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Generation and stabilization of Bell states viarepeated projective measurements on a drivenancilla qubitTo cite this article: L Magazzù et al 2018 Phys. Scr. 93 064001

 

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Generation and stabilization of Bell statesvia repeated projective measurements on adriven ancilla qubit

L Magazzù1 , J D Jaramillo1, P Talkner1 and P Hänggi1,2

1 Institute of Physics, University of Augsburg, D-86135 Augsburg, Germany2Nanosystems Initiative Munich, Schellingstraße 4, D-80799 München, Germany

E-mail: luca.magazzu@physik.uni-augsburg.de

Received 13 February 2018, revised 6 April 2018Accepted for publication 17 April 2018Published 18 May 2018

AbstractA protocol is proposed to generate Bell states in two non-directly interacting qubits by means ofrepeated measurements of the state of a central ancilla connected to both qubits. An optimalmeasurement rate is found that minimizes the time to stably encode a Bell state in the targetqubits, being of advantage in order to reduce detrimental effects from possible interactions withthe environment. The quality of the entanglement is assessed in terms of the concurrence and thedistance between the qubits state and the target Bell state is quantified by the fidelity.

Keywords: quantum entanglement, quantum information, quantum measurement

(Some figures may appear in colour only in the online journal)

1. Introduction

Preparing entangled states is a basic requirement for manyquantum technologies [1, 2], notably for quantum information[3] and quantum metrology [4, 5]. Being entanglement anexquisite non-classical feature, its quantification is of funda-mental interest [6–8]. Several protocols to generate entangledstates have been developed to date, including control ofquantum dynamics [9–12] and engineered dissipation[13–22]. An intriguing route towards this goal is to exploit thequantum backaction of measurements performed on a part oron the whole system. In this context, different schemes havebeen proposed [23–28, 55] and implemented [29–31] whichrely on the use of a parity meter on the collective state of twoqubits. The introduction of a feedback control based on thereadout of a continuous weak measurement of parity providesfurther means to entangle bipartite systems [32–35]. A paritymeter of the state of two qubits, α and β, discriminates if theyare in an even or odd parity collective state, associated to thetwo eigenvalues 1 and −1 of the parity operator z zs sÄa b,respectively. Consider the one-qubit state +ñ = ñ +∣ (∣

2ñ∣ ) expressed in the eigenbasis of σz. A parity mea-surement on the two-qubit system prepared in the separablejoint state + ñ + ña b∣ ∣ projects the system onto one of the Bell

states 2F ñ = ñ + ñ+∣ (∣ ∣ ) and Y ñ = ñ ++∣ (∣2 ñ∣ ) , corresponding to even and odd parity outcome,

respectively. An advantage of the class of schemes based onparity measurements is that they do not require direct inter-action between the qubits, a feature that makes them suitablefor linear optics setups, e.g., the scheme for quantum com-puting discussed in [36].

In practical implementations, the parity measurementsmay involve the coupling to an ancillary qubit upon whichmeasurements are performed, and the use of multi-qubit gatesto prepare the ancilla qubit as a parity meter [29, 30]. On theother hand, the action on the ancilla to drive the system state,allows for keeping the target qubits more isolated from theenvironment (which in general includes the measurementapparatus). In such a situation, the ancillary system can thusbe considered as a so-called quantum actuator (see [37] andreferences therein) accomplishing the indirect control of thesystem state.

In the present work we exploit the idea of a shared ancilladriven by the measurement backaction [38–40] to circumventthe use of collective unitary gates to generate Bell states in abipartite target system. Specifically, we encode and stabilizethe Bell states in a couple of mutually non-interacting qubits(B and C) by repeated projective measurements of the state of

Physica Scripta

Phys. Scr. 93 (2018) 064001 (9pp) https://doi.org/10.1088/1402-4896/aabeb8

0031-8949/18/064001+09$33.00 © 2018 IOP Publishing Ltd Printed in the UK1

a shared ancilla qubit (A) which may also driven by localcontrol fields (see figure 1). The sequence of measurements isperformed starting with the full system in the factorized statewith the three qubits in the same spin state. A readilyimplementable form of feedback, i.e., a ramp of the controlfield on A triggered by a specific outcome of the measure-ment, ensures that in the ideal case of perfect isolation fromthe environment the protocol yields a Bell state with prob-ability 1. Moreover, the sequence of outcomes of the mea-surements on A, unambiguously identifies the specific Bellstate in which the target qubits B and C are left asymptoti-cally. One can then switch among the four Bell states encodedin BC by addressing locally either B or C with a single qubitoperation [41], an action that does not require proximity orinteraction between the entangled qubits. The feasibility ofour scheme benefits from the progress in rapid high-fidelity,single-shot readout in circuit QED, superconducting qubits,and spins in solid state systems [42–50].

By analyzing the results of the protocol with differentinter-measurement times, we are able to identify the optimalrate of measurements to generate stable Bell states in aminimal amount of time. This is crucial for successfullyproducing a Bell state before the detrimental effects ofenvironmental noise spoil the protocol [51]. We then use thisoptimal rate to examine how stable Bell states in the targetqubits can be produced in a short time window, comprising∼20 measurements on ancilla A, before the latter is dis-connected leaving the target qubits isolated. We explicitlydepict sample time evolutions of the density matrix,corresponding to different realizations of the measurementsequences.

2. Setup

The model considered in our protocol consists of an openone-dimensional chain comprised of three 1/2-spins A, B, andC (the qubits, see figure 1). The central spin A plays the roleof an ancilla and is connected to the measurement apparatuswhich projectively monitors its spin state in the z-direction.The ancilla A can be driven by a control field along the z-axis.

The target spins B and C interact exclusively with A. Weassume that dephasing and relaxation effects fromthe environment do not affect the system, at least on the timescale of the protocol with optimal monitoring rate (seebelow). The inter-spin coupling and the control fieldare adjusted by control functions uJ(t) and uh(t), respectively,both assuming values in [0,1].

The Hamiltonian reads:

H t J u t h u t , 1x J xA

xB

xC

z h zAs s s s= + -( ) ( ) ( ) ( ) ( )

where σ ij denote the Pauli spin operators ( j=x, y, z) of spin i.

Here the couplings Jx and hz determine the magnitude ofinteraction and the field strength, respectively. The dynamicsof the three qubit system between consecutive measurementsis induced by the Hamiltonian in equation (1). Thus, the timeevolution of the total density matrix ρ of the tripartite systemA, B, C is governed by the Liouville–von Neumann equation

t H t ti

, . 2

r r= -˙ ( ) [ ( ) ( )] ( )

Throughout the present work we scale energies and timeswith the coupling Jx which can be in practice very small,favoring the isolation of the target qubits, but at the expenseof a longer duration of the protocol. Accordingly, we considerfor the control field driving the ancilla the (maximal)value h J50z x= .

The state of the three qubits (ancilla A and target qubitsB, C) are expressed in the basis nml n m lA B Cñ ñ ñ ñ∣ ≔ ∣ ∣ ∣ , wheren m l, , 0, 1Î { } are the eigenvalues of the operators

Z 1 2, 3i izis= +ˆ ( ) ( )

with i=A, B, and C, respectively. We consider repeatedprojective measurements of Z Aˆ which are assumed to beinstantaneous, meaning that they take place on a time scalewhich is much smaller than the dynamical time scales of thesystem. Each measurement on the ancilla projects the state ofthe full system into one of the eigensubspaces correspondingto the eigenvalues 0 and 1 of Z Aˆ .

In the next sections we study the time evolution of thesystem subject to repeated measurements of Z Aˆ , starting fromthe fully separable initial state 111 1110r = ñá∣ ∣ with hz set tozero. After each measurement, the density matrix evolvesunitarily with respect to H(t), until the next measurement isperformed. The cycle is repeated for an overall time com-prising several inter-measurement times.

3. Evolution under nonselective monitoring andoptimal inter-measurement time.

In our protocol, measurements of the spin state of the ancilla Aoccur at equally-spaced times instants tn=nτ, where τ is theinter-measurement time and n�1. The scope of the presentsection is to establish how the time needed to eventually reachan asymptotic state depends on τ. For this purpose, we study theevolution of the system undergoing a sequence of nonselectivemeasurements in the absence of external fields, i.e., uh(t)=0,and with uJ(t)=1. In a nonselective measurement the outcome

Figure 1. Two mutually non-interacting qubits (B and C) are coupledto a shared ancilla A whose state ( zs ) is projectively monitored.During the protocol, a control field acts on A triggered by a specificreadout. We assume the qubits to be isolated from the environment.

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Phys. Scr. 93 (2018) 064001 L Magazzù et al

is disregarded or simply not available, thus yielding only aprobabilistic information about the post-measurement state ofthe system. This is in distinct contrast to the case of selectivemeasurement where each measurement prepares the system inthe eigenstate corresponding to the outcome of the measuredoperator. A nonselective measurement of Z Aˆ reduces the stateof the entire system to a probabilistic mixture of projections intothe eigenstates 0Añ∣ and 1Añ∣ (see equation (3)). This transfor-mation is provided by the action of the projectors iP ºi i 1A A BCñá Ä∣ ∣ with i 0, 1Î { }. Indeed, immediately after thenth nonselective measurement, taking place at time tn=nτ, thedensity matrix of the total system can be written as

t U t U U t U , 4n n n n n n n0 1 0 1 1 1r r r= P P + P P- -( ) ( ) ( ) ( )† †

where Un is the time evolution operator from tn−1 to tn.In the presence of repeated nonselective measurements, the

density matrix at an arbitrary time t is calculated as follows.Starting at the initial time t0=0 in the state 111 1110r = ñá∣ ∣,the density matrix is propagated through equation (2) for a timespan τ after which the coherences between A and the target qubitsB and C are removed upon measuring the state of A, according toequation (4). Then, the post-measurement state ρ(t1) described byequation (4) is used as the initial condition for a further propa-gation up to time t2=2τ where a second measurement takesplace. The above sequence is repeated and the total number ofmeasurements occurring up to time t is given by the integer partof t/τ. A scheme of this sequence is depicted in figure 2. We findthat, for any non-pathological choice of the inter-measurementtime τ, i.e., for inter-measurement times that do not matchmultiples of the free system periodicity, the asymptotic state

1

21 1

1

41 1

1

40 0

5

A A BC BC A A BC BC

A A BC BC

r = ñá Ä F ñáF + ñá Ä F ñáF

+ ñá Ä Y ñáY

¥ - - + +

+ +

∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣

∣ ∣ ∣ ∣( )

is eventually reached, where the four Bell states are defined as

1

211 00

1

210 01 . 6

BC BC BC

BC BC BC

F ñ= ñ ñ

Y ñ= ñ ñ

∣ (∣ ∣ )

∣ (∣ ∣ ) ( )

Note that the Bell state BCY ñ-∣ does not appear in equation (5).This is due to the particular initial condition chosen, as explainedin appendix A. Inspection of equation (5) reveals that, once the

asymptotic state is attained, a further measurement on A, withavailable outcome, yields for the target qubits B and C the Bellstate BCY ñ+∣ conditioned on the readout 0, which occurs withprobability 1/4. On the other hand, the outcome 1 yields for thetarget qubits a probabilistic mixture of Bell states. In the fol-lowing section and in appendix B, a protocol that also yields theother entangled states BCY ñ-∣ and BCF ñ+∣ as pure states is specified.

To establish the optimal inter-measurement time τ* forwhich the asymptotic state r¥ is reached in the least amountof time, we consider the trace distance between the time-evolved density operator ρ(t) and the asymptotic state r¥.This quantity is defined as

D t t,1

2Tr

1

2, 7

iiå

r r r r

l

= -

=

¥ ¥( ( ) ) ∣ ( ) ∣

∣ ∣ ( )

with λi denoting the eigenvalues of the Hermitianmatrix tr r- ¥( ) .

The results for the trace distance as a function of time tand of the inter-measurement time τ are depicted in figure 3

Figure 2. Scheme of the unitary evolution of the full systeminterrupted by the sequence of equally-spaced measurements of thestate of qubit A. The time evolution between measurements isinduced by the Hamiltonian(1) according to the Liouville–vonNeumann equation (2).

Figure 3. Time evolution of the full density matrix under repeatednonselective measurements. (a) Trace distance D ,r r¥( ) betweenthe time-evolved state ρ≡ρ(t) and the asymptotic state r¥ (seeequation (5)) as a function of the actual time t and the inter-measurement time τ. For each value of τ the density matrix ispropagated up to the final time t J10 x= according to thesequence depicted in figure 2. The time evolution of the densitymatrix between measurements is obtained by numerically solving ofthe Liouville–von Neumann equation (2) with uh(t)=0 anduJ(t)=1 in the Hamiltonian(1). At the measurement times the stateof the system is transformed as prescribed by equation (4). Theinitial state is 0 111 111r = ñá( ) ∣ ∣. The horizontal line at τ=0.5 ÿ/Jxhighlights the optimal inter-measurement time to rapidly relax to theasymptotic state. In the limit 0t the system enters the quantumZeno regime, as witnessed by the freezing of the trace distance fromthe asymptotic state at its initial value. (b) Trace distance versus timefor three fixed values of τ. The red thick line corresponds to theoptimal inter-measurement time.

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Phys. Scr. 93 (2018) 064001 L Magazzù et al

for u t 0h =( ) (no external field on A) and uJ(t)=1 (constantqubits-ancilla coupling), see equation (1). The plot infigure 3(a) shows that an optimal inter-measurement timearound the value J0.5 x* t = exists which corresponds tothe minimal time t needed to relax to the final state r¥ (seethe horizontal dashed line). The curves depicted in figure 3(b)clearly show that, for τ=τ*, the asymptotic state is reachedafter few measurements whereas deviations from this optimalinter-measurement time entail a considerably larger numberof measurements. Indeed, in the limit of very small τ—wellbelow the optimal level—the system enters the quantum Zenoregime, as witnessed by the freezing of the trace distance fromthe asymptotic state at its initial value (see the lower part offigure 3(a))3. On the other hand, upon increasing the inter-measurement time τ above the optimal value τ* the mea-surements eventually synchronize with the periodicity of thefree system so that oscillations persist for long evolutiontimes around selected values of τ, as shown in the upper partof figure 3(a).

4. Selective evolution under projectivemeasurements and feedback

Next we focus on the selective evolution of the system underthe same sequence of repeated measurements on A detailed insection 3 (see figure 2). In this case the individual realizationsof the time evolution of ρ(t) determined by specific sequencesof random outcomes of the measurements on A are con-sidered. We show that the protocol converges to a factorizedstate with i iA A BCrñá Ä∣ ∣ , with the reduced system BC in a Bellstate.

The evolution under selective measurements is calculatedas follows: starting at t=0 with the system in the state

111 1110r = ñá∣ ∣, the full density matrix is evolved accordingto equation (2) for a time span τ. At time t1=τ the firstmeasurement takes place yielding a random outcomei 0, 1Î { } generated with probability Tr ir t P[ ( ) ]. The state ofthe system is then updated to the post-measurement statewhich is in turn used as the initial condition for a furtherunitary evolution of duration τ and so on. The process, withthe measurements taking place at times tn=nτ, is depicted infigure 2. The explicit expression for the state immediatelyafter the nth measurement with outcome i 0, 1n Î { } is

tU t U

U t UTr. 8n

i n n n i

n n n i

1

1

n n

n

rrr

=P P

P-

-( ) ( )

[ ( ) ]( )

†

†

The process described yields random realizations of the timeevolution of the density matrix. The expression for theprobability associated to a specific realization, i.e., to a spe-cific readout sequence i i, , ,...n1 ¼( ), is given in appendix B(see equation (B.8)). In the absence of a control field (hz=0),

we find two possible behaviors: either the repeated mea-surements on A yield an uninterrupted sequence of 1ʼs, andthe system is asymptotically left in the state 1A BCñ F ñ-∣ ∣ , or thesequence randomly flips between the two outcomes 1 and 0.In the latter case the state of the system flips between the twostates 1Añ F ñ+∣ ∣ and 0Añ Y ñ+∣ ∣ . In appendix B we account forthis behavior by explicitly considering actual realizations ofthe selective evolution.

To cope with the oscillating behavior taking place in asubset of the realizations of the protocol, a simple feedbackscheme can be implemented which does not require particularprecision in its execution. A ramp of the control field

h u tz h zAs- ( ) acting on A (see equation (1)) is turned on trig-

gered by the first readout of a 0 in the sequence of mea-surements. Specifically, the field is initially zero and is thenswitched on at the time t* when a first detection of the state ofA with outcome 0 occurs during the protocol. If no outcome 0is found the field stays off during the evolution (see forexample panel (a) of figure 4 below). Let tF be the totalduration of the protocol. The switching function is the smoothramp

u tt t t

t t t t t

t t t

1 cos 2

2, 2

1, 2

9h F F

F

*

* *

*

p

=Q - -- +

> +

⎧⎨⎪⎩⎪

( )( ){ [ (

) ]} ( )( )

( )

see figures 4(a)–(c). The stabilizing effect on the oscillatingsequences is due to the large value ( J50 x) of hz which makesthe field term dominating with respect to the interaction termin the Hamiltonian (1). As a result, a state with a definiteeigenvalue of Z Aˆ , such as a post-measurement state, isapproximately an eigenstate of the Hamiltonian and does notevolve on the time scales of the protocol. We note that thisstabilizing effect—and thus the results in the present work—does not depend on the precise form of the ramp uh(t). Thefinal part of the protocol consists in switching off the inter-action strength J u tx J ( ): the interaction is held constantu t 1J =[ ( ) ] throughout the protocol and is switched off at timetF, i.e., u t t 0J F =( ) , after the target qubits B and C havereached a steady state.

In figure 4, the time evolution of the reduced densitymatrix tBCr ( ) under selective measurements is shown for threesample realizations of the protocol. These realizations end upwith the target qubits left in different Bell states, corresp-onding to different sequences of the measurement readouts.The readout sequences are also shown for each sample timeevolution along with the behavior of the control field actingon A. In the numerical simulations, the inter-measurementtime τ is fixed at the optimal value J0.5 x* t = and theswitch off takes place at t J10F x= , namely, after 20measurements. A realization of the protocol leaves the targetqubits in the Bell state BCF ñ-∣ with probability 1/2 or in one ofthe Bells states BCY ñ+∣ and BCF ñ+∣ with total probability 1/2,(see the definitions in equation (6)). Although the protocolgenerates the Bell states non-deterministically, once thesteady state is reached they are unambiguously identified byreading the sequence of outcomes of the measurements on A.Having identified the Bell state encoded in the system BC,

3 In a different setup [52], measurements of the joint state of two qubits areproposed to entangle the qubits by exploiting the Zeno effect occurring atlarge measurement rates. Here, instead, we operate in the opposite regimewhere measurements at an appropriate, intermediate rate produce the fastestrelaxation to a target state.

4

Phys. Scr. 93 (2018) 064001 L Magazzù et al

one can act locally on qubit B or C, by applying a rotation ofthe state of the qubit, to switch to a different Bells state [41].

A scheme of the protocol is provided in the upper panelof figure 5, where the three realizations in figures 4(a)–(c) areassociated to the corresponding readout sequences and thefinal Bell state for the target qubits B and C. Note that, byusing a different initial condition, the final Bell states asso-ciated to definite readout sequence can differ from the onesdescribed here. This is exemplified in appendix C, where theprotocol is carried out with the three qubits initially preparedin the state 110ñ∣ .

To study the degree of entanglement during the protocol,we evaluate the concurrence of the reduced density matrix ofthe target system BC. This quantity is defined as tBC r =[ ( )]max 0, 1 2 3 4l l l l- - -( ), where the λiʼs are the orderedeigenvalues of the matrix t ty

ByC

yB

yCBC BC *r s s r s sÄ Ä( )( )[ ( )] ( )

[6, 7]. The concurrence takes values between zero and one,corresponding to non-entangled (zero) and maximally entangledstates (one), respectively. The results are depicted in figure 4(d)for the three sample realizations displayed in panels (a)–(c). Inaddition, we confirm that the reduced two-qubit system BC isleft in the specific Bell state corresponding to a particularreadout sequence. This is done by calculating the fidelity

[53, 54], given by the expression

F t t t, Tr , 10BC BC BC BC BCr r r r r=¥ ¥[ ( ) ] [ ( ) ( ) ] ( )

where the referential density matrix BCr¥ is chosen a posteriori,as the final state that corresponds to the readout sequence (seethe upper panel of figure 5). The time evolutions of the fidelityfor the sample evolutions in figure 4 are depicted in the lowerpanel of figure 5.

We conclude this section by noting that, in a realisticsituation, environmental effects are present, especially on theancilla qubit A which is connected to the meter. This makesthe choice of the optimal time crucial in order to conclude theprotocol before these detrimental effects spoil it. However,the realizations of the protocol in the classes depicted infigures 4(a) and (b) are expected to be more robust withrespect to the environmental influence, as compared to thosewhere the large control field acting on A freezes the system inthe higher energy state 0A BCñ Y ñ+∣ ∣ . In this latter case, the largeenergy splitting may cause the decay 0 1A Añ ñ∣ ∣ before theprotocol is completed.

Alternatively, one may think of freezing the oscillationsbetween 1Añ F ñ+∣ ∣ and 0Añ Y ñ+∣ ∣ by the Zeno effect, namely by

Figure 4. Time evolution of selected elements of the reduced density matrix tBCr ( ), for three sample realizations of the protocol with initialcondition 111ñ∣ . Each trajectory leads to a different Bell state for the target system BC. Specifically, (a) BCF ñ-∣ , (b) BCF ñ+∣ and (c) BCY ñ+∣ (seefigure 5 below). Persistent fluctuations in the trajectories (b) and (c) are suppressed by applying an external control field of the form h u tz h- ( ),with h J50z x= , to the ancilla A. Below each evolution we depict the corresponding sequence of outcomes from measuring the state of A andthe time dependent function uh(t) of the control field (see equation (1)) acting on A. (d) Time evolution of the concurrence of tBCr ( ) for thetrajectories (a)–(c). The inter-measurement time is set to the optimal value τ=0.5 ÿ/Jx.

5

Phys. Scr. 93 (2018) 064001 L Magazzù et al

suddenly let the inter-measurement time τ go to zero after ameasurement on the ancilla with outcome 0.

5. Conclusions

With this work a simple protocol is presented for generatingBell states in a couple of qubits which do not interact directly.The qubits are entangled by means of repeated projectivemeasurements of the state of a shared ancilla qubit. We haveshown that the protocol yields a definite and stable Bell state,starting from the completely factorized state 111ñ∣ of the fullsystem. This is attained by acting on the ancilla with a sui-table form of feedback, namely a ramp of a control field (seeequation (9)), triggered by a specific readout of the ancillastate. The backaction of the repeated measurements asymp-totically yields a stationary state of the form iA BCjñ ñ∣ ∣ , withi 0, 1Î { } and , ,BC BC BC BCj ñ Î F ñ Y ñ F ñ- + +∣ {∣ ∣ ∣ }, where theBell state is unambiguously identified by reading thesequence of measurement outcomes of the ancilla.

Acknowledgments

This work is dedicated to WolfgangP Schleich, a greatscientist and constant source of inspiration, on the occasion ofhis 60th birthday. PH and PT also thank Wolfgang for hisvivid and illuminating many discussions and constructivedebates during our yearly, alternating Augsburg-Ulm

workshops. In addition PH likes to thank Kathy and Wolf-gang for their lifetime-long wonderful friendship.

Appendix A. Asymptotic state for nonselectiverepeated measurements with different initialconditions

To gain insight in the reasons why, with the chosen initialseparable state

1111

21 , A.1A BC BCñ = ñ F ñ + F ñ+ -∣ ∣ (∣ ∣ ) ( )

the sequence of nonselective measurement considered insection 3 ends up with asymptotic state in equation (5), it issufficient to note that the Hamiltonian(1), in the absence ofcontrol fields and with constant interaction strength, can bewritten as

H t J2 . A.2x x BC BC BC BCAs= F ñáY + Y ñáF+ + + +( ) (∣ ∣ ∣ ∣) ( )

The state 1A BCñ F ñ-∣ ∣ is an eigenstate of the Hamiltonian and theonly states involved that are connected by a nonzero transitionamplitude are 1A BCñ F ñ+∣ ∣ and 0A BCñ Y ñ+∣ ∣ . Thus, with the choseninitial state, the probability to get the target qubits in theBell state BCY ñ-∣ is zero, as the dynamics induced byHamiltonian (A.2) is confined to the subspace spannedby 1 , 1 , 0A BC A BC A BCñ F ñ ñ F ñ ñ Y ñ- + +{∣ ∣ ∣ ∣ ∣ ∣ } and the measure-ments on A do not affect this feature. On the other hand,by starting with a different initial state, as for example110 1 2A BC BCñ = ñ Y ñ + Y ñ+ -∣ ∣ (∣ ∣ ) , the asymptotic state reads

1

21 1

1

41 1

1

40 0 , A.3

A A BC BC

A A BC BC

A A BC BC

r = ñá Ä Y ñáY

+ ñá Ä Y ñáY

+ ñá Ä F ñáF

¥ - -

+ +

+ +

∣ ∣ ∣ ∣

∣ ∣ ∣ ∣

∣ ∣ ∣ ∣ ( )

(see equation (5)). A complete list of the asymptotic statesattained by starting in each of the (computational) basis statesis shown in table A1.

Figure 5. Upper panel—schematics of the protocol to generate Bellstates in qubits B and C by a sequence of measurements on theancilla qubit A, starting in the state 111ñ∣ . The scheme shows threedifferent readout sequences with different resulting Bells statesencoded in BC, corresponding to the three sample realizations infigures 4(b), (c). A smooth ramp of the control field h u tz h- ( ),triggered by the first detection of a 0 in the outcomes’ sequence (seefigure 4), is applied to the ancilla in order to suppress the oscillatorybehavior between 1A BCñ F ñ+∣ ∣ and 0A BCñ Y ñ+∣ ∣ and converge to a definiteresult. Lower panel—fidelity of the trajectories depicted infigures 4(a)–(c) to the corresponding target Bell states.

Table A1. Repeated nonselective measurements in the absence ofexternal fields—asymptotic mixed density matrix for eight initialconfigurations.

Init. state r¥

111ñ∣ Equation (5)100ñ∣

110ñ∣ Equation (A.3)101ñ∣

011ñ∣ Equation (5) with 1 0A Añ « ñ∣ ∣000ñ∣

010ñ∣ Equation (A.3) with 1 0A Añ « ñ∣ ∣001ñ∣

6

Phys. Scr. 93 (2018) 064001 L Magazzù et al

To gain an intuition on how the entries in the table areobtained, let us introduce the map that propagates thedensity matrix for a time span τ and then applies a non-selective measurement on the ancilla. This map is defined bythe action

t U t U U t U

t . A.4n n n n n n n

n

0 1 0 1 1 1

1r r r

r=P P + P Pº

- -

-

( ) ( ) ( )[ ( )] ( )

† †

Then, the asymptotic state is given by

lim . A.5n

n0r r=¥

¥[ ] ( )

Now, let j be the operation that flips the state of spinj A B C, ,Î { } in the computational basis (the eigenbasis ofZ jˆ ). For j B C,Î { }, both the Hamiltonian and the projectorsΠi are invariant under such operation, we have

, A.6nB B

n0 0 r r=[ ] [ ] ( )

and similarly nC C

n0 0 r r=[ ] [ ] and n

B C 0 r =[ ]B C

n0 r[ ]. It follows that flipping the spin of either B or C

in the initial state 111 1110r = ñá∣ ∣ returns the asymptoticstate in equation (5) with the spin of B or C flipped, i.e.,equation (A.3). On the other hand, flipping both the spins of Band C in the initial state returns the asymptotic state(5) itself,

because B C r r=¥ ¥. The above reasoning accounts forthe first two rows of table A1.

In a similar way, by writing 0 1 1 0AA A A As = ñá + ñá∣ ∣ ∣ ∣ we

see that the Hamiltonian is invariant under the action of A andthat A 0 1 1 0 P = P( ) ( ), which entails n

A An

0 0 r r=[ ] [ ](see equation (A.4)). This accounts for the last two rows oftable A1.

Appendix B. Details of the selective evolution

Let us inspect the actual realizations of the time evolutiongiven by repeatedly measuring the state of A in a selectivefashion, starting from the state 111ñ∣ in the absence of controlfields, uh(t)=0. The action of the time evolution operatorinduced by the Hamiltonian(A.2) for a time span τ is

U ab

1111

21

20 ,

B.1

A BC BC A BCt ñ = ñ F ñ + F ñ + ñ Y ñ- + +( )∣ ∣ (∣ ∣ ) ∣ ∣

( )

with a b 12 2+ =∣ ∣ ∣ ∣ , where a and b depend on τ. The firstmeasurement on the ancilla collapses this evolved state into

Figure C1. Time evolution of selected elements of the reduced density matrix tBCr ( ) and scheme of the protocol. Each of the three samplerealizations of the protocol starts with the system in the state 110ñ∣ and leads to a different Bell state for the target system B, C. Specifically,(a) BCY ñ-∣ , (b) BCY ñ+∣ and (c) BCF ñ+∣ . Persistent fluctuations in the trajectories (b) and (c) are suppressed by applying an external control field ofthe form h u tz h- ( ), with h J50z x= , to the ancilla A. Below each evolution we depict the corresponding sequence of outcomes frommeasuring the state of A and the time dependent control function uh(t). The inter-measurement time is set to the optimal value τ=0.5 ÿ/Jx.(d) Scheme of the readout sequences and final Bell states corresponding to panels (a)–(c) (see figure 5).

7

Phys. Scr. 93 (2018) 064001 L Magazzù et al

one of the two alternative outcomes

a

b

1 prob 2

0 prob 2,

B.2

A BC BC

A BC

1 1 12

02

f

f

ñ = ñ F ñ + F ñ =

ñ= ñ Y ñ =

- +

+

∣ ∣ (∣ ∣ )

∣ ∣ ∣ ∣ ∣( )

( )

( )

where the indexes 1 0 ...( ) are used for bookkeeping thesequence of outcomes and a1n

n2 = + ∣ ∣ . Then, evolvingthe above states for another time span τ we get

U a ab1 0 ,

B.3A BC BC A BC1

21 1 t f ñ = ñ F ñ + F ñ + ñ Y ñ- + +( )∣ ∣ (∣ ∣ ) ∣ ∣

( )( )

U a b0 1 . B.4A BC A BC0t f ñ = ñ Y ñ + ñ F ñ+ +( )∣ ∣ ∣ ∣ ∣ ( )( )

Upon measuring again the state of A the following two cou-ples of alternative outcomes arise

a

ab

1 prob

0 prob

B.5

A BC BC

A BC

112

2 22

12

102

12

f

f

ñ = ñ F ñ + F ñ =

ñ = ñ Y ñ =

- +

+

⎪

⎪

⎧⎨⎩

∣ ∣ (∣ ∣ )

∣ ∣ ∣ ∣ ∣

( )

( )

( )

b

a

1 prob

0 probB.6

A BC

A BC

012

002

f

f

ñ = ñ F ñ =

ñ = ñ Y ñ =

+

+

⎪

⎪

⎧⎨⎩

∣ ∣ ∣ ∣ ∣

∣ ∣ ∣ ∣ ∣( )( )

( )

and so on. From this behavior it is clear that, for a 1¹ , i.e., ifτ is not a multiple of the period of the unitary evolution, along sequence of outcomes 1 will yield the stabilized state1A BCñ F ñ-∣ ∣ , meaning that a further measurement will leave thesystem in this same state with probability ∼1. On the otherhand, the presence of zeroes in the readout sequences entailsoscillations between 1A BCñ F ñ+∣ ∣ and 0A BCñ Y ñ+∣ ∣ which lastindefinitely. This behavior is very close to what is found in[55] by simulating the repeated parity measurements in acouple of double quantum dot qubits. The probability asso-ciated to a specific sequence of outcomes i i i, , , N1 2 ¼( ), wherei 0, 1n Î { }, is given by

P i i i

U U U U U U

, , ,

Tr ... ... ,

B.7

N

i N i i i i N i

1 2

2 1 0 1 2N N2 1 1 2r

¼

= P P P P P P

( )[ ]

( )

† † †

where, as in equation (4), Un is the time evolution operatorfrom tn−1 to tn. By extrapolating from the sequence inequations (B.1)–(B.6) one sees that the probability to obtainan interrupted sequence of n readouts 1 is

P 1, , 12

...1

2

1

2.

B.8

n

n

n

n

n1 2

1 1 1

1

¼ = =- -

( ) ⟶

( )

Appendix C. Realizations of the protocol with adifferent initial condition

In figure C1 we depict three sample realizations of the pro-tocol detailed in section 4 starting in the factorized state110 1 2A BC BCñ = ñ Y ñ + Y ñ+ -∣ ∣ (∣ ∣ ) . A schematics of the pro-tocol’s outcomes and readout sequences is also shown.

ORCID iDs

L Magazzù https://orcid.org/0000-0002-4377-8387

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