transformation V . B. Uploading info from M to C ¯ C For uploading states on the system C ¯ C , we again makeuse of the unitarity of W. Let us again first give a simplehand-waving argument why this is possible.Suppose you want to drive the system into the state | ψ i C ¯ C . To do this, you first use the downloading proto-col to make sure that the system is in the state | i C ¯ C (“cooling”). Then you bring the memories into the state | Φ( ψ ) i M they would have been ended up in case one wastrying to download | ψ i C ¯ C from C ¯ C into M as in Eq. (8).Now the quantum recurrence theorem [16] implies thatthere is a m such that | ψ i C ¯ C ⊗ | i M ≈ W m (cid:0) | ψ i C ¯ C ⊗ | i M (cid:1) ≈ W m − (cid:0) | i C ¯ C ⊗ | Φ( ψ ) i M (cid:1) , (10)where we have made use of Eq. (8). Hence by applying W m times you have approximately initialized | ψ i C ¯ C . Ofcourse it remains to be shown that unknown states canbe written to the system, too. This and the mathemati-cal details will be discussed in the next section. Anotherproblem with Eq. (10) is that the recurrence parameter m typically needs to be huge , scaling double exponentiallywith the number of qubits in the system. There are how-ever alternative, more efficient ways of implementing anuploading process from M to C ¯ C . The simplest one is ofcourse to apply the inverse transformation W − = W † to the state of Eq. (8). Indeed the protocol we presentedin Ref. [6] is based on this idea, which is a generaliza-tion of [4]. Unfortunately the inverse of W is generallyunphysical in the sense that it requires backward timeevolutions U − , i.e. one would have to wait negative time steps between the swaps (see however Ref. [15] forcases in which such an inverse time evolution can be im-plemented by clever external control techniques).To overcome this problem we introduce an extra hy-pothesis. Specifically we consider the case in which theinvariant property (6) holds also for the channel τ ′ ob-tained from Eq. (3) by replacing U with U † , i.e. τ ′ ( ρ ¯ C ) ≡ Tr C (cid:2) U † ( ρ ¯ C ⊗ | i C h | ) U (cid:3) . (11)Under this condition, similarly to the case of W discussedin the previous section, one can verify that in the limitof large L , i) the transformation W ′ ≡ S L U † S L − U † · · · S ℓ U † · · · S U † , (12)applied to | ψ i C ¯ C ⊗ | i M will converge to a vector of theform | i C ¯ C ⊗ | Φ ′ ( ψ ) i M ; ii) there exists a unitary trans-formation V ′ which does not depend upon | ψ i and whichapplied to M gives V ′† | Φ ′ ( ψ ) i M ≈ | ψ i M . (13)From this we can write | ψ i C ¯ C ⊗ | i M ≈ ( W ′ ) † V ′ (cid:0) | i C ¯ C ⊗ | ψ i M (cid:1) ≈ ( U S · · · U S ℓ · · · U S L − U S L ) V ′ (cid:0) | i C ¯ C ⊗ | ψ i M (cid:1) . (14)What it is relevant for us is the fact that now the unitarytransformation on the input state | i C ¯ C ⊗ | ψ i M does not V ρ M V † −→ ρ M −→ Λ( ρ M ) −→ V ′ Λ( ρ M ) V ′† V ′ Λ V † W Λ( ρ S ) ρ S W ′† − → − → FIG. 2: Summary of the scheme: any CP map Λ can be ap-plied to the system by acting on the memory instead throughthe transformations shown in the figure. The red and greenareas represent the downloading and uploading part of theprotocol, respectively. The unitary operators W and W ′† ofEqs. (2) and (15) are generated by acting on the memory and asmall subsystem of the system only; V † and V ′ are instead thedecoding and encoding unitary transformations introduced inEqs. (9) and (13), respectively — see also Sec. III. involve “time-reversal” evolutions U − but only “proper”time evolution U . Therefore, by imposing the condi-tion (6) on τ ′ , we are able to define an uploading pro-tocol which transfers an unknown state | ψ i from M to C ¯ C . Similarly to the downloading scheme it is composedby two stages: an encoding stage in which we apply theunitary transformation V ′ to “prepare” the memory M and a swapping stage in which we apply the unitary( W ′ ) † = U S · · · U S ℓ · · · U S L − U S L , (15)by recursively coupling C to the M through swaps.Two remarks are mandatory. On one hand, as in thecase of the downloading protocol, the convergence of thetransformation (14) only depends upon the invariant con-dition (6) of the channel τ ′ . On the other hand, thereexists a large class of physically relevant Hamiltonians H (e.g. nearest neighbors Heisenberg coupling Hamiltoni-ans) for which both τ and τ ′ verify the such condition —we refer the reader to Ref. [6] for details. For such Hamil-tonians, our analysis will yield both a simple downloadingand uploading mechanism. Putting these two elementstogether one can also realize more sophisticated controls.For instance, as shown in Fig. 2, one can perform anyquantum transformation Λ on C ¯ C by first downloadingits state on M , transforming it, and finally uploading thefinal state back into the system. III. CODING TRANSFORMATION
In this section we derive the decoding transformation V † that relates states on the memories M to the states on C ¯ C in the downloading protocol. To do so we exploit theformal decomposition of the evolved state of the systemafter L steps (see Appendix B). The encoding transfor-mation V ′ of the uploading protocol can be obtained ina similar way.Consider an orthonormal basis {| ψ k i C ¯ C } of H C ¯ C . Ac-cording to Eq. (B1) after L swaps it becomes W (cid:0) | ψ k i C ¯ C | i M (cid:1) (16)= | i C ⊗ h √ η k | i ¯ C | φ k i M + p − η k | ∆ k i ¯ CM i , where | ∆ k i ¯ CM is a vector orthogonal to | i ¯ C and η k ≈ W maps the orthonormal vec-tors | ψ k i C ¯ C into the vectors | φ k i M of the first L mem-ories. For any finite choice of L , the latter are typicallynot mutually orthogonal. However one can use Eq. (B6)to show that in the limit of large L the vectors | φ k i M become approximately orthogonal. Indeed from the uni-tarity of W and from Eq. (16) and (B3) we can establishthe following identity M h φ k | φ k ′ i M (17)= √ η k η k ′ δ kk ′ + p η k (1 − η k ′ ) ¯ CCM h ψ k | ˜∆ k ′ i ¯ CCM + p η k ′ (1 − η k ) ¯ CCM h ˜∆ k | ψ k ′ i ¯ CCM + p (1 − ˜ η k )(1 − ˜ η k ′ ) C ¯ CM h ˜∆ k | ˜∆ k ′ i C ¯ CM . To simplify this expression we define η ≡ min k η k . SinceEq. (B6) applies to all η k the parameter η must satisfythe inequality1 − η K ( L − d ¯ C κ L − . (18)Furthermore from Eq. (17) it follows that for k = k ′ onehas | M h φ k | φ k ′ i M | | p η k (1 − η k ′ ) | ¯ CCM h ψ k | ˜∆ k ′ i ¯ CCM | + p η k ′ (1 − η k ) | ¯ CCM h ˜∆ k | ψ k ′ i ¯ CCM | + p (1 − ˜ η k )(1 − ˜ η k ′ ) | C ¯ CM h ˜∆ k | ˜∆ k ′ i C ¯ CM | p − η + (1 − η ) p − η , (19)which according to Eq. (18) and using the fact that theparameter κ is strictly smaller than 1, shows that forlarge L the vectors | φ k i M and | φ k ′ i M become orthogonal.Define then the linear operator D on H M which per-forms the following transformation D | ψ k i M = | φ k i M , (20)with | ψ k i M being orthonormal vectors of M which rep-resent the states {| ψ k i C ¯ C } of H C ¯ C . Formally they areobtained by a partial isometry from ¯ CC to M and are“good” representations of the | ψ k i C ¯ C . The operator D in some sense “corrects” the non-orthogonality of the | φ k i M : indeed its inverse (when definable) allows us topass from these approximate images of the | ψ k i C ¯ C to thegood representations | ψ k i M . Therefore D − seems tobe a good candidate for defining our decoding transfor-mation V . Unfortunately however D is NOT unitary (itmaps an orthonormal set of states into a non-orthonormalone) and typically will not be even invertible. The idea is then to replace D with its best unitaryapproximation V [17, p 432]. The latter is obtained bytaking the polar decomposition of D , i.e. D = P V , (21)with P positive semidefinite. The unitary V minimizesthe norm distance from D yielding the inequality || D − V || = sX k hp λ k − i sX k | λ k − | √ d C ¯ C (1 − η ) / , (22)where we introduced the eigenvalues λ k of D † D andused Eq. (19) to bound them according to the inequal-ity | λ k − | d C ¯ C √ − η (in these expressions d C ¯ C = d C d ¯ C is the dimension of the system C ¯ C and k Θ k stands for pP kk ′ | Θ kk ′ | with Θ kk ′ being the ma-trix elements of the operator Θ). The inequalities (22)and (18) show that for L → ∞ , D can be approxi-mated arbitrarily well by the unitary V . We can hencedefine V † as our decoding transformation which invertsthe mapping (20) and transforms the “bad” representa-tions | φ k i M of the | ψ k i C ¯ C into the “good” representa-tions | ψ k i M . It is worth stressing that, by construction, V does not depend upon the input state | ψ i C ¯ C of thesystem C ¯ C .As mentioned in the introduction of this section, a sim-ilar procedure can used to defined the encoding protocolof the uploading protocol. Without entering into the de-tails we simply notice that in this case D and the vectors | φ k i M will be defined by replacing W of Eq. (16) withthe transformation W ′ of Eq. (12). Taking the polar de-composition of such new D it will yield the unitary V ′ which will be used as encoding for the uploading scheme.In the following section we will evaluate the transferfidelities associated with such a choice of decoding andencoding transformation, showing that they can arbitrar-ily increased by choosing L sufficiently high. A. Fidelity of the downloading protocol
Let | ψ i C ¯ C = P k α k | ψ k i C ¯ C be a generic input stateof C ¯ C . To evaluate the downloading fidelity F down as-sociated with our decoding scheme we need to comparethe state of M at the end of the protocol with the state | ψ i M = P k α k | ψ k i M , i.e. F down ( ψ ) ≡ M h ψ | V † R M V | ψ i M . (23)Here V † is the decoding transformation defined in theprevious section, and R M is the state of the memoryafter the application of the unitary W , i.e. R M ≡ Tr C ¯ C (cid:2) W ( | ψ i C ¯ C h ψ | ⊗ | i M h | ) W † (cid:3) = η | φ i M h φ | + (1 − η ) σ M . (24)In the above expression we used Eqs. (B1) and (B2) andintroduced the density matrix σ M ≡ Tr ¯ C [ | ∆ i ¯ CM h ∆ | ]. Bylinearity we get F down ( ψ ) = η | M h φ | V | ψ i M | + (1 − η ) M h ψ | V † σ M V | ψ i M > η | M h φ | V | ψ i M | . (25)We now bound the term on the right hand side as follows | M h φ | V | ψ i M | = | M h φ | V − D + D | ψ i M | (26) > | M h φ | D | ψ i M | − | M h φ | D − V | ψ i M | , and use the inequality (22) to write | M h φ | D − V | ψ i M | || D − V || √ d C ¯ C (1 − η ) / . If | ψ i M was a basis state | ψ k i M , then | M h φ | D | ψ i M | = 1by the definition Eq. (20) of D . For generic | ψ i M insteadwe can use the linearity to find after some algebra that √ η | M h φ | D | ψ i M | > √ η − d C ¯ C p − η . (27)Therefore Eq. (26) gives √ η | M h φ | V | ψ i M | > √ η − d C ¯ C (1 − η ) / , (28)which replaced in Eq. (25) yields F down ( ψ ) > η − d C ¯ C (1 − η ) / , (29)for all input states | ψ i C ¯ C . According to Eq. (18) it thenfollows that by choosing L sufficiently big our download-ing protocol will yield transferring fidelities arbitrarilyclose to one. B. Fidelity of the uploading protocol
Following the analysis of Sec. (II B) the fidelity for up-loading a state | ψ i M into ¯ CC is given by F up ( ψ ) ≡ (30) C ¯ C h ψ | Tr M (cid:2) W ′† V ′ ( | ψ i M h ψ | ⊗ | i ¯ CC h | ) V ′† W ′ (cid:3) | ψ i C ¯ C . A lower bound for this quantity is obtained by replacingthe trace over M with the expectation value on | i M , i.e. F up ( ψ ) > C ¯ C h ψ | M h | W ′† V ′ (cid:0) | ψ i M h ψ | ⊗ | i ¯ CC h | (cid:1) × V ′† W ′ | i M | ψ i C ¯ C = (cid:12)(cid:12) C ¯ C h | M h ψ | V ′† W ′ | i M | ψ i C ¯ C (cid:12)(cid:12) (31)= η ′ (cid:12)(cid:12) M h ψ | V ′† | φ i M (cid:12)(cid:12) = η ′ | M h φ | V ′ | ψ i M | . In deriving this equation we used Eq. (15) and a decom-position of the form of Eq. (B1) to simplify the vector W ′ | i M | ψ i C ¯ C . In this case η ′ is defined as in Eq. (B5)with τ being replaced by τ ′ of Eq. (11). Since we areassuming that this CP map satisfies the condition (6) it follows that also η ′ obeys an inequality of the form(B6) with K and κ replaced by new constants K ′ and κ ′ ∈ ]0 , F up ( ψ ) > η ′ − d C ¯ C (1 − η ′ ) / , (32)with 1 − η ′ K ′ ( L − d ¯ C ( κ ′ ) L − . (33)This shows that, as in the downloading case, also the up-loading fidelity converges to unity in the limit of large L . IV. EFFICIENCY OF COOLING
In this section we provide some numerical estimationof the quantities η of Eq. (B5) which measure the prob-ability of finding the state ¯ C in | i ¯ C . As seen in theprevious sections this is the fundamental parameter tobound the fidelities of both the downloading and upload-ing protocol. Moreover, given an initial state | ψ i C ¯ C , η measures the success probability of “cooling” it down tothe state | i C ¯ C during the downloading process. Accord-ing to Eq. (B6) the quantity η will asymptotically con-verge exponentially fast to unity. However Eq. (B6) doesnot tell us from which point onwards the convergence isexponentially fast, so it would be nice to have alternativeways to estimate the convergence speed.To simplify the analysis in the following, we will con-centrate on the spin network model of Fig. 1 assuming aHeisenberg Hamiltonian of the form H = X ( j,j ′ ) ∈ G d j,j ′ (cid:0) X j X j ′ + Y j Y j ′ + Z j Z j ′ (cid:1) , (34)which conserves the total magnetization along the z axis(here X j , Y j and Z j are the Pauli operators of the j -th spin and the summation is performed over all theedges of the weighted graph G associated with the net-work). Moreover we will take the vector | i C to be theconfiguration where all the qubits of C are in the spin-down state, i.e. | i C ≡ | · · · i C . For this choice ofthe controller state our main assumption of ergodicityEq. (6) is numerically found to be correct for the cou-pling graph depicted in Fig. 1. The fixed point is givenby | i ¯ C ≡ | · · · i ¯ C (more general conditions of ergod-icity for Heisenberg models are given in [6]). In this con-text η coincides then with the probability P ( L )0 of findingno excitations on the system after L steps of the proto-col. Some numerical examples showing the dependenceof η upon the initial state are presented in Fig. 3. Asexpected, asymptotically P ( L )0 is seen to converge expo-nentially fast.An approximate estimation of P ( L )0 can be easily ob-tained by looking at the average number of spin-up on G r ound S t a t e P r opab ili t y P ( L ) All flippedFully mixedGHZW
FIG. 3: Convergence of the cooling protocol for the weightedgraph of Fig. 1 where the couplings among the spins isgiven by the Hamiltonian (34) (the values of constants d j,j ′ have been chosen to be proportional to the length of thegraph edge). Four different states | ψ i C ¯ C are considered: thefully flipped state | i C ¯ C , a GHZ state ( | i C ¯ C + | i C ¯ C ) / √ , a fully mixed state ρ C ¯ C = 1 C ¯ C / , and aW state √ ( | i C ¯ C + | i C ¯ C + · · · + | i C ¯ C ). C ¯ C after L swaps, i.e. D ˆ N E ( L ) C ¯ C ≡ Tr C ¯ C h ˆ N ρ ( L ) C ¯ C i , (35)with ρ ( L ) C ¯ C ≡ Tr M (cid:2) W (cid:0) | ψ i C ¯ C h ψ | ⊗ | i M h | (cid:1) W † (cid:3) beingthe reduced density matrix of C ¯ C and with ˆ N ≡ P k ∈ C, ¯ C ( Z k + 1) / Z k is the z -Pauli matrix of the k -th spin). These quantities are related by P ( L )0 > − D ˆ N E ( L ) C ¯ C . (36)To get an approximation for the average number of exci-tations on the graph we assume now that the time inter-val t is chosen such that U shuffles the excitations on thegraph in a fully random way. For specific systems andspecific times intervals, this “classical” behavior mightnot be true due to interferences, but for general timesit is a good approximation (see Fig. 4). Let | C | be thenumber of edges on the graph controlled by Alice, and | ¯ C | the number of uncontrolled edges. On average, eachswap takes approximately a ratio | C | / ( | C | + | ¯ C | ) of exci-tations from the graph to the memory. We then get D ˆ N E ( L ) C ¯ C ≈ D ˆ N E (0) C ¯ C
11 + | C | / (cid:12)(cid:12) ¯ C (cid:12)(cid:12) ! L . (37)This is a reasonable result which shows that the fidelitydepends on the initial number of excitations and on therelative size of the controlled region with respect to theuncontrolled region. A v e r age nu m be r o f e xc i t a t i on s FIG. 4: Comparison of the approximation Eq. (37) with ex-act numerical results. Shown is the average number of excita-tions D ˆ N E C ¯ C on an open Heisenberg spin chain with 7 sitesand equal couplings as a function of the number of swaps tothe memory. The initial state is taken to be | i C ¯ C ,i.e. with a maximal number of excitations. The three curvescorrespond to different sizes of the region | C | controlled byAlice, and the time interval t has been chosen for each curveindependently to fit the approximation given in Eq. (37). V. CONCLUSION
We have given an explicit protocol for controlling andcooling a large permanently coupled system by accessinga small subsystem only. As we have shown, the applica-bility relies only on the invariant property (6) of a CPTmap. Since we had to assume a large quantum memoryin order to control the system, this protocol is not usefulfor replacing control in a homogeneous setup, but may
FIG. 5: A CCD-like application of our protocol could allow alight sensitive array of qubits to be read out coherently by aquantum computer without “disturbing” the qubits much. well have applications in inhomogeneous scenarios (whencontrol is harmful or expensive in some regions but easyin others). For example, we imagine a CCD-like appli-cation, in which a set of permanently coupled qubits isread out by a Quantum Computer in a coherent manner(see Fig. 5).
Acknowledgments
V.G. acknowledges the Quantum Information researchprogram of Centro di Ricerca Matematica Ennio De Giorgi of Scuola Normale Superiore for financial support.
APPENDIX A: EVOLUTION OF ¯ C Here we derive the evolution (4) of the subset ¯ C interms of the CP map (3). First rewrite the reduced den-sity matrix (4) as follows ρ ( L )¯ C = Tr CM (cid:2) W ( | Ψ i C ¯ C h Ψ | ⊗ | i M h | ) W † (cid:3) = Tr C h · · · Tr M h S U (cid:16) Tr M h S U (cid:16) | Ψ i C ¯ C h Ψ | ⊗ | i M h | (cid:17) U † S † i ⊗ | i M h | (cid:17) U † S † i · · · i . (A1)For the sake of clarity it is useful to explicitly denote the subsystems on which the various operators are acting on(e.g. Θ AB indicates that the operator Θ acts non trivially only on the subsystems A and B , while it is the identityelsewhere). By doing so and by using the properties [9] of the swap it is easy to verify the following identities:Tr M ℓ h S ℓ U (cid:16) ρ ¯ C ⊗ | i C h | ⊗ | i M ℓ h | (cid:17) U † S † ℓ i = Tr M ℓ h S CM ℓ U C ¯ C (cid:16) ρ ¯ C ⊗ | i C h | ⊗ | i M ℓ h | (cid:17) U † C ¯ C S † CM ℓ i = Tr M ℓ h S CM ℓ U C ¯ C (cid:16) S † CM ℓ S CM ℓ (cid:17)(cid:16) ρ ¯ C ⊗ | i C h | ⊗ | i M ℓ h | (cid:17)(cid:16) S † CM ℓ S CM ℓ (cid:17) U † C ¯ C S † CM ℓ i = Tr M ℓ h U M ℓ ¯ C (cid:16) ρ ¯ C ⊗ | i M ℓ h | ⊗ | i C h | (cid:17) U † M ℓ ¯ C i = τ ( ρ ¯ C ) ⊗ | i C h | , (A2)which holds for all ρ ¯ C and ℓ . Equation (4) then follows by replacing this into Eq. (A1) for all ℓ > M h S U (cid:16) | Ψ i C ¯ C h Ψ | ⊗ | i M h | (cid:17) U † S † i = Tr M h S CM U C ¯ C (cid:16) | Ψ i C ¯ C h Ψ | ⊗ | i M h | (cid:17) U † C ¯ C S † CM i (A3)= Tr M h U M ¯ C (cid:16) | Ψ i M ¯ C h Ψ | ⊗ | i C h | (cid:17) U † M ¯ C i = Tr M h U M ¯ C (cid:16) | Ψ i M ¯ C h Ψ | (cid:17) U † M ¯ C i ⊗ | i C h | ≡ ρ ′ ¯ C ⊗ | i C h | , with ρ ′ ¯ C as in Eq. (5). APPENDIX B: DECOMPOSITION EQUATIONS
Here we give a decomposition of the state after ap-plying the W operator of Eq. (2). This will allow us toestimate the fidelities for state transfer in terms of therelaxing properties of the map τ. Let | ψ i C ¯ C ∈ H C ¯ C be an arbitrary state. We noticethat the C component of W | ψ i C ¯ C | i M is always | i C .Therefore we can decompose it as follows W | ψ i C ¯ C | i M = | i C ⊗ h √ η | i ¯ C | φ i M + p − η | ∆ i ¯ CM i (B1)with η ∈ [0 ,
1] and with | ∆ i ¯ CM being a normalised vectorof ¯ C and M which satisfies the identity ¯ C h | ∆ i ¯ CM = 0 . (B2) It is worth stressing that in the above expression η , | φ i M and | ∆ i ¯ CM are depending on | ψ i C ¯ C . In a similar way wecan decompose the vector obtained by acting with W † on the first term of Eq. (B1), i.e. W † | i C ¯ C | φ i M = p ˜ η | ψ i C ¯ C | i M + p − ˜ η | ˜∆ i C ¯ CM , (B3)where | ˜∆ i C ¯ CM is the orthogonal complement of | ψ i C ¯ C | i M , i.e. ¯ CC h ψ | M h | ˜∆ i C ¯ CM = 0 . (B4)Multiplying Eq. (B3) from the left with C ¯ C h ψ | M h | andusing the conjugate of Eq. (B1) we find that η = ˜ η. Anexpression of η in terms of τ can be obtained by usingEq. (4). Therefore from Eq. (B1) and the orthogonalityrelation (B2) it follows that η = ¯ C h | τ L − (cid:0) ρ ′ ¯ C (cid:1) | i ¯ C , (B5)which, since τ is relaxing, shows that η → L → ∞ .Moreover we can use [18] to claim that1 − η = | ¯ C h | τ L − (cid:0) ρ ′ ¯ C (cid:1) | i ¯ C − | k τ L − (cid:0) ρ ′ ¯ C (cid:1) − | i ¯ C h |k K ( L − d ¯ C κ L − , (B6) where k Θ k = p Tr[Θ † Θ] is the trace norm of the opera-tor Θ, K is a constant which depends upon d ¯ C ≡ dim H ¯ C ,and where κ ∈ ]0 ,
1[ is the second largest of the moduli ofeigenvalues of τ. [1] L. Viola, S. Lloyd, and E. Knill, Phys. Rev. Lett. ,4888 (1999).[2] V. Scarani, M. Ziman, P. ˇStelmachoviˇc, N. Gisin, andV. Buˇzek. Phys. Rev. Lett. , 097905 (2002).[3] H. Nakazato, T. Takazawa, and K. Yuasa, Phys. Rev.Lett. , 060401 (2003).[4] T. Wellens, A. Buchleitner, B. K¨ummerer, andH. Maassen, Phys. Rev. Lett. , 3361 (2000).[5] V. Giovannetti and D. Burgarth, Phys. Rev. Lett. ,030501 (2006).[6] D. Burgarth and V. Giovannetti, Phys. Rev. Lett. ,012305 (2004).[8] B. K¨ummerer and H. Maassen, Inf. Dim. Anal. Quant.Prob. and Rel. Topics , 161 (2000).[9] Given two systems A and B of dimenstion d ,characterized by canonical basis {| i A , · · · , | d i A } and {| i B , · · · , | d i B } , respectively, the swap operators intro-duced in Sec. II A is the unitary transformation definedas S AB = S † AB ≡ P dj =1 P dk =1 | j i A h k | ⊗ | k i B h j | . Whenapplied to a joint state | Ψ i AB it exchanges the degreeof freedom of A with the degree of freedom of B . Moregenerally, for any Θ A and Ω A operators of A , and Θ B and Ω B their corresponding conterparts on B , we have S AB (Θ A ⊗ Ω B ) S AB = Ω A ⊗ Θ B .[10] M. Nielsen and I. Chuang, Quantum Computation andQuantum Information (Cambridge Univ. Press, Cam-bridge, 2000).[11] R. Gohm,
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