Entanglement transfer, accumulation and retrievalvia quantum-walk-based qubit-qudit dynamics

Taira Giordani,1 Luca Innocenti,2, 3 Alessia Suprano,1 Emanuele Polino,1 MauroPaternostro,3 Nicolo Spagnolo,1 Fabio Sciarrino,1, 4 and Alessandro Ferraro3

1Dipartimento di Fisica, Sapienza Universita di Roma, Piazzale Aldo Moro 5, I-00185 Roma, Italy2Department of Optics, Palacky University, 17. Listopadu 12, 771 46 Olomouc, Czech Republic

3Centre for �eoretical Atomic, Molecular, and Optical Physics,School of Mathematics and Physics, �een’s University Belfast, BT7 1NN Belfast, United Kingdom

4Consiglio Nazionale delle Ricerche, Istituto dei sistemi Complessi (CNR-ISC), Via dei Taurini 19, 00185 Roma, Italy

�e generation and control of quantum correlations in high-dimensional systems is a major challenge inthe present landscape of quantum technologies. Achieving such non-classical high-dimensional resourceswill potentially unlock enhanced capabilities for quantum cryptography, communication and computation.We propose a protocol that is able to a�ain entangled states of d-dimensional systems through a quantum-walk-based transfer & accumulate mechanism involving coin and walker degrees of freedom. �e choice ofinvestigating quantum walks is motivated by their generality and versatility, complemented by their successfulimplementation in several physical systems. Hence, given the cross-cu�ing role of quantum walks acrossquantum information, our protocol potentially represents a versatile general tool to control high-dimensionalentanglement generation in various experimental platforms. In particular, we illustrate a possible photonicimplementation where the information is encoded in the orbital angular momentum and polarization degreesof freedom of single photons.

I. INTRODUCTION

�antum entanglement underpins many of the advantagespromised by the technological advances in quantum informa-tion processors [1]. Despite considerable research e�orts havebeen devoted to achieving seamless generation and control oftwo-dimensional systems, it is known that two-dimensionalentanglement entails limitations in a variety of se�ings [2–4].When higher-dimensional entanglement is used — for exam-ple in the context of quantum communication [5] — higherchannel capacities can be achieved through superdense codingprotocols [6–8]. �antum cryptography protocols enhancedby higher-dimensional entangled states achieve be�er perfor-mances in terms of key rates, noise resilience, and security [9–22]. Signi�cant bene�ts can also be achieved in quantum errorcorrection [23–26] and fault-tolerant quantum computation[27–30].

�e potential bene�ts of high-dimensional entanglementhave stimulated a signi�cant e�ort towards its generation, ma-nipulation, and certi�cation in various platforms including, inparticular, optical systems [31, 32]. Despite signi�cant experi-mental advances, the implementation of such tasks remainsdemanding, especially in light of the di�culties associatedto controlling systems and transformations in large Hilbertspaces.

In this paper, we show how to leverage controllable low-dimensional systems, together with special quantum devicesacting as interfaces between systems of di�erent dimensions,to realize an e�ective entanglement-transfer protocol fromlow- to high-dimensional degrees of freedom. �antum corre-lations stored in two-dimensional degrees of freedom — suchas the polarizations of entangled photons — can thus be passedinto high-dimensional information carriers via suitable localinteractions and measurements.

We derive the general conditions under which such entan-

glement transfer is feasible. We then focus on the case of statesproducible by discrete-time one-dimensional quantum walks(QW) [33–37]. �ese model a natural type of interaction be-tween hetero-dimensional systems, and are widely available ina variety of physical systems. We study the conditions underwhich QW dynamics allow to transfer entanglement betweencoin and walker degrees of freedom, and prove the feasibil-ity of accumulating entanglement in the high-dimensionalsystem by repeatedly creating it and transfering it from thelow-dimensional one. �is scheme constitutes a promisingtwo-way interface to transfer reliably entanglement betweendi�erent information carriers [38–43].

A particularly suitable platform for the manipulation ofhigh-dimensional systems, which has also been successful indemonstrating control of the QW dynamics, is embodied bythe orbital angular momentum (OAM) of light. Recent experi-mental progress enabled by the growing capacity to prepare,manipulate and measure OAM states are opening up the pos-sibility to explore the richness of high-dimensional Hilbertspaces for the sake of quantum information processing [44]. Aprotocol allowing to generate high-dimensional OAM statesusing a simple dynamics such as the one o�ered by QWswould therefore be a signi�cant step forward towards theprovision of on demand high-dimensional entangled states.

�e remainder of this paper is organized as follows. In sec-tion II we overview the necessary background on QWs andOAM. In section III we formalise the general conditions for theoccurrence of entanglement transfer and study their solutions.We then specialise in section IV to the context set by QWs, andstudy – in section V – the possibility of accumulating entangle-ment in one degree of freedom by repeated applications of theentanglement-transfer protocol. We conclude in section VIIby detailing a possible experimental implementation of theprotocol in the framework of OAM-based implementation ofthe QW dynamics.

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II. BACKGROUND

Discrete-time QWs embody a widely studied type of inter-action between a two-dimensional “coin” degree of freedom,and a high-dimensional “walker” one [33–37]. Despite theirsimplicity, QWs allow to engineer e�ectively a broad rangeof evolutions [45–48]. Recently, some of us demonstrated thepotential of a QW-based architecture to �exibly implementquantum state engineering of a single OAM [49, 50], as wellas the machine-learning-enhanced classi�cation of hybridpolarization-OAM states of light [51]. A possible physical em-bodiment of such QW dynamics uses polarization and OAMof single photons, playing the roles of the coin and the walkerdegrees of freedom, respectively, with waveplates to imple-ment the coin operations and q-plates [52] to implement thecontrolled-shi�. State engineering protocols leveraging QWsin this se�ing were previously designed and demonstrated inRefs. [49, 51, 53].

More precisely, QWs are de�ned in a bipartite coin-walkerspace HC ⊗ HW , where HC(W) denotes the coin (walker)space. We assume dim(HC) = 2. �e evolution is de�ned bythe repeated action of a unitary walk operationWC ≡ S(C⊗I),which comprises the sequential action of a controlled-shi�operation S , and a coin �ipping operation C. �e coin �ippingoperation acts locally on the coin space, while the controlled-shi� changes the state of the walker conditionally to the stateof the coin:

S ≡∑

k

(P↑ ⊗ |k〉〈k|+ P↓ ⊗ |k + 1〉〈k|), (1)

where {|↑〉 , |↓〉} form a basis for HC , {|k〉}k≥0 spans HW ,and we introduced the notation Pψ ≡ |ψ〉〈ψ|.

�e state space we are interested in consists of two pairsof QWs, so that the overall system of coins and walkerslives in the four-partite space H ≡ H(1) ⊗ H(2), withH(i) ≡ H(i)

C ⊗H(i)W , andH(i)

C ,H(i)W accommodating coin and

walker of the ith party, respectively (i = 1, 2). Given |Ψ〉 ∈ H,we applyWC locally on H(1) and H(2). �is, in general, en-tangles each coin with the respective walker [54, 55]. In thenext sections, we will describe how to use this QW dynam-ics to transfer entanglement from the two-coin subspace tothe two-walker one, using only local operations on the coins.In an optical setup, this process will transfer the initial en-tanglement encoded in a polarization state to the two OAMdegrees of freedom. �e process can be iterated to transfermore entanglement from the polarizations to the OAMs.

III. ENTANGLEMENT TRANSFER VIA LOCALPROJECTIONS

We now address the challenge of transferring entanglementacross di�erent degrees of freedom using solely local projec-tions. More precisely, we consider four-partite states |Ψ〉 ∈ H,and ask when, via local projections, it is possible to transfer,or “focus”, the entanglement into the bipartitionH(1)

W ⊗H(2)W .

We thus look for conditions ensuring the existence of states

|γ〉 ∈ H(1)C and |δ〉 ∈ H(2)

C such that the entanglement of |Ψ〉in the bipartition H(1) ⊗H(2) is preserved in the projectedstate 〈γ, δ|Ψ〉 ∈ H(1)

W ⊗H(2)W . A schematic description of this

formal scenario is given in Fig. 1a. Note that such entangle-ment transfer is not always possible. It is therefore pivotal to�nd the conditions making such protocol viable. It is worthnoting that, when probabilistic operations are allowed (as inthe case of projections), even restricting to local operations,the amount of entanglement can be increased [56–58]. Suchprocess of e�ective entanglement distillation comes, however,at the expense of lowered success probabilities. We focushere on the case where we want to preserve, not enhance, theentanglement in a given state. In this case, it is also possi-ble to achieve entanglement transfer deterministically, whenthere is a complete basis of projections each element of whichachieves entanglement transfer.

We can break down the task at hand into two independentsub-problems, which we will refer to as transferability condi-tions: on the one hand, transferring the entanglement fromH(1) ⊗ H(2) to H(1)

W ⊗ H(2), and on the other hand, trans-ferring the entanglement from H(1)

W ⊗H(2) to H(1)W ⊗H

(2)W .

�e achievability of these two tasks will be referred to withTC1 and TC2, respectively. It is worth stressing that, whilethroughout the paper we will always focus our discussionon TC1, all results hold analogously for TC2 when instead ofprojecting in the spaceH(1)

C we project inH(2)C .

To frame the problem more precisely, consider a state |Ψ〉 ∈H with Schmidt decomposition

|Ψ〉 =∑

k

√pk |uk〉 |vk〉 , (2)

where∑k pk = 1, |uk〉 ∈ H(1) and |vk〉 ∈ H(2). To achieve

TC1 we want a state |γ〉 ∈ H(1)C such that the corresponding

projected state |Ψγ〉 ∈ H(1)W ⊗H(2) contains the same amount

of entanglement, in the bipartitionH(1)W ⊗H(2), as that initially

in |Ψ〉. In general, we have

|Ψγ〉 =1

√pproj

∑

k

√pkqk |uk〉 |vk〉 , (3)

where √qk |uk〉 = 〈γ|uk〉 ∈ H(1) and pproj =∑k pkqk. We

distinguish between three di�erent scenarios:

(1) If the states |uk〉 are not orthogonal, then someinformation about which k the state is in leaks throughthe coin projection, and some entanglement is thusdegraded. �is will be shown formally in Appendix A.

(2) If the states |uk〉 are orthogonal, but the correspondingprojection probabilities qk are uneven, then again theentanglement in |Ψγ〉 is smaller than that in |Ψ〉.

(3) If the states |uk〉 are orthogonal, and qk = pproj for allk, then projecting onto |γ〉 fully preserves the initialentanglement.

3

𝑈12 𝑈34

𝑈23Bell state source

1st itera�on: entanglement transfer

Entanglement transfer

N-th itera�on: entanglementaccumula�on

Entanglement transfer...

Entanglement transfer

High-dimensional entangled state

walker

coin

coin measurements

a b c

|Ψ⟩

2

ment in one degree of freedom by repeated applications of theentanglement-transfer protocol. We conclude in section VIIby detailing a possible experimental implementation of theprotocol in the framework of OAM-based implementation ofthe QW dynamics.

II. BACKGROUND

A widely studied type of interaction between a two-dimensional and a high-dimensional system is embodied bythe discrete-time QW [30–34]. By intertwining the evolutionof a two-dimensional coin and a high-dimensional walker,QWs allow to e�ectively engineer a broad range of evolu-tions. Recently, some of us demonstrated the potential of aQW-based architecture to �exibly implement quantum stateengineering of a single OAM [36, 37], as well as the machine-learning-enhanced classi�cation of hybrid polarization-OAMstates of light [38]. A QW dynamics on a bipartite coin-walkersystem – occurring in the Hilbert space HC⌦HW with HC(W)

the Hilbert space of the coin (walker) – is built by the repeatedaction of a unitary walk operation WC ⌘ S(C ⌦ I), whichdescribes the sequential action of a controlled-shi� operationS , and a coin �ipping operation C acting locally on the coinspace. �e controlled-shi� operation changes the state of thewalker conditionally to the state of the coin as

S ⌘X

k

(P" ⌦ |kihk| + P# ⌦ |k + 1ihk|), (1)

where {|"i , |#i} are a basis states of HC , {|ki} spans HW ,and we introduced the projectors P ⌘ | ih |.

A possible physical embodiment of such QW dynamics usespolarization and OAM of single photons, playing the roles ofthe coin and the walker degrees of freedom, respectively, withwaveplates to implement the coin operations and q-plates [39]to implement the controlled-shi�. State engineering protocolsleveraging QWs in this se�ing were previously designed anddemonstrated in Refs. [36, 38, 40].

�e state space we are interested in consists of two pairsof QWs, so that the overall system of coins and walkers livesin the four-partite space H ⌘ H(1) ⌦ H(2), with H(i) ⌘H(i)

C ⌦ H(i)W , and H(i)

C , H(i)W accommodating coin and walker

of the ith QW system, respectively (i = 1, 2). Given | i 2 H,we will apply the walk operation on the two QW systemsseparately. �is, in general, entangles the degrees of freedomof a coin with those of the respective walker. In the nextsections, we will describe how to use this QW dynamics totransfer entanglement that is possibly initially present in thetwo-coin subspace to the two-walker one, via local operationon their respective coin degrees of freedom. In an optical setup,this process will transfer the initial entanglement encoded ina polarization state to the two OAM degrees of freedom. �eprocess can be iterated to transfer more entanglement fromthe polarizations to the OAMs.

III. ENTANGLEMENT TRANSFER VIA LOCALPROJECTIONS

Let us consider the following question

Given a four-partite state in H, can we �nd lo-cal coin projections such that the entanglement inthe coin degrees of freedom is transferred to theprojected state of the walkers?

More precisely, given a state | i 2 H, we want to �nd suit-able states |�i 2 H(1)

C and |�i 2 H(2)C such that, upon per-

froming bi-local projections onto them, the coin entanglementpossibly present in | i is transferred to the projected stateh�, �| i 2 H(1)

W ⌦ H(2)W . A schematic description of this for-

mal scenario is given in �g. 1a [Note (Ale): make self-consistentthe notation used in the �gure and the text (| ini, | f i and thethree unitaries in the �gure; in panel (a) it might be worth toput symbols of coin/walker and maybe a curly braket for photon1 and 2 as well)]. It is worth stressing that such entanglementtransfer is not always possible — for example if | i is triviallyseparable with respect to the partition H(1)⌦H(2): no bilocaloperation, including projections, will be able to entangled thewalkers. It is therefore pivotal to �nd the conditions makingsuch protocol viable.

�e task at hand can be broken down into two indepen-dent subproblems, which we will refer to as transferibilityconditions: on the one hand, transferring the entanglementfrom H(1)

C ⌦ H(2)C to H(1)

W ⌦ H(2)C , and then transferring the

entanglement from H(1)W ⌦H(2)

C to H(1)W ⌦H(2)

W . Transferringentanglement from H(1)

C ⌦ H(2)C to H(1)

W ⌦ H(2)W is possible

only if these two subproblems are solvable.Let | i 2 H have Schmidt decomposition

| i =X

k

ppk |uki |vki . (2)

We want a projection |�i 2 H(1)C such that the corresponding

post-projection state | �i 2 H(1)W ⌦ H(2) contains the same

amount of entanglement, in the bipartition H(1)W ⌦ H(2), as

that initially contained in | i. In general, we have

| �i =1

ppproj

X

k

ppkqk |uki |vki , (3)

where pqk |uki = h�|uki and pproj =

Pk pkqk. We distin-

guish between three di�erent scenarios. If (1) the states |ukiare not orthogonal, then some information about which k thestate is in leaks through the polarization measurement, andsome amount of entanglement is therefore degraded. �is willbe shown formally in (�?). (2) If the states |uki are orthog-onal, but the corresponding projection probabilities qk areuneven, then again the entanglement in | �i is smaller thanthat in | i. Finally, if (3) the states |uki are orthogonal, andqk = pproj for all k, then projecting onto |�i fully preservesthe initial entanglement. Note that (3) is thus a necessary andsu�cient condition for entanglement transferability without

2

ment in one degree of freedom by repeated applications of theentanglement-transfer protocol. We conclude in section VIIby detailing a possible experimental implementation of theprotocol in the framework of OAM-based implementation ofthe QW dynamics.

II. BACKGROUND

A widely studied type of interaction between a two-dimensional and a high-dimensional system is embodied bythe discrete-time QW [30–34]. By intertwining the evolutionof a two-dimensional coin and a high-dimensional walker,QWs allow to e�ectively engineer a broad range of evolu-tions. Recently, some of us demonstrated the potential of aQW-based architecture to �exibly implement quantum stateengineering of a single OAM [36, 37], as well as the machine-learning-enhanced classi�cation of hybrid polarization-OAMstates of light [38]. A QW dynamics on a bipartite coin-walkersystem – occurring in the Hilbert space HC⌦HW with HC(W)

the Hilbert space of the coin (walker) – is built by the repeatedaction of a unitary walk operation WC ⌘ S(C ⌦ I), whichdescribes the sequential action of a controlled-shi� operationS , and a coin �ipping operation C acting locally on the coinspace. �e controlled-shi� operation changes the state of thewalker conditionally to the state of the coin as

S ⌘X

k

(P" ⌦ |kihk| + P# ⌦ |k + 1ihk|), (1)

where {|"i , |#i} are a basis states of HC , {|ki} spans HW ,and we introduced the projectors P ⌘ | ih |.

A possible physical embodiment of such QW dynamics usespolarization and OAM of single photons, playing the roles ofthe coin and the walker degrees of freedom, respectively, withwaveplates to implement the coin operations and q-plates [39]to implement the controlled-shi�. State engineering protocolsleveraging QWs in this se�ing were previously designed anddemonstrated in Refs. [36, 38, 40].

�e state space we are interested in consists of two pairsof QWs, so that the overall system of coins and walkers livesin the four-partite space H ⌘ H(1) ⌦ H(2), with H(i) ⌘H(i)

C ⌦ H(i)W , and H(i)

C , H(i)W accommodating coin and walker

of the ith QW system, respectively (i = 1, 2). Given | i 2 H,we will apply the walk operation on the two QW systemsseparately. �is, in general, entangles the degrees of freedomof a coin with those of the respective walker. In the nextsections, we will describe how to use this QW dynamics totransfer entanglement that is possibly initially present in thetwo-coin subspace to the two-walker one, via local operationon their respective coin degrees of freedom. In an optical setup,this process will transfer the initial entanglement encoded ina polarization state to the two OAM degrees of freedom. �eprocess can be iterated to transfer more entanglement fromthe polarizations to the OAMs.

III. ENTANGLEMENT TRANSFER VIA LOCALPROJECTIONS

Let us consider the following question

Given a four-partite state in H, can we �nd lo-cal coin projections such that the entanglement inthe coin degrees of freedom is transferred to theprojected state of the walkers?

More precisely, given a state | i 2 H, we want to �nd suit-able states |�i 2 H(1)

C and |�i 2 H(2)C such that, upon per-

froming bi-local projections onto them, the coin entanglementpossibly present in | i is transferred to the projected stateh�, �| i 2 H(1)

W ⌦ H(2)W . A schematic description of this for-

mal scenario is given in �g. 1a [Note (Ale): make self-consistentthe notation used in the �gure and the text (| ini, | f i and thethree unitaries in the �gure; in panel (a) it might be worth toput symbols of coin/walker and maybe a curly braket for photon1 and 2 as well)]. It is worth stressing that such entanglementtransfer is not always possible — for example if | i is triviallyseparable with respect to the partition H(1)⌦H(2): no bilocaloperation, including projections, will be able to entangled thewalkers. It is therefore pivotal to �nd the conditions makingsuch protocol viable.

�e task at hand can be broken down into two indepen-dent subproblems, which we will refer to as transferibilityconditions: on the one hand, transferring the entanglementfrom H(1)

C ⌦ H(2)C to H(1)

W ⌦ H(2)C , and then transferring the

entanglement from H(1)W ⌦H(2)

C to H(1)W ⌦H(2)

W . Transferringentanglement from H(1)

C ⌦ H(2)C to H(1)

W ⌦ H(2)W is possible

only if these two subproblems are solvable.Let | i 2 H have Schmidt decomposition

| i =X

k

ppk |uki |vki . (2)

We want a projection |�i 2 H(1)C such that the corresponding

post-projection state | �i 2 H(1)W ⌦ H(2) contains the same

amount of entanglement, in the bipartition H(1)W ⌦ H(2), as

that initially contained in | i. In general, we have

| �i =1

ppproj

X

k

ppkqk |uki |vki , (3)

where pqk |uki = h�|uki and pproj =

Pk pkqk. We distin-

guish between three di�erent scenarios. If (1) the states |ukiare not orthogonal, then some information about which k thestate is in leaks through the polarization measurement, andsome amount of entanglement is therefore degraded. �is willbe shown formally in (�?). (2) If the states |uki are orthog-onal, but the corresponding projection probabilities qk areuneven, then again the entanglement in | �i is smaller thanthat in | i. Finally, if (3) the states |uki are orthogonal, andqk = pproj for all k, then projecting onto |�i fully preservesthe initial entanglement. Note that (3) is thus a necessary andsu�cient condition for entanglement transferability without

E|γ⟩

|δ⟩

{{

2

ment in one degree of freedom by repeated applications of theentanglement-transfer protocol. We conclude in section VIIby detailing a possible experimental implementation of theprotocol in the framework of OAM-based implementation ofthe QW dynamics.

II. BACKGROUND

A widely studied type of interaction between a two-dimensional and a high-dimensional system is embodied bythe discrete-time QW [30–34]. By intertwining the evolutionof a two-dimensional coin and a high-dimensional walker,QWs allow to e�ectively engineer a broad range of evolu-tions. Recently, some of us demonstrated the potential of aQW-based architecture to �exibly implement quantum stateengineering of a single OAM [36, 37], as well as the machine-learning-enhanced classi�cation of hybrid polarization-OAMstates of light [38]. A QW dynamics on a bipartite coin-walkersystem – occurring in the Hilbert space HC⌦HW with HC(W)

the Hilbert space of the coin (walker) – is built by the repeatedaction of a unitary walk operation WC ⌘ S(C ⌦ I), whichdescribes the sequential action of a controlled-shi� operationS , and a coin �ipping operation C acting locally on the coinspace. �e controlled-shi� operation changes the state of thewalker conditionally to the state of the coin as

S ⌘X

k

(P" ⌦ |kihk| + P# ⌦ |k + 1ihk|), (1)

where {|"i , |#i} are a basis states of HC , {|ki} spans HW ,and we introduced the projectors P ⌘ | ih |.

A possible physical embodiment of such QW dynamics usespolarization and OAM of single photons, playing the roles ofthe coin and the walker degrees of freedom, respectively, withwaveplates to implement the coin operations and q-plates [39]to implement the controlled-shi�. State engineering protocolsleveraging QWs in this se�ing were previously designed anddemonstrated in Refs. [36, 38, 40].

�e state space we are interested in consists of two pairsof QWs, so that the overall system of coins and walkers livesin the four-partite space H ⌘ H(1) ⌦ H(2), with H(i) ⌘H(i)

C ⌦ H(i)W , and H(i)

C , H(i)W accommodating coin and walker

of the ith QW system, respectively (i = 1, 2). Given | i 2 H,we will apply the walk operation on the two QW systemsseparately. �is, in general, entangles the degrees of freedomof a coin with those of the respective walker. In the nextsections, we will describe how to use this QW dynamics totransfer entanglement that is possibly initially present in thetwo-coin subspace to the two-walker one, via local operationon their respective coin degrees of freedom. In an optical setup,this process will transfer the initial entanglement encoded ina polarization state to the two OAM degrees of freedom. �eprocess can be iterated to transfer more entanglement fromthe polarizations to the OAMs.

III. ENTANGLEMENT TRANSFER VIA LOCALPROJECTIONS

Let us consider the following question

Given a four-partite state in H, can we �nd lo-cal coin projections such that the entanglement inthe coin degrees of freedom is transferred to theprojected state of the walkers?

More precisely, given a state | i 2 H, we want to �nd suit-able states |�i 2 H(1)

C and |�i 2 H(2)C such that, upon per-

froming bi-local projections onto them, the coin entanglementpossibly present in | i is transferred to the projected stateh�, �| i 2 H(1)

W ⌦ H(2)W . A schematic description of this for-

mal scenario is given in �g. 1a [Note (Ale): make self-consistentthe notation used in the �gure and the text (| ini, | f i and thethree unitaries in the �gure; in panel (a) it might be worth toput symbols of coin/walker and maybe a curly braket for photon1 and 2 as well)]. It is worth stressing that such entanglementtransfer is not always possible — for example if | i is triviallyseparable with respect to the partition H(1)⌦H(2): no bilocaloperation, including projections, will be able to entangled thewalkers. It is therefore pivotal to �nd the conditions makingsuch protocol viable.

�e task at hand can be broken down into two indepen-dent subproblems, which we will refer to as transferibilityconditions: on the one hand, transferring the entanglementfrom H(1)

C ⌦ H(2)C to H(1)

W ⌦ H(2)C , and then transferring the

entanglement from H(1)W ⌦H(2)

C to H(1)W ⌦H(2)

W . Transferringentanglement from H(1)

C ⌦ H(2)C to H(1)

W ⌦ H(2)W is possible

only if these two subproblems are solvable.Let | i 2 H have Schmidt decomposition

| i =X

k

ppk |uki |vki . (2)

We want a projection |�i 2 H(1)C such that the corresponding

post-projection state | �i 2 H(1)W ⌦ H(2) contains the same

amount of entanglement, in the bipartition H(1)W ⌦ H(2), as

that initially contained in | i. In general, we have

| �i =1

ppproj

X

k

ppkqk |uki |vki , (3)

where pqk |uki = h�|uki and pproj =

Pk pkqk. We distin-

guish between three di�erent scenarios. If (1) the states |ukiare not orthogonal, then some information about which k thestate is in leaks through the polarization measurement, andsome amount of entanglement is therefore degraded. �is willbe shown formally in (�?). (2) If the states |uki are orthog-onal, but the corresponding projection probabilities qk areuneven, then again the entanglement in | �i is smaller thanthat in | i. Finally, if (3) the states |uki are orthogonal, andqk = pproj for all k, then projecting onto |�i fully preservesthe initial entanglement. Note that (3) is thus a necessary andsu�cient condition for entanglement transferability without

2

ment in one degree of freedom by repeated applications of theentanglement-transfer protocol. We conclude in section VIIby detailing a possible experimental implementation of theprotocol in the framework of OAM-based implementation ofthe QW dynamics.

II. BACKGROUND

A widely studied type of interaction between a two-dimensional and a high-dimensional system is embodied bythe discrete-time QW [30–34]. By intertwining the evolutionof a two-dimensional coin and a high-dimensional walker,QWs allow to e�ectively engineer a broad range of evolu-tions. Recently, some of us demonstrated the potential of aQW-based architecture to �exibly implement quantum stateengineering of a single OAM [36, 37], as well as the machine-learning-enhanced classi�cation of hybrid polarization-OAMstates of light [38]. A QW dynamics on a bipartite coin-walkersystem – occurring in the Hilbert space HC⌦HW with HC(W)

the Hilbert space of the coin (walker) – is built by the repeatedaction of a unitary walk operation WC ⌘ S(C ⌦ I), whichdescribes the sequential action of a controlled-shi� operationS , and a coin �ipping operation C acting locally on the coinspace. �e controlled-shi� operation changes the state of thewalker conditionally to the state of the coin as

S ⌘X

k

(P" ⌦ |kihk| + P# ⌦ |k + 1ihk|), (1)

where {|"i , |#i} are a basis states of HC , {|ki} spans HW ,and we introduced the projectors P ⌘ | ih |.

A possible physical embodiment of such QW dynamics usespolarization and OAM of single photons, playing the roles ofthe coin and the walker degrees of freedom, respectively, withwaveplates to implement the coin operations and q-plates [39]to implement the controlled-shi�. State engineering protocolsleveraging QWs in this se�ing were previously designed anddemonstrated in Refs. [36, 38, 40].

�e state space we are interested in consists of two pairsof QWs, so that the overall system of coins and walkers livesin the four-partite space H ⌘ H(1) ⌦ H(2), with H(i) ⌘H(i)

C ⌦ H(i)W , and H(i)

C , H(i)W accommodating coin and walker

of the ith QW system, respectively (i = 1, 2). Given | i 2 H,we will apply the walk operation on the two QW systemsseparately. �is, in general, entangles the degrees of freedomof a coin with those of the respective walker. In the nextsections, we will describe how to use this QW dynamics totransfer entanglement that is possibly initially present in thetwo-coin subspace to the two-walker one, via local operationon their respective coin degrees of freedom. In an optical setup,this process will transfer the initial entanglement encoded ina polarization state to the two OAM degrees of freedom. �eprocess can be iterated to transfer more entanglement fromthe polarizations to the OAMs.

III. ENTANGLEMENT TRANSFER VIA LOCALPROJECTIONS

Let us consider the following question

Given a four-partite state in H, can we �nd lo-cal coin projections such that the entanglement inthe coin degrees of freedom is transferred to theprojected state of the walkers?

More precisely, given a state | i 2 H, we want to �nd suit-able states |�i 2 H(1)

C and |�i 2 H(2)C such that, upon per-

froming bi-local projections onto them, the coin entanglementpossibly present in | i is transferred to the projected stateh�, �| i 2 H(1)

W ⌦ H(2)W . A schematic description of this for-

mal scenario is given in �g. 1a [Note (Ale): make self-consistentthe notation used in the �gure and the text (| ini, | f i and thethree unitaries in the �gure; in panel (a) it might be worth toput symbols of coin/walker and maybe a curly braket for photon1 and 2 as well)]. It is worth stressing that such entanglementtransfer is not always possible — for example if | i is triviallyseparable with respect to the partition H(1)⌦H(2): no bilocaloperation, including projections, will be able to entangled thewalkers. It is therefore pivotal to �nd the conditions makingsuch protocol viable.

�e task at hand can be broken down into two indepen-dent subproblems, which we will refer to as transferibilityconditions: on the one hand, transferring the entanglementfrom H(1)

C ⌦ H(2)C to H(1)

W ⌦ H(2)C , and then transferring the

entanglement from H(1)W ⌦H(2)

C to H(1)W ⌦H(2)

W . Transferringentanglement from H(1)

C ⌦ H(2)C to H(1)

W ⌦ H(2)W is possible

only if these two subproblems are solvable.Let | i 2 H have Schmidt decomposition

| i =X

k

ppk |uki |vki . (2)

We want a projection |�i 2 H(1)C such that the corresponding

post-projection state | �i 2 H(1)W ⌦ H(2) contains the same

amount of entanglement, in the bipartition H(1)W ⌦ H(2), as

that initially contained in | i. In general, we have

| �i =1

ppproj

X

k

ppkqk |uki |vki , (3)

where pqk |uki = h�|uki and pproj =

Pk pkqk. We distin-

guish between three di�erent scenarios. If (1) the states |ukiare not orthogonal, then some information about which k thestate is in leaks through the polarization measurement, andsome amount of entanglement is therefore degraded. �is willbe shown formally in (�?). (2) If the states |uki are orthog-onal, but the corresponding projection probabilities qk areuneven, then again the entanglement in | �i is smaller thanthat in | i. Finally, if (3) the states |uki are orthogonal, andqk = pproj for all k, then projecting onto |�i fully preservesthe initial entanglement. Note that (3) is thus a necessary andsu�cient condition for entanglement transferability without

EBell state source

FIG. 1. a Entanglement transfer unit. �e system is composed by two particles, 1 and 2, equipped with a `-dimensional degree of freedom,which will be instrumental to the protocol, and an additional d-dimensional degree of freedom. �e entanglement transfer protocol requires a�rst operation E that generates entanglement between the `-dimensional subsystems. �en we have two local operations – with respectto the 1-vs-2 bipartition – that correlates the inner degrees of freedoms of each particle and realizes the `-d dynamics. In the end, localmeasurements allow to transfer the entanglement stored in the initial state to the reduced state of the d-dimensional sub-systems. Weconsider explicitly the case of ` = 2 (qubits) and local operations embodied by the walk operationsWC . Indeed, a discrete-time quantumwalks framework o�ers a very natural encoding of this dynamics: in such embodiment, the coin particle would codify the ` = 2-dimensionalsystem, with the qudit being provided by the position degrees of freedom of the walker. Assuming initially maximally entangled states ofthe qubits, a single iteration of our protocol would be able to transfer one ebit of entanglement at most. By repeating the use of this unit,high-dimensional entangled states can be generated in the d-dimensional degrees of freedom. Furthermore the entanglement stored in suchdegrees of freedom can be retrieved by same operations and transferred back to the two-qubit state. b Conceptual scheme for the transferfrom a Bell state in the coin degree of freedom to the two walkers position space a�er quantum walks and local coin measurements. c Protocoliteration and entanglement accumulation in the high-dimensional space of the two quantum walkers.

Note that situation (3) is thus a necessary and su�cient condi-tion for entanglement transferability without degradation, asif 〈uj |uk〉 = δjk and qk = pproj then Eq. (3) is the Schmidtdecomposition of |Ψγ〉, and therefore the Schmidt coe�cientsof |Ψγ〉 are (in the relevant bipartition) the same as those of|Ψ〉. On the other hand, if (3) is not satis�ed, then the projec-tion results in the degradation of some of the entanglement,as shown in Appendix A.

�erefore, we achieve transferability if |γ〉 is such that〈γ|uk〉 /√pproj are orthonormal vectors. An equivalent — ifless explicit — condition for transferability is the requirement

σ(tr2(PΨγ )) = σ (tr2(PΨ)) , (4)

where σ(A) ≡ σ(A) \ {0} and σ(A) is the set of eigenvaluesof A. �is is a necessary and su�cient condition for transfer-ability, as Eq. (4) is equivalent to requesting that the Schmidtcoe�cients of |Ψγ〉 are the same as those of |Ψ〉. In Fig. 2 wepresent a pictorial description of what TC1 allows to achieve.It is worth noting that, while Eq. (4) is required to fully trans-fer entanglement, it is still possible to transfer some degree ofentanglement if the vectors 〈γ|uk〉 are not fully orthogonal,or the projection probabilities are unequal.

�is problem can be understood as a more restrictive ver-sion of entanglement swapping. Such protocol [59] deals

with a four-partite system in the Hilbert space ⊗jHj (j =A,B,C,D), whose state is separable in the bipartition (AB)-vs-(CD) but entangled in the subsystems A−B and C −D.�e goal of entanglement swapping is to achieve entangle-ment in the state of the A − D compound by performingprojective measurements on B − C . �is is possible for in-stance by implementing a Bell measurement over the jointstate of B and C . Clearly, the problem is analogous to ours,except that we only allow local operations on B and C . No-tably, the use of a Bell measurement is not available in ourse�ing.

IV. ENTANGLEMENT TRANSFER THROUGHQUANTUM-WALK DYNAMICS

In section III we discussed the general problem of transfer-ring entanglement by means of local projections. Most notablywe made no assumption on the inner structure of correlationsin H(i), nor we speci�ed the dimensionality of the entan-glement in the bipartitionH(1) ⊗H(2). �e framework andresults set up so far thus also apply to cases where some pre-existing entanglement exists between the walkers’ degrees offreedom. We now specialize to the case dimH(i)

C = 2, which

4

applies directly to QWs with two-dimensional coins. Moreprecisely, in section IV A we consider states in which H(1)

and H(2) are only entangled through their coin spaces (asin �g. 3). In section IV B we then apply these results to theoutput states obtained from QW dynamics.

A. Entanglement transfer via two-dimensional coins

Consider a state |Ψ〉 ∈ H(1) ⊗ H(2) which is entangledonly via its coin spaces (or more generally, a state having rank2), as in Fig 3. �e corresponding reduced state reads

ρ = p1Pu + p2Pv, p1 + p2 = 1, (5)

for a pair of orthonormal states {|u〉 , |v〉} ∈ H(1). As dis-cussed in section III, to achieve maximal entanglement transferwe need a projection onto a state |γ〉 satisfying TC1, i.e. ful-�lling Eq. (4). �is is equivalent to requiring 〈u|v〉 = 0 where〈γ|j〉 =

√pproj

∣∣j⟩

(j = u, v). Explicitly, these amount tothe conditions

〈γ|trW(|u〉〈v|)|γ〉 = 0, (6)

and 〈γ|trW(|u〉〈u|)|γ〉 = 〈γ|trW(|v〉〈v|)|γ〉 = pproj. Weshow in Appendix B that it is always possible to �nd a state|γ〉 that preserves the orthogonality. To satisfy condition TC1,one then only has to verify that the projection probabilitiesare equal.

B. Entanglement transfer with coined QWs

We now apply the results of the previous section to thespeci�c quantum states resulting from coined QWs. As in sec-tion IV A, we �rst assume that the overall state is entangledwith respect to the bipartitionH(1) ⊗H(2) only via its coinspaces (see Fig. 3). We thus take the initial full state of theform

|Ψ〉 =√p1 |↑, 1〉 ⊗ |↑, 1〉+

√p2 |↓, 1〉 ⊗ |↓, 1〉 , (7)

for some coe�cients p1, p2 ≥ 0 with p1 + p2 = 1. Focusingon H(1), we thus see that the initial states upon which theQW operates are |↑, 1〉 and |↓, 1〉.

H(1)W H(2)

W

H(1)C H(2)

C

〈γ|

H(1)W

H(2)W

H(2)C

FIG. 2. Pictorial representation of the �rst transferability procedure.Given a state which is entangled with respect to the bipartitionH(1) ⊗ H(2), we apply a local projection |γ〉 which preserves theentanglement between the two spaces. Condition (4) determineswhen such a projection exists.

A single QW step with coin operation C amounts to theevolution

|↑, 1〉 → |Ψ↑,1〉 ≡ c11 |↑, 1〉+ c21 |↓, 2〉 ,|↓, 1〉 → |Ψ↓,1〉 ≡ c12 |↑, 1〉+ c22 |↓, 2〉 ,

(8)

where cij are the entries of the unitary matrix representingC. By projecting onto |γ〉 ≡ γ↑ |↑〉 + γ↓ |↓〉 (γ↑,↓ ∈ C) andimposing 〈Ψ↑,1|Ψ↓,1〉 = 0, we get

|γ↑|2c∗11c12 + |γ↓|2c∗21c22 = 0, (9)

which is satis�ed for |γ〉 = (|↑〉+ eiφ |↓〉)/√

2 for any φ ∈ R.�e corresponding projection probabilities are both equal to1/2, as follows from

2|〈γ|Ψ↑,1〉|2 = |c11|2 + |c21|2 = 1,

2|〈γ|Ψ↓,1〉|2 = |c12|2 + |c22|2 = 1.(10)

We conclude that TC1 is always achievable for this class ofstates. Remarkably, the freedom in the choice of the phaseφ means that projections onto |±〉 = (|↑〉 ± |↓〉)/

√2 (as

well as any other orthonormal basis of balanced states) aresuitable to achieve entanglement transfer. �is results in anoverall transfer success probability of 1: measuring in the |±〉basis, both of the possible outcomes achieve TC1, albeit withdi�erent post-projection states.

Consider now the state a�er multiple QW steps. �e �nalreduced state onH(1) is a mixture of |Ψ↑〉 and |Ψ↓〉, where

|Ψs〉 = cos(θs) |↑,Ψs,↑〉+ sin(θs) |↓,Ψs,↓〉 , (11)

with θs and |Ψs,p〉 depending on the number of steps andchoice of coin operators, and s, p ∈ {↑, ↓}. To assess theachievability of TC1 we consider, as in section IV A, the matrixM ≡ trW(|Ψ↑〉〈Ψ↓|). �is has the form

M =

(cos(θ↑) cos(θ↓)O↑↑ cos(θ↑) sin(θ↓)O↓↑cos(θ↓) sin(θ↑)O↑↓ sin(θ↑) sin(θ↓)O↓↓

). (12)

with Osp ≡ 〈Ψ↓s|Ψ↑p〉. Such M is not in general Hermitian,nor normal. Consequently, while it is always possible to �nda state |γ〉 upon which to perform a projection, the corre-sponding projection probabilities are not in general equal, asshown in Fig. 4. It is worth stressing that this does not imply

H(1)W H(2)

W

H(1)C H(2)

C

〈γ|

H(1)W

H(2)W

H(2)C

FIG. 3. Like Fig. 2, but for states in which the entanglement is onlydue to pre-shared entanglement between the coins. �ese are thetypes of states at the �rst entanglement accumulation step.

5

the impossibility of accumulating entanglement using thesetypes of QWs. Rather, this result tells us that this is only pos-sible via entanglement distillation, and thus there cannot bea deterministic protocol achieving such entanglement trans-fer. In other words, Fig. 4 shows that, in such cases, there isno projection preserving entanglement in the residual spaceH(1)W ⊗H(2). Nonetheless, there might still be a |δ〉 ∈ H(2)

Csuch that the second projection recovers the original amountof entanglement, but this can only be done probabilistically,as shown in Refs. [56, 58]. To further highlight this point, weprovide in Fig. 5 numerical results regarding the possibility ofprobabilistic entanglement transfer when both projections areconsidered. In these cases, probabilistic entanglement transferis possible despite TC1 and TC2 are not satis�ed.

�ere are nonetheless QW dynamics in which TC1 isachievable. For example, consider a QW in which the coin isalways taken to be identity: C = I at all steps. �en, a�er nsteps, the evolution amounts to

|↑, 1〉 → |↑, 1〉 , |↓, 1〉 → |↓, n〉 , (13)

where |k〉 denotes the k-th walker position. �e matrix M isthus in this case easily seen to be M = 0, implying that theorthogonality requirement is always satis�ed, making TC1

achievable as long as the projection probabilities are equal.�is constraint is satis�ed by any balanced projection of theform |γ〉 = (|↑〉+ eiφ |↓〉)/

√2, φ ∈ R.

V. ENTANGLEMENT ACCUMULATION

Here we investigate whether the entanglement transferprocedure can be applied iteratively, accumulating more andmore entanglement into the state of the walkers’ degrees offreedom. For this purpose, a�er each successful entanglementtransfer stage, which produces a state of the form

(|γ1〉 ⊗ |γ2〉)C ⊗ |Ψ〉W , (14)

we apply an operation restoring the entanglement betweenthe coins, thus producing a state of the form |Ψ〉W ⊗ |Φ〉C ,with |Φ〉 ∈ H(1)

C ⊗ H(2)C some entangled state — usually a

maximally entangled one. �e QW evolution is then used tocorrelate each coin and walker degree of freedom locally, in or-der to make transfering the entanglement via local projectionspossible.

Suppose one round of entanglement transfer was executedsuccessfully. We therefore have entanglement in the biparti-tion H(1)

W ⊗ H(2)W , while the coin spaces are separated. Can

we perform another round of QW evolutions to transfer evenmore entanglement to the walkers?

Let us consider, as an example, the case where |Ψ〉W hasentanglement dimension 2, and the full state has the form

|Ψ〉 = |↑↑〉C ⊗ (|ψ1ψ2〉+ |ψ2ψ1〉)W/√

2, (15)

for some walker states |ψi〉 with 〈ψi|ψj〉 = δij . Restoring theentanglement between the coins we get

|Ψ′〉 = (|↑↑〉+ |↓↓〉)C ⊗ (|ψ1ψ2〉+ |ψ2ψ1〉)W/2. (16)

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40

1

2

3

4

5

6

0.02

0.04

0.06

0.08

0.10

0.12

θ

φ

Oa

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40

1

2

3

4

5

6

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

θ

Ob

FIG. 4. Final overlap and projection probabilities for the di�erentpossible projections |γ〉 = cos(θ) |0〉 + sin(θ)eiφ |1〉, computedon the output of a Hadamard (a) or random QW (b) with 4 steps.For the random QW, a random coin is used at each step. In eachcase, we consider the input states |↑, 1〉 and |↓, 1〉, and verify thesatis�ability of TC1 on the corresponding outputs. We �rst plotthe squared overlap O ≡ ‖ 〈Ψ↑|γ〉 〈γ|Ψ↓〉 ‖2 for all possible |γ〉,where |Ψ↑〉 , |Ψ↓〉 denote the output states. We �nd that there aretwo orthogonal projections |γ1〉 , |γ2〉 such that this quantity is zero,represented in the �gure with black dots. As discussed in section III,the vanishing overlap is only a necessary, not su�cient condition. Toachieve TC1, we also require the projection probabilities being equal,i.e. p↑ = p↓ where ps = ‖ 〈Ψs|γ〉 ‖2, s ∈ {↑, ↓}. We represent(θ, φ) for which this condition is satis�ed with the magenta regionbounded by dashed black lines. More precisely, the magenta regionoutlines the set of (θ, φ) such that the entropy of the projectionsprobabilities, S((p↑, p↓)), is larger than 0.693 (remembering that− ln 2 ' 0.6931). It is worth noting that, while it is not in generaltrue that p↑ + p↓ = 1 for an arbitrary unitary evolution, this isalways the case for QWs, which allows us to quantify how close p↑and p↓ via the corresponding entropy. As clear from the �gure, inthese two cases, TC1 cannot be achieved for any |γ〉, as the twonecessary conditions cannot be simultaneously satis�ed.

Let us, as in section III, focus on the transferability in H(1).�e reduced state ρ(1) ≡ tr2 PΨ′ has the form

ρ(1) = (P↑ + P↓)C ⊗ (Pψ1+ Pψ2

)W/4. (17)

A QW evolution WS then gives P[WS |↑, ψ1〉] +P[WS |↑, ψ2〉] + P[WS |↓, ψ1〉] + P[WS |↓, ψ2〉] =PΨ1

+ PΨ2+ PΨ3

+ PΨ4, where 〈Ψi|Ψj〉 = δij , and

thus WSρ(1)W†S has rank 4. Achieving entanglementtransfer now entails �nding |γ〉 ∈ H(1)

C such that

〈Ψi|Pγ ⊗ IW |Ψj〉 = δijpproj. (18)

Each successive entanglement transfer iteration involves adoubling of the number of orthogonal states to preserve, asfollows from observing that if A has rank r and B has rankr′, then A⊗B has rank rr′.

Consider now a QW in which each coin operation is theidentity: C = I . We will show that, with this particulartype of dynamics, we can accumulate arbitrary amounts ofentanglement into the walkers’ degrees of freedom, using thecoins as mediators. �e unitary evolution corresponding to nsteps with C = I isWS,n = Sn with S the controlled-shi�

6

a b

FIG. 5. (a) Log-negativity N of the output state a�er 4-stepHadamard QW and coin projections along |γ〉 = cos θ |↑〉 +eiφ sin θ |↓〉, with θ ∈ [0, π/2] and φ ∈ [0, 2π]. States with maxi-mum entanglementN = 1 are found for values of θ and φ identi�edby the red spots in the �gure. In this scenario, such states are gen-erated with probability p, reported in panel (b) �e probability oftransfer is p = 0.43 (see the red spots highlighted in the map). �isestimate takes into account that the projections on

∣∣γ⊥⟩ producestates with the same log-negativity.

operation. �e action on the basis states is then

Sn = P↑ ⊗ I + P↓ ⊗ En+, (19)

where E+ ≡∑k |k + 1〉〈k| is the operation shi�ing the

walker’s position, and En+ is thus the operator moving thewalker n positions forward. Consider an initial state

(|↑, ↑〉+ |↓, ↓〉)C ⊗ |Ψ〉W (20)

with |Ψ〉W an entangled state of the walkers in which thedi�erence between �nal and initial occupied positions is ` ∈ N(for example, if

√2 |Ψ〉W = |1, 1〉 + |3, 3〉, then ` = 2). If

|Ψ〉W has rank r, the reduced state onH(1) has the form

tr2(PΨ) =1

2(P↑ + P↓)⊗

r∑

k=1

pkPψk (21)

for some set of orthonormal states {|ψk〉}k ⊂ H(1)W . Evolving

through S`+1, we get

tr2(PΨ)→ 1

2

(P↑ ⊗

r∑

k=1

pkPψk + P↓ ⊗r∑

k=1

pkPψ′k

),

(22)with

⟨ψ′j∣∣ψ′k⟩

= δjk and⟨ψ′j∣∣ψk⟩

= 0. �en, any balancedprojection |γ〉 = 1√

2(|↑〉 + eiφ |↓〉) achieves TC1, which

means that the entanglement can be transferred determin-istically from coins to walkers.

In light of these �ndings, we can now propose the fol-lowing explicit protocol, which allows to accumulate deter-ministically entanglement into the walkers degrees of free-dom using the coins as mediators. Starting from the state(|↑, ↑〉 + |↓, ↓〉)/

√2 ⊗ |1, 1〉 ∈ H, we apply the conditional

shi� operation to both QWs and then measure both coinsin the basis |±〉. �e possible states a�er the projection arethen (|1, 1〉± |2, 2〉)/

√2, where the sign is + if the two coins

a b

FIG. 6. (a) Trends ofN for QWs with C = I (purple) and HadamardQWs (green) in the entanglement accumulation protocol. �e num-bers near the markers specify the number of QW steps needed tostore the entanglement in the walkers subspaces. �e �rst casecorresponds to the deterministic optimal transfer described in themain text, in which one ebit is transferred at each iteration. In theHadamard QWs this optimal transfer it is not achievable a�er the�rst iteration. (b) Accumulation probability for the two cases.

are found in the same state, and − otherwise. Restoring theentanglement between the coins, we then re-apply the QWevolution, now for two steps, and project again in the basis|±〉, resulting in an output state of the form

1

2[(|1, 1〉 ± |2, 2〉)± (|3, 3〉 ± |4, 4〉)]. (23)

�is procedure can be iterated to accumulate more and moreentanglement inH(1)

W ⊗H(2)W . At the n-th iteration, we evolve

both systems through 2n QW steps with C = I , that is,through the unitary S2n ⊗S2n , and then project onto the |±〉basis, resulting in a maximally entangled state of the form

2−n/22n∑

k=1

(−)σk |k, k〉 , (24)

with (−)σk ∈ {1,−1} for all k. In Fig.6 we report the trend ofthe deterministic transfer and accumulation described above.Notice that this goal cannot always be achieved, as for examplein the case of the Hadamard QW reported in the �gure. Hereit is not possible to transfer one-ebit of entanglement periteration, not even probabilistically.

VI. ENTANGLEMENT RETRIEVAL

�e arguments of section III do not make assumptions onthe dimensions of H(i)

C and H(i)W . �is means that they can

be used not only to study the transfer of entanglement fromcoins to positions, but also the other way around. For exam-ple, if the initial reduced state onH(1) is [P↑ ⊗ (P1 + P2)]/2,with |1〉 , |2〉 a pair of orthonormal walker’s states, then ap-plying a Hadamard operation to the coin, and two steps ofQW evolution with C = I , we obtain the state

1

4[P↑ ⊗ (P1 + P2) + P↓ ⊗ (P3 + P4)]. (25)

7

�en, measuring in the Hadamard four-dimensional basis —i.e., the basis formed by the columns of the 4× 4 Hadamardmatrix — we achieve TC1.

VII. EXPERIMENTAL PROPOSAL

QWs have been previously demonstrated on photonic plat-forms [60–69]. We propose here a scheme to implement ourentanglement transfer and accumulation protocol in an opti-cal platform, encoding coin and walker degrees of freedom incircular polarization and OAM degrees of freedom of singlephotons.

Implementing the entanglement transfer protocol involvesas starting point two polarization-entangled photons, that canbe generated via photon sources based on parametric downconversion. �en, each photon of the pair evolves through aQW evolution in the polarization and OAM degrees of free-dom, followed by a projective measurement on the polariza-tions. �e coin operators are realized on the polarizationthrough suitable sets of waveplates. �e shi� operator, whichinvolves an interaction between OAM and polarization, is nat-urally implemented by the inhomogeneous and birefringentdevices known as q-plates [52, 53]. Projective measurementson the coins are realized using waveplates and a polarizingbeam spli�er.

For the entanglement accumulation protocol, one alsoneeds a way to “reload” the entanglement into the photons’polarization without a�ecting their OAMs. As discussed in sec-tion V, the �rst entanglement transfer procedure results inone of the states

|++〉 ⊗ (|1, 1〉 ± |2, 2〉)/√

2, (26)

. where |1〉 and |2〉 label OAM states, while |+〉 are diago-nal polarization states. It is straightforward to show that theaction of a polarizing beam-spli�er combined with two half-waveplates can restore, with probability 1/2, the entangledstate in polarization needed to achieve accumulation. Express-ing the state of Eq. (26) in terms of creation operators a† andb† of the two photons, we have:

1√2

(a†+,1b

†+,1 ± a†+,2b†+,2

)|vac〉 , (27)

where |vac〉 is the vacuum state in the Fock representation.�e two photons are injected in the input ports, labelled by{a, b}, of a polarizing beam-spli�er, a�er a polarization rota-tion made by two half-waveplates of angles θa and θb respec-tively. �e creation operators a�er the overall transformationbecome

a†+,1/2 → cos θaa†+,1/2 + ı sin θab

†−,1/2

b†+,1/2 → cos θbb†+,1/2 + ı sin θba

†−,1/2

(28)

Substituting such expression in Eq. (27) and choosing theorientation of the two half-waveplates θa = θb = π/4 we ob-tain that the output state is composed by two terms. �e �rstterm corresponds to the two photons exiting from di�erent

output ports of the polarizing beam-spli�er, while the secondterm corresponds to the case where the photons exit from thesame port. �e �rst part of this state embodies the resourceneeded for the protocol accumulation, and has the followingform:

(|++〉 ± |−−〉)√2

⊗ (|1, 1〉 ∓ |2, 2〉)√2

. (29)

We can discard the second term where two photon exit fromthe same port by post-selecting two-fold coincidences be-tween single-photon detectors at the end of the second itera-tion. It is worth noting that the probabilistic generation of thesecond maximally entangled state is due to the choice of en-coding qubits in photons. However, we remark that we couldalso consider the state produced by the projection |−−〉 a�erthe �rst operation. Indeed, this projection produces stateswith the same symmetry properties. In this way it is possibleto double the probability of generating states with more thanone-ebit of entanglement.

As a �nal remark, we note that the whole protocol, beingessentially based upon the general QW dynamics, can beimplemented in various experimental platforms [70–74].

VIII. CONCLUSIONS

We have addressed the generation of high-dimensional en-tangled states through a protocol of entanglement transferfrom a low-dimensional resources. We have identi�ed gen-eral transfer conditions that, if met, guarantee the successfulpouring of any entanglement initially contained in the stateof the resource to the high-dimensional receiver. �is has thenallowed us to draw a speci�c analysis aimed at the dynamicsentailed by a QW, where low-dimensional resources and high-dimensional receivers are naturally embodied by coin andwalker degrees of freedom respectively. While characterizingthe performance of the entanglement transfer scheme, wehave been able to design schemes for entanglement accumu-lation and retrieval, thus drawing a complete picture for themanipulation of entanglement through a hetero-dimensionalinterface of great experimental potential. Indeed, the QW-based protocols addressed and studies in this paper are fullyamenable to an implementation making use of polarizationand OAM encoding. �e scenario set by our schemes sets apromising framework for the use of low-dimensional entan-glement as a resource to achieve otherwise complex entangledstructures and states that can be experimentally synthesisedand exploited.

ACKNOWLEDGMENTS

�is work is supported by MIUR via PRIN 2017 (Proge�odi Ricerca di Interesse Nazionale): project QUSHIP (2017SRN-BRK), 387439, and by the ERC Advanced Grant QU-BOSS(Grant agreement no. 884676). �e authors acknowledge �-nancial support from H2020 through the Collaborative ProjectTEQ (Grant Agreement No. 766900), the DfE-SFI Investigator

8

Programme (Grant No. 15/IA/2864), the Leverhulme TrustResearch Project Grant Ultra�te (grant nr. RGP-2018-266),COST Action CA15220, and the Royal Society Wolfson Re-search Fellowship scheme (RSWF\R3\183013). T.G. acknowl-edges La Sapienza University of Rome via the grant for jointresearch projects for the mobility n.2289/2018 Prot. n.50074

Appendix A: Entanglement decreases if orthogonality is notpreserved

Let us show that if 〈uj |uk〉 6= δjk then the Schmidt coe�-cients must change upon projection. Indeed, in this case, |Ψγ〉has the form |Ψγ〉 =

∑k

√pk |uk〉 |vk〉 where 〈vj |vk〉 = δjk

and∑k pk = 1. Denoting with Ψγ the matrix whose vec-

torization is |Ψγ〉, this Schmidt decomposition amounts tothe singular value decomposition Ψγ = U

√DV †, with

D = diag(p1, ..., pn), V the unitary matrix whose columnsare |vk〉, and U the (non-unitary) matrix with columns |uk〉.�en

ΨγΨ†γ = UDU† =∑

k

pkPuk , (A1)

where Puk = |uk〉〈uk| are in general non-orthogonal rank-1 projectors. Let us then prove that if a matrix is a convexcombination of rank-1 projections, then it always majorizesthe vector of coe�cients of the convex combination. In ourcase, this translates to

∑k pkPuk � p.

Let Pk be rank-1 projections, pk ≥ 0 coe�cients such that∑nk=1 pk = 1, and A ≡∑n

k=1 pkPk. We want to prove thatA � p, where p = (pk)nk=1 is the vector of coe�cients, andthe majorization relation is de�ned on Hermitian matricesvia the corresponding vector of eigenvalues, that is, A �p ⇐⇒ σ(A) � p where σ(A) is the vector of eigenvaluesof A. If A has dimension larger than n, we de�ne λ(A) asthe vector of the n largest eigenvalues, in order to make themajorization relation well-de�ned. Without loss of generality,let us assume that the pk are in decreasing order: p1 ≥ p2 ≥... ≥ pn. De�ne the partial sums A` ≡

∑`k=1 pkPk, so that

A = An. Observe that A` ≥ Ar whenever ` ≥ r. Becauserank(Pk) = 1 for all k, we must also have rank(A`) ≤ `.Denoting with λ↓j (A) the j-th largest eigenvalue of A, thisimplies that

∑

k=1

λ↓k(A`) = tr(A`) =∑

k=1

pk. (A2)

Using A = An ≥ A` for all 1 ≤ ` < n, we thus conclude that

∑

k=1

λ↓k(A) ≥∑

k=1

λ↓k(A`) =∑

k=1

pk ≡∑

k=1

p↓k, (A3)

that is, λ(A) � p, which is the de�nition of A � p.Assuming |uk〉 are orthogonal, then the Schmidt coe�-

cients of |Ψγ〉 are√pk = p

−1/2proj

√pkqk .

Appendix B: Finding projections preserving orthogonality

We prove in this section that, for any state of the form|Ψ〉 =

√p1 |u, u′〉 +

√p2 |v, v′〉, with 〈u|v〉 = 〈u′|v′〉 =

0, there is some |γ〉 such that the post-projected states areorthogonal, i.e. such that 〈u|v〉 = 0 where √pu |u〉 = 〈γ|u〉and√pv |v〉 = 〈γ|v〉.

Here, |Ψ〉 ∈ H(1) ⊗H(2), |u〉 , |v〉 ∈ H(1), and |u′〉 , |v′〉 ∈H(2). Moreover, H(1) = H(1)

C ⊗H(1)W , and |γ〉 ∈ H(1)

C . Notethat here we assume dim(H(1)

C ) = 2, while the only require-ment onH(1)

W andH(2) is that their dimension must be largerthan 2, in order to accommodate |Ψ〉.

De�ne M ≡ trW(|u〉〈v|) ∈ Lin(H(1)C ). Note that this is

a 2 × 2 traceless matrix, as follows from 〈u|v〉 = 0. Ourobjective is then to �nd |γ〉 such that 〈γ|M |γ〉 = 0. For thepurpose, we consider di�erent scenarios:

1. If M is normal, then

M = λ(|v1〉〈v1| − |v2〉〈v2|), (B1)

for some λ ∈ C and 〈vi|vj〉 = δij . �en,√

2 |γφ〉 = |v1〉+ eiφ |v2〉 , φ ∈ R (B2)

are all suitable projections such that 〈γφ|M |γφ〉 = 0.Note that this also implies that we can �nd orthogonalstates that both correspond to valid projections.

2. Consider now a generic 2 × 2 M . Given a two-dimensional M with tr(M) = 0, provided M 6= 0, wemust always haveM2 = −det(M)I . �is follows fromobserving that the eigenvalues of M are ±

√−detM ,

and therefore (M +√−detM)(M −

√−detM) = 0

Writing its singular value decomposition as M =UDV †, this implies that UDV †UDV † = −det(M)I,and therefore

DV †U = −det(M)(V †U)†D−1. (B3)

IfD = d1P1 +d2P2 and V †U = |1〉〈w1|+ |2〉〈w2|, then

d1 |1〉〈w1|+ d2 |2〉〈w2| =−eiφ(d2 |w1〉〈1|+ d1 |w2〉〈2|),

(B4)

where det(M) = |det(M)|eiφ and we observed that|det(M)| = d1d2. �ere are then two possibilities:either d1 = d2, which implies M is normal, and thiscase was covered above, or d1 6= d2, which implies bythe uniqueness of the singular value decomposition that|w1〉 = |2〉 and |w2〉 = |1〉 up to phases. Consequently,we would have

M = d1 |u1〉〈v1|+ d2 |u2〉〈v2| , (B5)

where 〈ui|uj〉 = 〈vi|vj〉 = δij and 〈u1|v2〉 =〈u2|v1〉 = 0. We can then use |γ〉 = |vi〉 as suitableprojections, as 〈vi|M |vi〉 = 0.

9

Appendix C: Entanglement transfer toy examples

We give in this section a few toy examples showcasing theuse of the results presented in appendices A and B.

1. Example with di�erent projection probabilities

Suppose

2 |u〉 ≡(√

2 |↑〉 ⊗ |2〉+ |↓〉 ⊗ (|1〉+ |2〉)),

2 |v〉 ≡(|↑〉 ⊗ (|1〉+ |2〉)−

√2 |↓〉 ⊗ |2〉

).

(C1)

�en, M = 12√

2

(1 −

√2√

2 −1

). �e singular values of M are

√2 |γ±〉 = |↑〉 ± |↓〉, which are therefore also the projections

that maintain the orthogonality. �e corresponding projectionprobabilities are

pu± = |〈γ±|u〉|2 = (2±√

2)/4,

pv± = |〈γ±|v〉|2 = (2∓√

2)/4.(C2)

and the projected states read

√pu± |u±〉 ≡ 〈γ±|u〉 ,

√pv± |v±〉 ≡ 〈γ±|v〉 . (C3)

It follows that, despite |γ±〉 preserving the orthogonality of|u〉 , |v〉, the two states correspond to di�erent projectionprobabilities, and therefore entanglement is necessarily de-graded. More explicitly, using |γ+〉 as an example, the corre-sponding projection probabilities are pu+ = (2 +

√2)/4 and

pv+ = (2−√

2)/4. �is means that if we have an entangled

state of the form

|Ψ〉 =√p1 |u〉 ⊗ |0〉+

√p2 |v〉 ⊗ |1〉 , (C4)

where |0〉 , |1〉 is an arbitrary pair of orthonormal states in anauxiliary space, then projecting onto |γ+〉 gives the state

N−1/2(√p1pu+ |u+〉 ⊗ |0〉+

√p2pv+ |v+〉 ⊗ |1〉) (C5)

with probabilityN ≡ p1pu++p2p

v+. Clearly, because pu+ 6= pv+,

the Schmidt coe�cients of this states are di�erent, and thusthe state is less entangled.

2. Example with same projection probabilities

Suppose

2 |u〉 =√

2 |↑〉 ⊗ |2〉+ |↓〉 ⊗ (|1〉+ |2〉),2 |v〉 = |↑〉 ⊗ (|1〉 − |2〉) +

√2 |↓〉 ⊗ |1〉 .

(C6)

�en, M = 12√

2

(−1 00 1

)is normal with eigenvectors

|λ+〉 = |↑〉, |λ−〉 = |↓〉. It follows that any balanced state ofthe form

√2 |γφ〉 = |↑〉+ eiφ |↓〉 preserves the orthogonality

of |u〉 , |v〉. Correspondingly, we have

2√

2eiφ 〈γφ|u〉 = |1〉+ (√

2eiφ + 1) |2〉 ,2√

2eiφ 〈γφ|v〉 = (√

2 + eiφ) |1〉 − eiφ |2〉 ,(C7)

with probabilities

puφ = pvφ =1

4(2 +

√2 cosφ). (C8)

It follows that any |γφ〉 achieves entanglement transfer. More-over, we can choose two orthogonal states, e.g. |γ0〉 and |γπ〉,so that entanglement transfer is achieved deterministically.

[1] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki,Reviews of Modern Physics 81, 865 (2009).

[2] A. D. A.D. Greentree, S. G. Schirmer, F. Green, L. C. L. Hol-lenberg, A. R. Hamilton, and R. G. Clark, Phys. Rev. Le�. 92,097901 (2004).

[3] M. Fujiwara, M. Takeoka, J. Mizuno, and M. Sasaki, Phys. Rev.Le�. 90, 167906 (2003).

[4] D. Kaszlikowski, P. Gnacinski, M. Zukowski, W. Miklaszewski,and A. Zeilinger, Phys. Rev. Le�. 85, 4418 (2000).

[5] D. Cozzolino, B. Da Lio, D. Bacco, and L. K. Oxenløwe, Ad-vanced �antum Technologies 2, 1900038 (2019).

[6] X. Liu, G. Long, D. Tong, and F. Li, Physical Review A 65,022304 (2002).

[7] A. Grudka and A. Wojcik, Phys. Rev. A 66, 014301 (2002).[8] X.-M. Hu, Y. Guo, B.-H. Liu, Y.-F. Huang, C.-F. Li, and G.-C.

Guo, Sci. Adv. 4, eaat9304 (2018).[9] H. Bechmann-Pasquinucci and A. Peres, Phys. Rev. Le�. 85,

3313 (2000).

[10] N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, Phys. Rev.Le�. 88, 127902 (2002).

[11] D. Bruß and C. Macchiavello, Phys. Rev. Le�. 88, 127901 (2002).[12] V. Karimipour, A. Bahraminasab, and S. Bagherinezhad, Phys.

Rev. A 65, 052331 (2002).[13] A. Acin, N. Gisin, and V. Scarani, �ant. Inf. Comp. 3, 563

(2003).[14] V. Karimipour, A. Bahraminasab, and S. Bagherinezhad, Phys.

Rev. A 65, 052331 (2002).[15] T. Durt, D. Kaszlikowski, J.-L. Chen, and L. C. Kwek, Phys. Rev.

A 69, 032313 (2004).[16] S. Groblacher, T. Jennewein, A. Vaziri, G. Weihs, and

A. Zeilinger, New J. Phys. 8, 75 (2006).[17] M. Huber and M. Pawłowski, Phys. Rev. A 88, 032309 (2013).[18] J. Nunn, L. J. Wright, C. Soller, L. Zhang, I. A. Walmsley, and

B. J. Smith, Optics Express 21, 15959 (2013).[19] J. Mower, Z. Zhang, P. Desjardins, C. Lee, J. H. Shapiro, and

D. Englund, Phys. Rev. A 87, 062322 (2013).

10

[20] C. Lee, Z. Zhang, G. R. Steinbrecher, H. Zhou, J. Mower,T. Zhong, L. Wang, X. Hu, R. D. Horansky, V. B. Verma, A. E.Lita, R. P. Mirin, F. Marsili, M. D. Shaw, S. W. Nam, G. W. Wor-nell, F. N. C. Wong, J. H. Shapiro, and D. Englund, Phys. Rev. A90, 062331 (2014).

[21] T. Zhong, H. Zhou, R. D. Horansky, C. Lee, V. B. Verma, A. E.Lita, A. Restelli, J. C. Bienfang, R. P. Mirin, T. Gerrits, S. W. Nam,F. Marsili, M. D. Shaw, Z. Zhang, L. Wang, D. Englund, G. W.Wornell, J. H. Shapiro, and F. N. C. Wong, New J. Phys. 17,022002 (2015).

[22] M. Mirhosseini, O. S. Magana-Loaiza, M. N. O’Sullivan, B. Ro-denburg, M. Malik, M. P. J. Lavery, M. J. Padge�, D. J. Gauthier,and R. W. Boyd, New J. Phys. 17, 033033 (2015).

[23] I. L. Chuang, D. W. Leung, and Y. Yamamoto, Phys. Rev. A 56,1114 (1997).

[24] E. T. Campbell, H. Anwar, and D. E. Browne, Phys. Rev. X 2,041021 (2012).

[25] G. Duclos-Cianci and D. Poulin, Phys. Rev. A 87, 062338 (2013).[26] M. H. Michael, M. Silveri, R. T. Brierley, V. V. Albert, J. Salmile-

hto, L. Jiang, and S. M. Girvin, Phys. Rev. X 6, 031006 (2016).[27] S. D. Bartle�, H. d. Guise, and B. C. Sanders, Phys. Rev. A 65,

052316 (2002).[28] T. C. Ralph, K. J. Resch, and A. Gilchrist, Phys. Rev. A 75, 022313

(2007).[29] B. P. Lanyon, M. Barbieri, M. P. Almeida, T. Jennewein, T. C.

Ralph, K. J. Resch, G. J. Pryde, J. L. O’Brien, A. Gilchrist, andA. G. White, Nat. Phys. 5, 134 (2009).

[30] E. T. Campbell, Phys. Rev. Le�. 113, 230501 (2014).[31] N. Friis, G. Vitagliano, M. Malik, and M. Huber, Nat. Rev. Phys.

1, 72 (2019).[32] M. Erhard, M. Krenn, and A. Zeilinger, Nat. Rev. Phys. 2, 365

(2020).[33] Y. Aharonov, L. Davidovich, and N. Zagury, Phys. Rev. A 48,

1687 (1993).[34] A. Nayak and A. Vishwanath, arXiv (2000), arXiv:quant-

ph/0010117.[35] A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, and J. Watrous,

in Proceedings of the thirty-third annual ACM symposium on�eory of computing (ACM, 2001) pp. 37–49.

[36] J. Kempe, Contemp. Phys. 44, 307 (2003).[37] S. E. Venegas-Andraca, �antum Inf. Process. 11, 1015 (2012).[38] M. Paternostro, W. Son, and M. S. Kim, Phys. Rev. Le�. 92,

197901 (2004).[39] M. Paternostro, G. Falci, M. Kim, and G. Massimo Palma, Phys.

Rev. B 69, 214502 (2004).[40] M. Paternostro, W. Son, M. S. Kim, G. Falci, and G. M. Palma,

Phys. Rev. A 70, 022320 (2004).[41] A. Sera�ni, M. Paternostro, M. S. Kim, and S. Bose, Phys. Rev.

A 73, 022312 (2006).[42] G. Adesso, S. Campbell, F. Illuminati, and M. Paternostro, Phys.

Rev. Le�. 104, 240501 (2010).[43] M. Paternostro, G. Adesso, and S. Campbell, Phys. Rev. A 80,

062318 (2009).[44] M. Erhard, R. Fickler, M. Krenn, and A. Zeilinger, Light: Science

& Applications 7, 17146 (2018).[45] T. Kitagawa, M. A. Broome, A. Fedrizzi, M. S. Rudner, E. Berg,

I. Kassal, A. Aspuru-Guzik, E. Demler, and A. G. White, NatureCommunications 3, 882 (2012).

[46] L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi,R. Ramponi, and R. Osellame, Phys. Rev. Le�. 108, 010502(2012).

[47] F. Cardano, A. D’Errico, A. Dauphin, M. Ma�ei, B. Piccirillo,C. de Lisio, G. D. Filippis, V. Cataudella, E. Santamato, L. Mar-rucci, M. Lewenstein, and P. Massignan, Nat. Commun. 8, 15516

(2017).[48] I. Pitsios, L. Banchi, A. S. Rab, M. Bentivegna, D. Caprara,

A. Crespi, N. Spagnolo, S. Bose, P. Mataloni, R. Osellame, andF. Sciarrino, “Photonic simulation of entanglement growth a�era spin chain quench,” (2016), arXiv:1603.02669.

[49] L. Innocenti, H. Majury, T. Giordani, N. Spagnolo, F. Sciarrino,M. Paternostro, and A. Ferraro, Phys. Rev. A 96 (2017).

[50] T. Giordani, F. Flamini, M. Pompili, N. Viggianiello, N. Spagnolo,A. Crespi, R. Osellame, N. Wiebe, M. Walschaers, A. Buchleitner,and et al., Nat. Photon. 12, 173 (2018).

[51] T. Giordani, A. Suprano, E. Polino, F. Acanfora, L. Innocenti,A. Ferraro, M. Paternostro, N. Spagnolo, and F. Sciarrino, Phys.Rev. Le�. 124 (2020).

[52] L. Marrucci, C. Manzo, and D. Paparo, Phys. Rev. Le�. 96,163905 (2006).

[53] T. Giordani, E. Polino, S. Emiliani, A. Suprano, L. Innocenti,H. Majury, L. Marrucci, M. Paternostro, A. Ferraro, N. Spagnolo,and et al., Phys. Rev. Le�. 122 (2019).

[54] R. Vieira, E. P. M. Amorim, and G. Rigolin, Phys. Rev. Le�. 111,180503 (2013).

[55] A. Gratsea, M. Lewenstein, and A. Dauphin, �antum Scienceand Technology 5, 025002 (2020).

[56] M. A. Nielsen and G. Vidal, �antum Info. Comput. 1, 76 (2001).[57] M. A. Nielsen, Phys. Rev. Le�. 83, 436 (1999).[58] G. Vidal, Phys. Rev. Le�. 83, 1046 (1999).[59] M. Zukowski, A. Zeilinger, M. A. Horne, and A. K. Ekert, Phys.

Rev. Le�. 71, 4287 (1993).[60] H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morando�i, and

Y. Silberberg, Phys. Rev. Le�. 100, 170506 (2008).[61] A. Peruzzo, M. Lobino, J. C. F. Ma�hews, N. Matsuda, A. Politi,

K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Worho�,Y. Bromberg, Y. Silberberg, M. G. �ompson, and J. L. O’Brien,Science 329, 1500 (2010).

[62] M. A. Broome, A. Fedrizzi, B. P. Lanyon, I. Kassal, A. Aspuru-Guzik, and A. G. White, Phys. Rev. Le�. 104, 153602 (2010).

[63] P. P. Rohde, A. Schreiber, M. Stefanak, I. Jex, and C. Silberhorn,New J. Phys. 13, 013001 (2011).

[64] A. Schreiber, K. N. Cassemiro, V. Potocek, A. Gabris, P. J. Mosley,E. Andersson, I. Jex, and C. Silberhorn, Phys. Rev. Le�. 104,050502 (2010).

[65] L. Sansoni, F. Sciarrino, G. Vallone, P. Mataloni, A. Crespi,R. Ramponi, and R. Osellame, Phys. Rev. Le�. 108, 010502(2012).

[66] J. Boutari, A. Feizpour, S. Barz, C. Di Franco, M. S. Kim, W. S.Kolthammer, and I. A. Walmsley, Journal of Optics 18, 094007(2016).

[67] F. Cardano, F. Massa, H. Qassim, E. Karimi, S. Slussarenko,D. Paparo, C. de Lisio, F. Sciarrino, E. Santamato, R. W. Boyd,and L. Marrucci, Sci. Adv. 1, e1500087 (2015).

[68] F. Cardano, M. Ma�ei, F. Massa, B. Piccirillo, C. de Lisio, G. D.Filippis, V. Cataudella, E. Santamato, and L. Marrucci, Nat.Commun. 7, 11439 (2016).

[69] F. Caruso, A. Crespi, A. G. Ciriolo, F. Sciarrino, and R. Osellame,Nat. Commun. 7, 11682 (2016).

[70] K. Manouchehri and J. Wang, Physical Implementation of �an-tum Walks (Springer Berlin Heidelberg, Berlin, Heidelberg,2014).

[71] R. Cote, A. Russell, E. E. Eyler, and P. L. Gould, New J. Phys. 8,156 (2006).

[72] H. Schmitz, R. Matjeschk, C. Schneider, J. Glueckert, M. En-derlein, T. Huber, and T. Schaetz, Phys. Rev. Le�. 103, 090504(2009).

[73] F. Zahringer, G. Kirchmair, R. Gerritsma, E. Solano, R. Bla�,and C. F. Roos, Phys. Rev. Le�. 104, 100503 (2010).

11

[74] F. Meinert, M. J. Mark, E. Kirilov, K. Lauber, P. Weinmann,M. Grobner, A. J. Daley, H.-C. Nagerl, N. Ismail, K. Worho�,

Y. Bromberg, Y. Silberberg, M. G. �ompson, and J. L. OBrien,Science 344, 1259 (2014).