Frequency-encoded linear cluster states with coherent Raman photons

# Frequency-encoded linear cluster states with coherent Raman photons

Dale Scerri    Ralph N. E. Malein    Brian D. Gerardot    Erik M. Gauger SUPA, Institute of Photonics and Quantum Sciences, Heriot-Watt University, EH14 4AS, United Kingdom.
August 22, 2019
###### Abstract

Entangled multi-qubit states are an essential resource for quantum information and computation. Solid-state emitters can mediate interactions between subsequently emitted photons via their spin, thus offering a route towards generating entangled multi-photon states. However, existing schemes typically rely on the incoherent emission of single photons and suffer from severe practical limitations, for self-assembled quantum dots most notably the limited spin coherence time due to Overhauser magnetic field fluctuations. We here propose an alternative approach of employing spin-flip Raman scattering events of self-assembled quantum dots in Voigt geometry. We argue that weakly driven hole spins constitute a promising platform for the practical generation of frequency-entangled photonic cluster states.

Introduction – Robust highly-entangled ‘cluster’ states are of paramount importance for measurement-based (‘one-way’) quantum computation Briegel2001b ; Briegel2001a ; Raussendorf2007 ; OBrien2009 . The experimental challenges of obtaining high-dimensional cluster states can be significantly reduced by probabilistically ‘fusing’ qubits from adjacent 1D linear cluster (LC) states TerryFusion2005 ; Herrera2010 ; Weinstein2011 , or ‘glueing’ together micro-clusters Nielsen2004 . Several platforms for generating photonic LC states have been proposed, varying from condensed matter emitters such as quantum dots Terry2009 ; Denning2017 ; Barrett2005 ; Lin2008 ; Herrera2010 ; Schwartz2016 and crystal defects Economou2016 ; Barrett2005 to parametric downconversion Vallone2007 ; Zou2005 , all presenting their own sets of advantages and challenges. Solid-state-based protocols typically make use of pulsed excitations to drive optical transitions in a matter qubit to entangle the emitter’s spin degree of freedom with the polarisation of subsequently emitted photons. Encouragingly, a photonic LC of length two (LC) has recently been demonstrated experimentally, showing that the entanglement in this setup could persist for up to five consecutively emitted photonsSchwartz2016 .

Whilst conceptually elegant and ostensibly deterministic, real-world imperfections pose significant barriers to the experimental realisation of protocols such as the ones introduced by Refs. Terry2009, ; Economou2010, ; Denning2017, ; Terry2018, . These include phonon-dephasing of excited states Iles-Smith2017 , modified selection rules as a consequence of hole mixing as well as a transverse (Voigt) component of the Overhauser field Loss2002 ; Testelin2009 ; Hansom2014 ; Malein2016 , and limited spin lifetimes due to Overhauser field fluctuations Loss2002 ; Merkulov2002 ; Braun2002 ; Chekhovich2013 ; Urbaszek2013 . Decoupling techniques Viola1999 ; Witzel2007 ; Zhang2007 ; Uhrig2007 ; Uhrig2008 ; Bluhm2010 ; Stockill2016 and control of the nuclear environment Eble2006 ; Petta2008 ; Urbaszek2013 ; Majcher2017 overcome the latter but provide no remedy for other error sources. Unavoidable shortcomings of real quantum dots thus severely limit the size of cluster state achievable and render genuinely deterministic operation impractical for the current experimental state-of-the-art.

In contrast to direct pulsed excitation, we here propose employing a weak continuous wave (c.w.) laser to drive the Zeeman-detuned transitions of a hole-spin for entangling the spin with the frequency of Raman scattered photon. We show such a setup overcomes the experimental barriers suffered by previous schemes: in particular, our protocol is impervious to phonon dephasing, robust against fluctuations of the Overhauser field, and unaffected by heavy-hole (hh) light-hole (lh) mixing. This comes at the cost making the protocol probabilistic, however, we show that LC states of sufficient length to serve as building blocks for fusion TerryFusion2005 can be produced at high rates and with high fideliy based on current experimental capabilities. Further, extended versions of our protocol mitigating its probabilistic limitations (whilst keeping its robustness) are possible. Our work thus shows that the significant divide between elegant theoretical proposals and experimental progress in the generation of linear cluster states can be overcome.

Model – Despite its many attractive features for quantum metrology and quantum informationLoss1998 ; Merkulov2002 , the spin of an electron trapped in an epitaxial quantum dot suffers from rapid ensemble dephasing due to the hyperfine interaction with randomly fluctuating nuclear spins of the host material. This typically results in a loss of coherence on the order of nanosecondsMerkulov2002 ; Braun2002 ; Chekhovich2013 ; Malein2016 . By contrast, the -orbital-like wavefunction of hole spin states vanishes at the location of the nuclear spins, which suppresses the Fermi-contact interaction, leaving only the much weaker dipole-dipole interaction as the main source of dephasing Fischer2008 ; Testelin2009 ; Chekhovich2011 ; Fallahi2010 . Strain lifts the degeneracy of the hole states, resulting in energetically split heavy () and light () holes; the former being closer to the valence band edge (see Fig. 1). However, chiefly due to strain anisotropy in the QD, a finite admixture of these states is always present (the effects on hole-based multi-photon entanglement schemes are briefly discussed in Appendix Sec F.3). In the following, we denote the (Zeeman) spin state of the heavy hole as and whereas the electron spin states are and . In this notation, the positively charged X transition couples to polarised light and to polarised light. In the absence of an external magnetic field, both and transitions are dipole-forbidden. However, an external magnetic field in Voigt geometry unlocks those diagonal Raman transitions (see Fig. 1)Emary2007 . For weakly off-resonantly driven hole spins, the width of these Raman transitions is solely limited by the laser linewidth and ground state spin dephasing Imamoglu2009 ; Sun2016 , making them attractive candidates for single photon sources, as well as being attractive spin-spin qubit entanglers due to the spin’s rich level scheme and selection rules Delteil2015 ; Stockill2017 .

Wishing to exploit such Raman photons for LC generation we consider a self-assembled quantum dot in the Voigt geometry, with the applied magnetic field strong enough to dominate over nuclear Overhauser field fluctuations (see Appendix Sec B). The applied -field (w.l.o.g. along the -axis) then defines the basis of spin eigenstates. We also include a c.w. laser field that is resonant with the unperturbed transition of the QD (Fig. 1a). In a frame rotating with the laser frequency (after performing the RWA), the Hamiltonian in the Zeeman basis reads

 H=δh(\Ket⇑\Bra⇑−\Ket⇓\Bra⇓)+δe(\KetT↓\BraT↓−\KetT↑\BraT↑)−(ΩH2\KetT↑\Bra⇓+ΩH2\KetT↓\Bra⇑+ΩV2\KetT↓\Bra⇓+ΩV2\KetT↑\Bra⇑+H.c.) , (1)

where are the electron and hole Zeeman splittings, respectively, are the Rabi frequencies for the horizontally/vertically-polarised transitions, and denotes the Hermitian conjugate.

Protocol – Fig. 1b shows that the emission of blue and red-detuned Raman spin-flip photons from a single quantum dot must alternate, provided that the scattering rate is faster than the hole spin-flip time. We build on this correlation between spin and photon colour to develop a protocol for generating an entangled LC state. As an intrinsic drawback of Raman spin-flips, the time at which a photon is scattered is not known prior to its detection. In the following, we assume that there is exactly one Raman scattering event per time-bin (albeit at a random time within the bin, see Fig. S5.). The overall probability and ways of circumventing this limitation111In practice, this assumption limits the size of the LCs that can be produced in this approach to less than ten. will be discussed later. Fig. 2 contains a diagrammatic representation of a successful run of our protocol. Let us trace the evolution of the joint spin-photon-state step by step: we start with the hole spin initialised in the superposition state (ignoring normalisation factors) and precessing at its Larmor frequency. Let the accumulated phase prior to the first scattering event be , then a Raman spin flip evolves the state to

 e−iϕ12\Ket⇑+eiϕ12\Ket⇓→e−iϕ12\Ket⇓B1+eiϕ12\Ket⇑R1 , (2)

where the labels inside the ket denote the first emitted blue (red) photon. A subsequent period of free precession until the end of the time-bin results in a phase . We now apply a -rotation (), yielding the state

 e−iχ12\Ket⇑B1+e−iχ12\Ket⇓B1+eiχ12\Ket⇑R1−eiχ12\Ket⇓R1 , (3)

where . The next Raman scattering event will have been preceded by another spin precession angle resulting in

 e−iϕ32e−iχ12\Ket⇓B1B2+eiϕ32e−iχ12\Ket⇑B1R2+e−iϕ32eiχ12\Ket⇓R1B2−eiϕ32eiχ12\Ket⇑R1R2 . (4)

The spin precesses further by before we apply the next rotation, yielding

 e−iϕ32eiϕ42e−iχ12\Ket⇓B1B2+eiϕ32e−iϕ42e−iχ12\Ket⇑B1R2+e−iϕ32eiϕ42eiχ12\Ket⇓R1B2−eiϕ32e−iϕ42eiχ12\Ket⇑R1R2\coloneqqe−iχ22e−iχ12\Ket⇓B1B2+eiχ22e−iχ12\Ket⇑B1R2+e−iχ22eiχ12\Ket⇓R1B2−eiχ22eiχ12\Ket⇑R1R2 . (5)

Let us stop at this point and, for clarity, consider the resulting state without its free precession phases

 \Ket⇓B1B2+\Ket⇑B1R2+\Ket⇓R1B2−\Ket⇑R1R2 . (6)

Using the photon qubit encoding , , the state following the final Y rotation it is given by

 \Ket⇑1112+\Ket⇓1112+\Ket⇑1102−\Ket⇓1102+\Ket⇑0112+\Ket⇓0112−\Ket⇑0102+\Ket⇓0102 . (7)

In Appendix Sec D, we show that, whether the spin is measured to be in the or state, the resulting photonic state ( or , respectively) indeed corresponds to LC. Further, we show that the above protocol generalises trivially to the production of LC states of arbitrary length. Crucially, reintroducing the above precession phases keeps the state local-unitarily (LU) equivalent to LC. The phases become known post-measurement through the timestamps of the detection clicks, and in the Appendix, we discuss how to make allowances for them for a tomographic reconstruction of the LC state.

Results – We now analyse the quality and success probability of our protocol. We begin with the rate for Raman scattering events followed by the success probability of a string of Raman photons with one per time-bin. Fig. 3a shows the Raman scattering rate and its dependence on both and the (vertically-polarised) driving . Comparison with numerical simulations shows that this rate is well-approximated by the transition probability obtained by treating the weak driving field perturbatively to second order (see Appendix Sec A)

 γpert=18Ω2VγΔ2 , (8)

provided mT and . We proceed to determine the optimal duration (i.e. the free precession time between -rotations) for maximising the probability of obtaining a single Raman event per time bin. Adopting  mT and (taking ns as the spontaneous emission rate), we calculate the number of successful trials with one Raman photon per interval in successive time-bins. Fig. 3c illustrates the results of Monte-Carlo simulations for to scattering events, suggesting that is close to optimal. We have the relation between the success probability for a single bin and that of bins.

Apart from addressing the possibility of having no Raman events within a time-bin, we also need to account for the possibility of ‘false-positives’, i.e. detecting only one of multiple Raman events occurring in a single time-bin, due to a photon detection efficiency 222We assume is the probability of obtaining a detector click if a photon was produced by the QD, i.e. it also includes any photon losses in the setup.). The probability of such photon false positives, , is given by the simple relation:

 Pfp(n)=Pnd(n)×Pd(1)×Ps(n+1)=Cn+1n(1−η)n×η×Ps(n+1) , (9)

where is the binomial coefficient, [] denotes the probability of detecting [not detecting] photons. We find that remains optimal after taking this into account. Fig. 3d shows the rate of LC generation for to for different detector efficiencies.

To demonstrate the robustness of our protocol against nuclear environment fluctuations, we calculate the fidelity between the state obtained with and without Overhauser field (both for the the same set of precession phases determined by randomly chosen scattering times). For a pure hh, only the Overhauser component perpendicular that is perpendicular to the applied -field affects the protocol [by randomly modifying direction and magnitude of the total -field by ]. By contrast, a mixed hh–lh system suffers predominantly from the parallel component, to an extent determined by the mixing factor . This is also exemplified in a decreased spin coherence time from the ideal hh limit, as shown in Fig. 3b. Only considering this term, the following analytical expression (see Appendix Sec E) captures the fidelity decay as a function of :

 ¯F(1)=12+√2π4TBδBxNerf(TBδBxN√2) , (10)

where denotes the average fidelity for a state of scattered photons. For a single scattered photon, we obtain for large as expected. More generally, Eqn. (10) represents an upper bound on the maximally achievable fidelity in the case of finite hh-lh mixing. To fully account for the effects of the stochastically varying net -field vector, we show numerically obtained333The numerical calculations were performed using the Overhauser ensemble-averaged matrix operations defined in Appendix Sec C. fidelity overlaps of desired vs the ensemble-average of realised LC states in Fig. 4. In the presence of the Overhauser field with fluctuations mT, near unit fidelity remains possible in the region with (moderately) strong magnetic field  T and relatively short  s (Fig. 4a). Conversely, large LC generation rates demand  s and  T (Fig. 4b), so that a trade-off situation arises. Encouragingly, there is a wide middle-ground where high fidelity operation is possible at respectable rates.

Another important figure of merit of our protocol is the localisable entanglement (LE) Popp2005 ; Schwartz2016 between any two qubits of the LC state (and also the spin). The LE represents the maximum negativity of the reduced density matrix of two qubits of interest (indexed and ), after all others have measured out projectively. Choosing the set of projectors as our measurement defines an ensemble , where is the probability of obtaining the two-spin density matrix for the outcome having measured the remaining qubits. The LE is then defined as the maximum negativity after averaging over all the outcomes for each measurement, that is

 LENj,k=maxM∑spM,s N(ρj,kM,s) , (11)

where is the negativity of . We choose a quasi-uniformly distributed basis on the Bloch sphere of each qubit (see points in insets of Fig. 4c,d). The computational unwieldiness of Eqn. (11) restricts the number of projectors, and we can only obtain a lower-bound of the true LE for LC and LC (Fig. 4c,d). Within the variance of the sample over which the optimisation was performed, the LE falls off with qubit distance, but encouragingly it remains remarkably high overall, and is thus unlikely to be a limiting factor in the length of the LC that could be generated using this protocol.

Conclusion – We have presented a novel scheme for generating frequency-encoded linear cluster states, which could serve as a stepping stone towards measurement-based quantum computation. Unlike current rival schemes, our protocol does not rely on incoherent photon emission, and is therefore only sensitive to ground-state hole-spin dephasing, at the cost of being limited by its intrinsic probabilistic nature. Based on experimentally informed properties of real epitaxial quantum dots, we have shown that linear cluster states of sufficient length and high fidelity for fusion into larger cluster states can nevertheless be produced at respectable rates. Our protocol takes full account of unmitigated Overhauser field fluctuations. It is inherently impervious to hole-mixing induced modifications of the optical selection rules, but, like other approaches, it stands to gain from dynamic decoupling. Whilst the probabilistic nature of the Raman scattering events limits our protocol as described in the main text to LC states of length , our approach can, in principle, be made deterministic if embedded in a QD molecule, as discussed in Appendix Sec G. We believe such a Raman hole-spin emitter is a viable, practical alternative in the quest for realising non-classical multi-photon states, and importantly one which can be straightforwardly implemented with current expertise and devices.

###### Acknowledgements.
We thank David Gershoni, Emil Denning, and Jake Iles-Smith for insightful and stimulating discussions. D.S. thanks SUPA for financial support. We acknowledge support from EPSRC (EP/M013472/01) and the ERC (no. 307392). B. D. G. thanks the Royal Society for a Wolfson Merit Award, and E. M. G. acknowledges support from the Royal Society of Edinburgh and the Scottish Government.

## Appendix A Second-order perturbation rate

It can be easily shown that, after moving to a rotating frame with respect to the unperturbed transition frequency, the amplitude of the Raman-flip transition is given by

 T⇓→⇑=\Bra⇑;ωRHI\KetT↓;0\BraT↓;0HI\Ket⇓;ωRayℏΔ(1)1+\Bra⇑;ωRayHI\KetT↑;0\BraT↑;0HI\Ket⇓;ωBℏΔ(1)2 , (12)

where , , and , and are the red-, blue-detuned and Rayleigh scattered photon frequencies, respectively. The first term in Eqn. (12) gives the amplitude of a red Raman photon event: the system, initially in the state, scatters a photon, after which the final state is given by (that is, the system in the state and a red-detuned Raman photon ( polarised) is scattered). Similarly, the transition giving rise to the blue-detuned photon scattering event occurs with amplitude

 T⇑→⇓=\Bra⇓;ωBHI\KetT↑;0\BraT↑;0HI\Ket⇑;ωRayℏΔ(2)1+\Bra⇓;ωRayHI\KetT↓;0\BraT↓;0HI\Ket⇑;ωRℏΔ(2)2 , (13)

where , .

The second term in each of the transition amplitudes does not contribute to the Raman processes, and vanish as the driving field can only drive vertically-polarised transitions. After performing the necessary solid angle integrals, we arrive at the scattering rate given by Eqn. (8) in the main text.

## Appendix B Overhauser field for hole-spin systems

Vanishing wavefunctions at the nuclear sites means that the Fermi-contact hyperfine term for the nuclear–hole spin interaction is effectively zero, leaving only the dipole-dipole interaction term as the dominant source of dephasing. For an idealised pure hh, this term is of Ising-nature, with just the ZZ component being present. In most epitaxially grown QDs, however, some degree of hh and lh mixing is always present Prechtel2016 ; Testelin2009 , breaking the Ising-like nature of the dipole-dipole term and introducing XX and YY terms in the Hamiltonian. This means that the eigenstates of the Hamiltonian are no longer given separately by the hh or lh states, but a linear combination of both (the consequences of this mixing in quantum dot-based LC protocols is further discussed in Appendix Sec F). Without going into too much detail, the hyperfine coupling Hamiltonian for the hh–lh states is given by:

 Hddhf=V∑jCj|Ψ(Rj)|2[α(IjxSx+IjySy)+IjzSz] (14)

where are dipole-dipole hyperfine constants, is the unit cell volume, and is a parameter depending on the deformation potentials for the valence band, and the strain tensor Prechtel2016 ; Testelin2009 . In the ‘frozen-fluctuation’ model Merkulov2002 , this results in an effective magnetic field with mean (which, due to the finite size of the spin bath, is not necessarily zero), and a fluctuation (which is the source of the spin’s loss of coherence), and is assumed to follow normal statisticsTestelin2009 :

 P(BN)=(12π)321δB∥2NδB⊥N×exp⎡⎣−ΔBx2N2δB∥2N−ΔBy2N2δB∥2N−ΔBz2N2δB⊥2N⎤⎦ , (15)

where , and . Experimentally, Overhauser field fluctuations of 10–30mT have been measured Chekhovich2013 ; Urbaszek2013 , putting a lower-bound on the applied external field required to screen these fluctuations.

## Appendix C Matrix operations

Consider a single scattering process that can be described by the action of the product of matrices:

 \Ket⇑\KetRayk→e−iϕ(k)12eiϕ(k)22(\Ket⇑+\Ket⇓)\KetBk=UrUp(ϕ(k)2)T(k)sUp(ϕ(k)1)\Ket⇑\KetRayk=Q(k)\Ket⇑\KetRayk ,\Ket⇓\KetRayk→eiϕ(k)12e−iϕ(k)22(\Ket⇑−\Ket⇓)\KetRk=UrUp(ϕ(k)2)T(k)sUp(ϕ(k)1)\Ket⇓\KetRayk=Q(k)\Ket⇓\KetRayk , (16)

where is the free spin precession transformation before () and after () the scattering event (prior to the Y rotation), with the resulting matrix of events being . The scattering matrix is given by

 (17)

with and written in the basis , which simultaneously flips the spin state , and applies the local transformations

 T(k)B:\KetRayk↦\KetBkT(k)R:\KetRayk↦\KetRk , (18)

where we have omitted the unaffected photon states for brevity. Hence and take the form:

 T(k)R=I⨂k−13⊗⎛⎜⎝000001000⎞⎟⎠⊗I⨂n−k3T(k)B=I⨂k−13⊗⎛⎜⎝001000000⎞⎟⎠⊗I⨂n−k3 , (19)

and and are simply given by given by

 Ur=exp(iπ4σy)⊗I⨂n3Up(ϕ)=⎛⎜⎝e−iϕ200eiϕ2⎞⎟⎠⊗I⨂n3 , (20)

where the first matrices act on the spin state and have been written in the basis. Unfortunately, the matrix product describing -photon scattering events becomes unwieldy with increasing . In Appendix Sec D, however, we show that this protocol does indeed generalise to a LC state, up to free precession phases.

## Appendix D Generalisation to n-photons

### d.1 Preliminary lemmas

In this section, we will show that the general form of the -photon state obtained using our protocol can be written recursively. In fact,

###### Lemma D.1.

, the -photon state can be decomposed into the recursive relations

 S(n)+=S(n−1)+\Ket1n+S(n−1)−\Ket0n ,S(n)−=S(n−1)+\Ket1n−S(n−1)−\Ket0n , (21)

depending whether the spin is measured to be in the or state, respectively.

###### Proof.

We will, w.l.o.g., ignore the spin precession, although the proof is the same for the general case:

Basis case: For , and . After the next scattering event, we get

 S(2)+=\Ket1112+\Ket1102+\Ket0112−\Ket0102=(\Ket11+\Ket01)\Ket12+(\Ket11−\Ket01)\Ket02=S(1)+\Ket12+S(1)−\Ket02 . (22)

Similarly,

 S(2)−=\Ket1112−\Ket1102+\Ket0112+\Ket0102=(\Ket11+\Ket01)\Ket12−(\Ket11−\Ket01)\Ket02=S(1)+\Ket12−S(1)−\Ket02 . (23)

Induction step: Assume statement holds for , and consider the scattering event:

 S(n+1)+= UrT(n+1)scat(\Ket⇑S(n)++\Ket⇓S(n)−)\KetRayn+1= (\Ket⇑+\Ket⇓)S(n)+\Ket1n+1+(\Ket⇑−\Ket⇓)S(n)−\Ket0n+1= \Ket⇑(S(n)+\Ket1n+1+S(n)−\Ket0n+1) +\Ket⇓(S(n)+\Ket1n+1−S(n)−\Ket0n+1) . (24)

Therefore and , so the statement holds . ∎

It is then easy to see that we also have that

###### Lemma D.2.
 σ(n)zS(n)±=−S(n)∓∀n∈N , (25)

which we shall use to prove that the -photon state we generate is indeed a linear cluster state.

### d.2 Equivalence to LCn states

In order to show that the states are indeed LCs, we have to show that they both satisfy the set of eigenvalue equations

 K(a)nS(n)±=(−1)k(a)±S(n)± , (26)

with

 K(a)n=σ(a)x⨂b∈N(a)σ(b)z , (27)

where , is the set of direct neighbours of photon along the state, and , depending on the particular realisation of LC. The subscript on the operator denotes the state tensor-length of , and hence the length of the state it acts upon. In fact we shall show the following statement

###### Theorem D.3.

The -photon state satisfies the set of LC-eigenvalue equations for

 (28)
###### Proof.

The proof follows, once again, by induction, as well as the use of Lemma D.1

Basis case: For , .

 S(2)+=(\Ket11+\Ket02)\Ket1n+(\Ket11−\Ket01)\Ket02 ,S(2)+=(\Ket11+\Ket02)\Ket1n−(\Ket11−\Ket01)\Ket02 , (29)

and the statement holds when applying and .

Induction step: Suppose the statement holds for , and consider .Then

If :

 K(a)n+1S(n+1)+=(K(a)n⊗I2)(S(n)+\Ket1n+1+S(n)−\Ket0n+1)=(−1)k(1)+S(n)+\Ket1n+1+(−1)k(1)−S(n)−\Ket0n+1=−(S(n)+\Ket1n+1+S(n)−\Ket0n+1)=−S(n+1)+ , (30)

with being the identity matrix. The penultimate step holds due the induction hypothesis. Similarly, for ,

 K(a)n+1S(n+1)−=(K(a)n⊗I2)(S(n)+\Ket1n+1−S(n)−\Ket0n+1)=(−1)k(1)+S(n)−\Ket1n+1−(−1)k(1)−S(n)−\Ket0n+1=−(S(n)+\Ket1n+1−S(n)−\Ket0n+1)=−S(n+1)− . (31)

If :

 K(a)n+1S(n+1)+=(K(a)n⊗I2)(S(n)+\Ket1n+1+S(n)−\Ket0n+1)=(−1)k(a)+S(n)+\Ket1n+1+(−1)k(a)−S(n)−\Ket0n+1=S(n)+\Ket1n+1+S(n)−\Ket0n+1=S(n+1)+ , (32)
 K(a)n+1S(n+1)−=(K(a)n⊗I2)(S(n)+\Ket1n+1−S(n)−\Ket0n+1)=(−1)k(a)+S(n)−\Ket1n+1−(−1)k(a)−S(n)−\Ket0n+1=S(n)+\Ket1n+1−S(n)−\Ket0n+1=S(n+1)− . (33)

If :

 K(a)n+1S(n+1)+=(K(a)n⊗σ(n+1)z)(S(n)+\Ket1n+1+S(n)−\Ket0n+1)=−(−1)k(n)+S(n)+\Ket1n+1+(−1)k(n)−S(n)−\Ket0n+1=S(n)+\Ket1n+1+S(n)−\Ket0n+1=S(n+1)+ , (34)
 K(a)n+1S(n+1)−=(K(a)n⊗σ(n+1)z)(S(n)+\Ket1n+1−S(n)−\Ket0n+1)=−(−1)k(n)+S(n)−\Ket1n+1−(−1)k(n)−S(n)−\Ket0n+1=S(n)+\Ket1n+1−S(n)−\Ket0n+1=S(n+1)− . (35)

For the case, we shall make use of Lemma D.2. The operator can be decomposed as , and hence we get that

If :

 K(a)n+1S(n+1)+=−S(n)−\Ket0n+1−S(n)+\Ket1n+1=−S(n+1)+ , (36)
 K(a)n+1S(n+1)−=−S(n)−\Ket0n+1+S(n)+\Ket1n+1=S(n+1)− . (37)

Therefore, the states satisfy the eigenvalue conditions (26) for the set of parameters (28), meaning that the the state obtained by our protocol is an LC state. ∎

## Appendix E Average fidelity

Consider a single scattering event in which the spin precesses for a time prior to the scattering event and a subsequent precession time followed by a Y rotation marking the end of the run (such that ). In the presence of the component, the rotation matrix in (20) picks up a stochastic term , that is

 Up((ωB+ωN)t)=⎛⎝e−i1