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

The generation and control of quantum correlations in high-dimensional systems is a major challenge in the present landscape of quantum technologies. Achieving such non-classical high-dimensional resources will potentially unlock enhanced capabilities for quantum cryptography, communication and computation. We propose a protocol that is able to attain entangled states of $d$-dimensional systems through a quantum-walk-based {\it transfer \&accumulate} mechanism involving coin and walker degrees of freedom. The choice of investigating quantum walks is motivated by their generality and versatility, complemented by their successful implementation in several physical systems. Hence, given the cross-cutting role of quantum walks across quantum information, our protocol potentially represents a versatile general tool to control high-dimensional entanglement generation in various experimental platforms. In particular, we illustrate a possible photonic implementation where the information is encoded in the orbital angular momentum and polarization degrees of freedom of single photons.


I. INTRODUCTION
antum entanglement underpins many of the advantages promised by the technological advances in quantum information processors [1]. Despite considerable research e orts have been devoted to achieving seamless generation and control of two-dimensional systems, it is known that two-dimensional entanglement entails limitations in a variety of se ings [2][3][4]. When higher-dimensional entanglement is used -for example in the context of quantum communication [5] -higher channel capacities can be achieved through superdense coding protocols [6][7][8].
e potential bene ts of high-dimensional entanglement have stimulated a signi cant e ort towards its generation, manipulation, and certi cation in various platforms including, in particular, optical systems [31,32]. Despite signi cant experimental advances, the implementation of such tasks remains demanding, especially in light of the di culties associated to controlling systems and transformations in large Hilbert spaces.
In this paper, we show how to leverage controllable lowdimensional systems, together with special quantum devices acting as interfaces between systems of di erent dimensions, to realize an e ective entanglement-transfer protocol from low-to high-dimensional degrees of freedom.
antum correlations stored in two-dimensional degrees of freedom -such as the polarizations of entangled photons -can thus be passed into high-dimensional information carriers via suitable local interactions and measurements.
We derive the general conditions under which such entan-glement transfer is feasible. We then focus on the case of states producible by discrete-time one-dimensional quantum walks (QW) [33][34][35][36][37]. ese model a natural type of interaction between hetero-dimensional systems, and are widely available in a variety of physical systems. We study the conditions under which QW dynamics allow to transfer entanglement between coin and walker degrees of freedom, and prove the feasibility of accumulating entanglement in the high-dimensional system by repeatedly creating it and transfering it from the low-dimensional one. is scheme constitutes a promising two-way interface to transfer reliably entanglement between di erent information carriers [38][39][40][41][42][43].
A particularly suitable platform for the manipulation of high-dimensional systems, which has also been successful in demonstrating control of the QW dynamics, is embodied by the orbital angular momentum (OAM) of light. Recent experimental progress enabled by the growing capacity to prepare, manipulate and measure OAM states are opening up the possibility to explore the richness of high-dimensional Hilbert spaces for the sake of quantum information processing [44]. A protocol allowing to generate high-dimensional OAM states using a simple dynamics such as the one o ered by QWs would therefore be a signi cant step forward towards the provision of on demand high-dimensional entangled states. e remainder of this paper is organized as follows. In section II we overview the necessary background on QWs and OAM. In section III we formalise the general conditions for the occurrence of entanglement transfer and study their solutions. We then specialise in section IV to the context set by QWs, and study -in section V -the possibility of accumulating entanglement in one degree of freedom by repeated applications of the entanglement-transfer protocol. We conclude in section VII by detailing a possible experimental implementation of the protocol in the framework of OAM-based implementation of the QW dynamics. arXiv:2010.07127v1 [quant-ph] 14 Oct 2020 2 II. BACKGROUND Discrete-time QWs embody a widely studied type of interaction between a two-dimensional "coin" degree of freedom, and a high-dimensional "walker" one [33][34][35][36][37]. Despite their simplicity, QWs allow to engineer e ectively a broad range of evolutions [45][46][47][48]. Recently, some of us demonstrated the potential of a QW-based architecture to exibly implement quantum state engineering of a single OAM [49,50], as well as the machine-learning-enhanced classi cation of hybrid polarization-OAM states of light [51]. A possible physical embodiment of such QW dynamics uses polarization and OAM of single photons, playing the roles of the coin and the walker degrees of freedom, respectively, with waveplates to implement the coin operations and q-plates [52] to implement the controlled-shi . State engineering protocols leveraging QWs in this se ing were previously designed and demonstrated in Refs. [49,51,53].
More precisely, QWs are de ned in a bipartite coin-walker space H C ⊗ H W , where H C(W) denotes the coin (walker) space. We assume dim(H C ) = 2. e evolution is de ned by the repeated action of a unitary walk operation W C ≡ S(C⊗I), which comprises the sequential action of a controlled-shi operation S, and a coin ipping operation C. e coin ipping operation acts locally on the coin space, while the controlledshi changes the state of the walker conditionally to the state of the coin: where {|↑ , |↓ } form a basis for H C , {|k } k≥0 spans H W , and we introduced the notation P ψ ≡ |ψ ψ|. e state space we are interested in consists of two pairs of QWs, so that the overall system of coins and walkers lives in the four-partite space H ≡ H (1) ⊗ H (2) , with W accommodating coin and walker of the i th party, respectively (i = 1, 2). Given |Ψ ∈ H, we apply W C locally on H (1) and H (2) . is, in general, entangles each coin with the respective walker [54,55]. In the next sections, we will describe how to use this QW dynamics to transfer entanglement from the two-coin subspace to the two-walker one, using only local operations on the coins. In an optical setup, this process will transfer the initial entanglement encoded in a polarization state to the two OAM degrees of freedom. e process can be iterated to transfer more entanglement from the polarizations to the OAMs.

III. ENTANGLEMENT TRANSFER VIA LOCAL PROJECTIONS
We now address the challenge of transferring entanglement across di erent degrees of freedom using solely local projections. 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 bipartition H (1) 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 projected state γ, δ|Ψ ∈ H (1) W ⊗ H (2) W . A schematic description of this formal scenario is given in Fig. 1a. Note that such entanglement transfer is not always possible. It is therefore pivotal to nd the conditions making such protocol viable. It is worth noting that, when probabilistic operations are allowed (as in the case of projections), even restricting to local operations, the amount of entanglement can be increased [56][57][58]. Such process of e ective entanglement distillation comes, however, at the expense of lowered success probabilities. We focus here on the case where we want to preserve, not enhance, the entanglement in a given state. In this case, it is also possible to achieve entanglement transfer deterministically, when there is a complete basis of projections each element of which achieves entanglement transfer.
We can break down the task at hand into two independent sub-problems, which we will refer to as transferability conditions: on the one hand, transferring the entanglement from (2) , and on the other hand, transferring the entanglement from H (1) W . e achievability of these two tasks will be referred to with TC 1 and TC 2 , respectively. It is worth stressing that, while throughout the paper we will always focus our discussion on TC 1 , all results hold analogously for TC 2 when instead of projecting in the space H (1) To frame the problem more precisely, consider a state |Ψ ∈ H with Schmidt decomposition where k p k = 1, |u k ∈ H (1) and |v k ∈ H (2) . To achieve TC 1 we want a state |γ ∈ H (1) C such that the corresponding projected state |Ψ γ ∈ H (1) W ⊗H (2) contains the same amount of entanglement, in the bipartition H (1) W ⊗H (2) , as that initially in |Ψ . In general, we have where √ q k |ũ k = γ|u k ∈ H (1) and p proj = k p k q k . We distinguish between three di erent scenarios: (1) If the states |ũ k are not orthogonal, then some information about which k the state is in leaks through the coin projection, and some entanglement is thus degraded. is will be shown formally in Appendix A.
(2) If the states |ũ k are orthogonal, but the corresponding projection probabilities q k are uneven, then again the entanglement in |Ψ γ is smaller than that in |Ψ .
(3) If the states |ũ k are orthogonal, and q k = p proj for all k, then projecting onto |γ fully preserves the initial entanglement. QWs allow to eectively engineer a broad range of evolutions. Recently, some of us demonstrated the potential of a QW-based architecture to exibly implement quantum state engineering of a single OAM [36,37], as well as the machinelearning-enhanced classication of hybrid polarization-OAM states of light [38]. A QW dynamics on a bipartite coin-walker system -occurring in the Hilbert space H C ⌦H W with H C(W) the Hilbert space of the coin (walker) -is built by the repeated action of a unitary walk operation W C ⌘ S(C ⌦ I), which describes the sequential action of a controlled-shi operation S, and a coin ipping operation C acting locally on the coin space. e controlled-shi operation changes the state of the walker conditionally to the state of the coin as where {|"i , |#i} are a basis states of H C , {|ki} spans H W , and we introduced the projectors P ⌘ | ih |.
A possible physical embodiment of such QW dynamics uses polarization and OAM of single photons, playing the roles of the coin and the walker degrees of freedom, respectively, with waveplates to implement the coin operations and q-plates [39] to implement the controlled-shi. State engineering protocols leveraging QWs in this seing were previously designed and demonstrated in Refs. [36,38,40]. e state space we are interested in consists of two pairs of QWs, so that the overall system of coins and walkers lives in the four-partite space H ⌘ H (1) W accommodating coin and walker of the i th QW system, respectively (i = 1, 2). Given | i 2 H, we will apply the walk operation on the two QW systems separately. is, in general, entangles the degrees of freedom of a coin with those of the respective walker. In the next sections, we will describe how to use this QW dynamics to transfer entanglement that is possibly initially present in the two-coin subspace to the two-walker one, via local operation on their respective coin degrees of freedom. In an optical setup, this process will transfer the initial entanglement encoded in a polarization state to the two OAM degrees of freedom. e process can be iterated to transfer more entanglement from the polarizations to the OAMs.
W W mal scenario is given in g. 1a [Note (Ale): make self-consistent the notation used in the gure and the text (| in i, | f i and the three unitaries in the gure; in panel (a) it might be worth to put symbols of coin/walker and maybe a curly braket for photon 1 and 2 as well)]. It is worth stressing that such entanglement transfer is not always possible -for example if | i is trivially separable with respect to the partition H (1) ⌦ H (2) : no bilocal operation, including projections, will be able to entangled the walkers. It is therefore pivotal to nd the conditions making such protocol viable. e task at hand can be broken down into two independent subproblems, which we will refer to as transferibility conditions: on the one hand, transferring the entanglement from H (1) C , and then transferring the entanglement from H (1) W is possible only if these two subproblems are solvable.
Let | i 2 H have Schmidt decomposition We want a projection | i 2 H (1) C such that the corresponding post-projection state | i 2 H  W ⌦ H (2) , as that initially contained in | i. In general, we have where p q k |ũ k i = h |u k i and p proj = P k p k q k . We distinguish between three dierent scenarios. If (1) the states |ũ k i are not orthogonal, then some information about which k the state is in leaks through the polarization measurement, and some amount of entanglement is therefore degraded. is will be shown formally in (?). (2) If the states |ũ k i are orthogonal, but the corresponding projection probabilities q k are uneven, then again the entanglement in | i is smaller than that in | i. Finally, if (3) the states |ũ k i are orthogonal, and q k = p proj for all k, then projecting onto | i fully preserves the initial entanglement. Note that (3) is thus a necessary and sucient condition for entanglement transferability without A widely studied type of interaction between a twodimensional and a high-dimensional system is embodied by the discrete-time QW [30][31][32][33][34]. By intertwining the evolution of a two-dimensional coin and a high-dimensional walker, QWs allow to eectively engineer a broad range of evolutions. Recently, some of us demonstrated the potential of a QW-based architecture to exibly implement quantum state engineering of a single OAM [36,37], as well as the machinelearning-enhanced classication of hybrid polarization-OAM states of light [38]. A QW dynamics on a bipartite coin-walker system -occurring in the Hilbert space H C ⌦H W with H C(W) the Hilbert space of the coin (walker) -is built by the repeated action of a unitary walk operation W C ⌘ S(C ⌦ I), which describes the sequential action of a controlled-shi operation S, and a coin ipping operation C acting locally on the coin space. e controlled-shi operation changes the state of the walker conditionally to the state of the coin as where {|"i , |#i} are a basis states of H C , {|ki} spans H W , and we introduced the projectors P ⌘ | ih |.
A possible physical embodiment of such QW dynamics uses polarization and OAM of single photons, playing the roles of the coin and the walker degrees of freedom, respectively, with waveplates to implement the coin operations and q-plates [39] to implement the controlled-shi. State engineering protocols leveraging QWs in this seing were previously designed and demonstrated in Refs. [36,38,40]. e state space we are interested in consists of two pairs of QWs, so that the overall system of coins and walkers lives in the four-partite space W accommodating coin and walker of the i th QW system, respectively (i = 1, 2). Given | i 2 H, we will apply the walk operation on the two QW systems separately. is, in general, entangles the degrees of freedom of a coin with those of the respective walker. In the next sections, we will describe how to use this QW dynamics to transfer entanglement that is possibly initially present in the two-coin subspace to the two-walker one, via local operation on their respective coin degrees of freedom. In an optical setup, this process will transfer the initial entanglement encoded in a polarization state to the two OAM degrees of freedom. e process can be iterated to transfer more entanglement from the polarizations to the OAMs. able states | i 2 H (1) C and | i 2 H (2) C such that, upon perfroming bi-local projections onto them, the coin entanglement possibly present in | i is transferred to the projected state W . A schematic description of this formal scenario is given in g. 1a [Note (Ale): make self-consistent the notation used in the gure and the text (| in i, | f i and the three unitaries in the gure; in panel (a) it might be worth to put symbols of coin/walker and maybe a curly braket for photon 1 and 2 as well)]. It is worth stressing that such entanglement transfer is not always possible -for example if | i is trivially separable with respect to the partition H (1) ⌦ H (2) : no bilocal operation, including projections, will be able to entangled the walkers. It is therefore pivotal to nd the conditions making such protocol viable. e task at hand can be broken down into two independent subproblems, which we will refer to as transferibility conditions: on the one hand, transferring the entanglement from H (1) C , and then transferring the entanglement from H W is possible only if these two subproblems are solvable.
Let | i 2 H have Schmidt decomposition We want a projection | i 2 H (1) C such that the corresponding post-projection state | i 2 H W ⌦ H (2) , as that initially contained in | i. In general, we have where p q k |ũ k i = h |u k i and p proj = P k p k q k . We distinguish between three dierent scenarios. If (1) the states |ũ k i are not orthogonal, then some information about which k the state is in leaks through the polarization measurement, and some amount of entanglement is therefore degraded. is will be shown formally in (?). (2) If the states |ũ k i are orthogonal, but the corresponding projection probabilities q k are uneven, then again the entanglement in | i is smaller than that in | i. Finally, if (3) the states |ũ k i are orthogonal, and q k = p proj for all k, then projecting onto | i fully preserves the initial entanglement. Note that (3) is thus a necessary and sucient condition for entanglement transferability without A widely studied type of interaction between a two-dimensional and a high-dimensional system is embodied by the discrete-time QW [30][31][32][33][34]. By intertwining the evolution of a two-dimensional coin and a high-dimensional walker, QWs allow to eectively engineer a broad range of evolutions. Recently, some of us demonstrated the potential of a QW-based architecture to exibly implement quantum state engineering of a single OAM [36,37], as well as the machinelearning-enhanced classication of hybrid polarization-OAM states of light [38]. A QW dynamics on a bipartite coin-walker system -occurring in the Hilbert space H C ⌦H W with H C(W) the Hilbert space of the coin (walker) -is built by the repeated action of a unitary walk operation W C ⌘ S(C ⌦ I), which describes the sequential action of a controlled-shi operation S, and a coin ipping operation C acting locally on the coin space. e controlled-shi operation changes the state of the walker conditionally to the state of the coin as where {|"i , |#i} are a basis states of H C , {|ki} spans H W , and we introduced the projectors P ⌘ | ih |.
A possible physical embodiment of such QW dynamics uses polarization and OAM of single photons, playing the roles of the coin and the walker degrees of freedom, respectively, with waveplates to implement the coin operations and q-plates [39] to implement the controlled-shi. State engineering protocols leveraging QWs in this seing were previously designed and demonstrated in Refs. [36,38,40]. e state space we are interested in consists of two pairs of QWs, so that the overall system of coins and walkers lives in the four-partite space W accommodating coin and walker of the i th QW system, respectively (i = 1, 2). Given | i 2 H, we will apply the walk operation on the two QW systems separately. is, in general, entangles the degrees of freedom of a coin with those of the respective walker. In the next sections, we will describe how to use this QW dynamics to transfer entanglement that is possibly initially present in the two-coin subspace to the two-walker one, via local operation on their respective coin degrees of freedom. In an optical setup, this process will transfer the initial entanglement encoded in a polarization state to the two OAM degrees of freedom. e process can be iterated to transfer more entanglement from the polarizations to the OAMs. able states | i 2 H C and | i 2 H C suc froming bi-local projections onto them, the c possibly present in | i is transferred to th W . A schematic desc mal scenario is given in g. 1a [Note (Ale): m the notation used in the gure and the text (| three unitaries in the gure; in panel (a) it put symbols of coin/walker and maybe a curl 1 and 2 as well)]. It is worth stressing that s transfer is not always possible -for exampl separable with respect to the partition H (1) operation, including projections, will be abl walkers. It is therefore pivotal to nd the c such protocol viable. e task at hand can be broken down i dent subproblems, which we will refer to conditions: on the one hand, transferring from H We want a projection | i 2 H (1) (2) amount of entanglement, in the bipartitio that initially contained in | i. In general, w p q k |ũ k i = h |u k i and p proj = P guish between three dierent scenarios. If are not orthogonal, then some information state is in leaks through the polarization m some amount of entanglement is therefore d be shown formally in (?). (2) If the state onal, but the corresponding projection pr uneven, then again the entanglement in | that in | i. Finally, if (3) the states |ũ k i ar q k = p proj for all k, then projecting onto | the initial entanglement. Note that (3) is thu sucient condition for entanglement trans A widely studied type of interaction between a two-dimensional and a high-dimensional system is embodied by the discrete-time QW [30][31][32][33][34]. By intertwining the evolution of a two-dimensional coin and a high-dimensional walker, QWs allow to eectively engineer a broad range of evolutions. Recently, some of us demonstrated the potential of a QW-based architecture to exibly implement quantum state engineering of a single OAM [36,37], as well as the machinelearning-enhanced classication of hybrid polarization-OAM states of light [38]. A QW dynamics on a bipartite coin-walker system -occurring in the Hilbert space H C ⌦H W with H C(W) the Hilbert space of the coin (walker) -is built by the repeated action of a unitary walk operation W C ⌘ S(C ⌦ I), which describes the sequential action of a controlled-shi operation S, and a coin ipping operation C acting locally on the coin space. e controlled-shi operation changes the state of the walker conditionally to the state of the coin as where {|"i , |#i} are a basis states of H C , {|ki} spans H W , and we introduced the projectors P ⌘ | ih |.
A possible physical embodiment of such QW dynamics uses polarization and OAM of single photons, playing the roles of the coin and the walker degrees of freedom, respectively, with waveplates to implement the coin operations and q-plates [39] to implement the controlled-shi. State engineering protocols leveraging QWs in this seing were previously designed and demonstrated in Refs. [36,38,40]. e state space we are interested in consists of two pairs of QWs, so that the overall system of coins and walkers lives in the four-partite space H ⌘ H (1) ⌦ H (2) , with W accommodating coin and walker of the i th QW system, respectively (i = 1, 2). Given | i 2 H, we will apply the walk operation on the two QW systems separately. is, in general, entangles the degrees of freedom of a coin with those of the respective walker. In the next sections, we will describe how to use this QW dynamics to transfer entanglement that is possibly initially present in the two-coin subspace to the two-walker one, via local operation on their respective coin degrees of freedom. In an optical setup, this process will transfer the initial entanglement encoded in a polarization state to the two OAM degrees of freedom. e process can be iterated to transfer more entanglement from the polarizations to the OAMs. able states | i 2 H C an froming bi-local projection possibly present in | i is W . mal scenario is given in g the notation used in the g three unitaries in the gu put symbols of coin/walker 1 and 2 as well)]. It is wor transfer is not always poss separable with respect to t operation, including proje walkers. It is therefore pi such protocol viable. e task at hand can b dent subproblems, which conditions: on the one ha from H (1) We want a projection | i post-projection state | amount of entanglement that initially contained in p q k |ũ k i = h |u k guish between three die are not orthogonal, then s state is in leaks through t some amount of entanglem be shown formally in ( onal, but the correspond uneven, then again the en that in | i. Finally, if (3) q k = p proj for all k, then the initial entanglement. N sucient condition for en  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 respect to the 1-vs-2 bipartition -that correlates the inner degrees of freedoms of each particle and realizes the -d dynamics. In the end, local measurements allow to transfer the entanglement stored in the initial state to the reduced state of the d-dimensional sub-systems. We consider explicitly the case of = 2 (qubits) and local operations embodied by the walk operations WC. Indeed, a discrete-time quantum walks framework o ers a very natural encoding of this dynamics: in such embodiment, the coin particle would codify the = 2-dimensional system, with the qudit being provided by the position degrees of freedom of the walker. Assuming initially maximally entangled states of the 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 such degrees of freedom can be retrieved by same operations and transferred back to the two-qubit state. b Conceptual scheme for the transfer from a Bell state in the coin degree of freedom to the two walkers position space a er quantum walks and local coin measurements. c Protocol iteration and entanglement accumulation in the high-dimensional space of the two quantum walkers.
Note that situation (3) is thus a necessary and su cient condition for entanglement transferability without degradation, as if ũ j |ũ k = δ jk and q k = p proj then Eq. (3) is the Schmidt decomposition of |Ψ γ , and therefore the Schmidt coe cients of |Ψ γ are (in the relevant bipartition) the same as those of |Ψ . On the other hand, if (3) is not satis ed, then the projection results in the degradation of some of the entanglement, as shown in Appendix A.
erefore, we achieve transferability if |γ is such that γ|u k / √ p proj are orthonormal vectors. An equivalent -if less explicit -condition for transferability is the requirement σ(tr 2 (P Ψγ )) =σ (tr 2 (P Ψ )) , whereσ(A) ≡ σ(A) \ {0} and σ(A) is the set of eigenvalues of A. is is a necessary and su cient condition for transferability, as Eq. (4) is equivalent to requesting that the Schmidt coe cients of |Ψ γ are the same as those of |Ψ . In Fig. 2 we present a pictorial description of what TC 1 allows to achieve. It is worth noting that, while Eq. (4) is required to fully transfer entanglement, it is still possible to transfer some degree of entanglement if the vectors γ|u k are not fully orthogonal, or the projection probabilities are unequal. is problem can be understood as a more restrictive version of entanglement swapping. Such protocol [59] deals with a four-partite system in the Hilbert space ⊗ j H j (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 entanglement in the state of the A − D compound by performing projective measurements on B − C. is is possible for instance by implementing a Bell measurement over the joint state of B and C. Clearly, the problem is analogous to ours, except that we only allow local operations on B and C. Notably, the use of a Bell measurement is not available in our se ing.

IV. ENTANGLEMENT TRANSFER THROUGH QUANTUM-WALK DYNAMICS
In section III we discussed the general problem of transferring entanglement by means of local projections. Most notably we made no assumption on the inner structure of correlations in H (i) , nor we speci ed the dimensionality of the entanglement in the bipartition H (1) ⊗ H (2) . e framework and results set up so far thus also apply to cases where some preexisting entanglement exists between the walkers' degrees of freedom. We now specialize to the case dim H (i) C = 2, which 4 applies directly to QWs with two-dimensional coins. More precisely, in section IV A we consider states in which H (1) and H (2) are only entangled through their coin spaces (as in g. 3). In section IV B we then apply these results to the output states obtained from QW dynamics.

A. Entanglement transfer via two-dimensional coins
Consider a state |Ψ ∈ H (1) ⊗ H (2) which is entangled only via its coin spaces (or more generally, a state having rank 2), as in Fig 3. e corresponding reduced state reads for a pair of orthonormal states {|u , |v } ∈ H (1) . As discussed in section III, to achieve maximal entanglement transfer we need a projection onto a state |γ satisfying TC 1 , i.e. fullling Eq. (4). is is equivalent to requiring ũ|ṽ = 0 where γ|j = √ p proj j (j = u, v). Explicitly, these amount to the conditions γ|tr W (|u v|)|γ = 0, and γ|tr W (|u u|)|γ = γ|tr W (|v v|)|γ = p proj . We show in Appendix B that it is always possible to nd a state |γ that preserves the orthogonality. To satisfy condition TC 1 , one then only has to verify that the projection probabilities are equal.

B. Entanglement transfer with coined QWs
We now apply the results of the previous section to the speci c quantum states resulting from coined QWs. As in section IV A, we rst assume that the overall state is entangled with respect to the bipartition H (1) ⊗ H (2) only via its coin spaces (see Fig. 3). We thus take the initial full state of the form for some coe cients p 1 , p 2 ≥ 0 with p 1 + p 2 = 1. Focusing on H (1) , we thus see that the initial states upon which the QW operates are |↑, 1 and |↓, 1 .
Pictorial representation of the rst transferability procedure. Given a state which is entangled with respect to the bipartition H (1) ⊗ H (2) , we apply a local projection |γ which preserves the entanglement between the two spaces. Condition (4) determines when such a projection exists.
A single QW step with coin operation C amounts to the evolution where c ij are the entries of the unitary matrix representing C. By projecting onto |γ ≡ γ ↑ |↑ + γ ↓ |↓ (γ ↑,↓ ∈ C) and imposing Ψ ↑,1 |Ψ ↓,1 = 0, we get which is satis ed for |γ = (|↑ + e iφ |↓ )/ √ 2 for any φ ∈ R. e corresponding projection probabilities are both equal to 1/2, as follows from We conclude that TC 1 is always achievable for this class of states. Remarkably, the freedom in the choice of the phase φ means that projections onto |± = (|↑ ± |↓ )/ √ 2 (as well as any other orthonormal basis of balanced states) are suitable to achieve entanglement transfer. is results in an overall transfer success probability of 1: measuring in the |± basis, both of the possible outcomes achieve TC 1 , albeit with di erent post-projection states.
Consider now the state a er multiple QW steps. e nal reduced state on H (1) is a mixture of |Ψ ↑ and |Ψ ↓ , where with θ s and |Ψ s,p depending on the number of steps and choice of coin operators, and s, p ∈ {↑, ↓}. To assess the achievability of TC 1 we consider, as in section IV A, the matrix with O sp ≡ Ψ ↓s |Ψ ↑p . Such M is not in general Hermitian, nor normal. Consequently, while it is always possible to nd a state |γ upon which to perform a projection, the corresponding projection probabilities are not in general equal, as shown in Fig. 4. It is worth stressing that this does not imply FIG. 3. Like Fig. 2, but for states in which the entanglement is only due to pre-shared entanglement between the coins. ese are the types of states at the rst entanglement accumulation step. 5 the impossibility of accumulating entanglement using these types of QWs. Rather, this result tells us that this is only possible via entanglement distillation, and thus there cannot be a deterministic protocol achieving such entanglement transfer. In other words, Fig. 4 shows that, in such cases, there is no projection preserving entanglement in the residual space W ⊗ H (2) . Nonetheless, there might still be a |δ ∈ H (2) C such that the second projection recovers the original amount of entanglement, but this can only be done probabilistically, as shown in Refs. [56,58]. To further highlight this point, we provide in Fig. 5 numerical results regarding the possibility of probabilistic entanglement transfer when both projections are considered. In these cases, probabilistic entanglement transfer is possible despite TC 1 and TC 2 are not satis ed. ere are nonetheless QW dynamics in which TC 1 is achievable. For example, consider a QW in which the coin is always taken to be identity: C = I at all steps. en, a er n steps, the evolution amounts to where |k denotes the k-th walker position. e matrix M is thus in this case easily seen to be M = 0, implying that the orthogonality requirement is always satis ed, making TC 1 achievable as long as the projection probabilities are equal. is constraint is satis ed by any balanced projection of the form |γ = (|↑ + e iφ |↓ )/ √ 2, φ ∈ R.

V. ENTANGLEMENT ACCUMULATION
Here we investigate whether the entanglement transfer procedure can be applied iteratively, accumulating more and more entanglement into the state of the walkers' degrees of freedom. For this purpose, a er each successful entanglement transfer stage, which produces a state of the form we apply an operation restoring the entanglement between the coins, thus producing a state of the form |Ψ W ⊗ |Φ C , with |Φ ∈ H (1) C some entangled state -usually a maximally entangled one. e QW evolution is then used to correlate each coin and walker degree of freedom locally, in order to make transfering the entanglement via local projections possible.
Suppose one round of entanglement transfer was executed successfully. We therefore have entanglement in the bipartition H (1) W , while the coin spaces are separated. Can we perform another round of QW evolutions to transfer even more entanglement to the walkers?
Let us consider, as an example, the case where |Ψ W has entanglement dimension 2, and the full state has the form for some walker states |ψ i with ψ i |ψ j = δ ij . Restoring the entanglement between the coins we get |Ψ = (|↑↑ + |↓↓ ) C ⊗ (|ψ 1 ψ 2 + |ψ 2 ψ 1 ) W /2. For the random QW, a random coin is used at each step. In each case, we consider the input states |↑, 1 and |↓, 1 , and verify the satis ability of TC1 on the corresponding outputs. We rst plot the squared overlap O ≡ Ψ ↑ |γ γ|Ψ ↓ 2 for all possible |γ , where |Ψ ↑ , |Ψ ↓ denote the output states. We nd that there are two 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. To achieve 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 region bounded by dashed black lines. More precisely, the magenta region outlines the set of (θ, φ) such that the entropy of the projections probabilities, S((p ↑ , p ↓ )), is larger than 0.693 (remembering that − ln 2 0.6931). It is worth noting that, while it is not in general true that p ↑ + p ↓ = 1 for an arbitrary unitary evolution, this is always the case for QWs, which allows us to quantify how close p ↑ and p ↓ via the corresponding entropy. As clear from the gure, in these two cases, TC1 cannot be achieved for any |γ , as the two necessary conditions cannot be simultaneously satis ed.
Let us, as in section III, focus on the transferability in H (1) . e reduced state ρ (1) ≡ tr 2 P Ψ has the form A QW evolution W S then gives P[W S |↑, ψ 1 ] + P[W S |↑, ψ 2 ] + P[W S |↓, ψ 1 ] + P[W S |↓, ψ 2 ] = P Ψ1 + P Ψ2 + P Ψ3 + P Ψ4 , where Ψ i |Ψ j = δ ij , and thus W S ρ (1) W † S has rank 4. Achieving entanglement transfer now entails nding |γ ∈ H (1) Each successive entanglement transfer iteration involves a doubling of the number of orthogonal states to preserve, as follows from observing that if A has rank r and B has rank r , then A ⊗ B has rank rr . Consider now a QW in which each coin operation is the identity: C = I. We will show that, with this particular type of dynamics, we can accumulate arbitrary amounts of entanglement into the walkers' degrees of freedom, using the coins as mediators. e unitary evolution corresponding to n steps with C = I is W S,n = S n with S the controlled-shi operation. e action on the basis states is then where E + ≡ k |k + 1 k| is the operation shi ing the walker's position, and E n + is thus the operator moving the walker n positions forward. Consider an initial state with |Ψ W an entangled state of the walkers in which the di 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 on H (1) has the form for some set of orthonormal states {|ψ k } k ⊂ H W . Evolving through S +1 , we get with ψ j ψ k = δ jk and ψ j ψ k = 0. en, any balanced projection |γ = 1 √ 2 (|↑ + e iφ |↓ ) achieves TC 1 , which means that the entanglement can be transferred deterministically from coins to walkers.
In light of these ndings, we can now propose the following explicit protocol, which allows to accumulate deterministically entanglement into the walkers degrees of freedom using the coins as mediators. Starting from the state  6. (a) Trends of N for QWs with C = I (purple) and Hadamard QWs (green) in the entanglement accumulation protocol. e numbers near the markers specify the number of QW steps needed to store the entanglement in the walkers subspaces. e rst case corresponds to the deterministic optimal transfer described in the main text, in which one ebit is transferred at each iteration. In the Hadamard 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 the entanglement between the coins, we then re-apply the QW evolution, now for two steps, and project again in the basis |± , resulting in an output state of the form is procedure can be iterated to accumulate more and more entanglement in H W . At the n-th iteration, we evolve both systems through 2 n QW steps with C = I, that is, through the unitary S 2 n ⊗ S 2 n , and then project onto the |± basis, resulting in a maximally entangled state of the form with (−) σ k ∈ {1, −1} for all k. In Fig.6 we report the trend of the deterministic transfer and accumulation described above. Notice that this goal cannot always be achieved, as for example in the case of the Hadamard QW reported in the gure. Here it is not possible to transfer one-ebit of entanglement per iteration, not even probabilistically.
VI. ENTANGLEMENT RETRIEVAL e arguments of section III do not make assumptions on the dimensions of H W . is means that they can be used not only to study the transfer of entanglement from coins to positions, but also the other way around. For example, if the initial reduced state on H (1) is [P ↑ ⊗ (P 1 + P 2 )]/2, with |1 , |2 a pair of orthonormal walker's states, then applying a Hadamard operation to the coin, and two steps of QW evolution with C = I, we obtain the state en, measuring in the Hadamard four-dimensional basisi.e., the basis formed by the columns of the 4 × 4 Hadamard matrix -we achieve TC 1 .
Implementing the entanglement transfer protocol involves as starting point two polarization-entangled photons, that can be generated via photon sources based on parametric down conversion. en, each photon of the pair evolves through a QW evolution in the polarization and OAM degrees of freedom, followed by a projective measurement on the polarizations.
e coin operators are realized on the polarization through suitable sets of waveplates. e shi operator, which involves an interaction between OAM and polarization, is naturally implemented by the inhomogeneous and birefringent devices known as q-plates [52,53]. Projective measurements on the coins are realized using waveplates and a polarizing beam spli er.
For the entanglement accumulation protocol, one also needs a way to "reload" the entanglement into the photons' polarization without a ecting their OAMs. As discussed in section V, the rst entanglement transfer procedure results in one of the states . where |1 and |2 label OAM states, while |+ are diagonal polarization states. It is straightforward to show that the action of a polarizing beam-spli er combined with two halfwaveplates can restore, with probability 1/2, the entangled state in polarization needed to achieve accumulation. Expressing the state of Eq. (26) in terms of creation operators a † and b † of the two photons, we have: 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 rotation made by two half-waveplates of angles θ a and θ b respectively. e creation operators a er the overall transformation become Substituting such expression in Eq. (27) and choosing the orientation of the two half-waveplates θ a = θ b = π/4 we obtain that the output state is composed by two terms. e rst term corresponds to the two photons exiting from di erent output ports of the polarizing beam-spli er, while the second term corresponds to the case where the photons exit from the same port. e rst part of this state embodies the resource needed for the protocol accumulation, and has the following form: We can discard the second term where two photon exit from the same port by post-selecting two-fold coincidences between single-photon detectors at the end of the second iteration. It is worth noting that the probabilistic generation of the second maximally entangled state is due to the choice of encoding qubits in photons. However, we remark that we could also consider the state produced by the projection |−− a er the rst operation. Indeed, this projection produces states with the same symmetry properties. In this way it is possible to double the probability of generating states with more than one-ebit of entanglement. As a nal remark, we note that the whole protocol, being essentially based upon the general QW dynamics, can be implemented in various experimental platforms [70][71][72][73][74].

VIII. CONCLUSIONS
We have addressed the generation of high-dimensional entangled states through a protocol of entanglement transfer from a low-dimensional resources. We have identi ed general transfer conditions that, if met, guarantee the successful pouring of any entanglement initially contained in the state of the resource to the high-dimensional receiver. is has then allowed us to draw a speci c analysis aimed at the dynamics entailed by a QW, where low-dimensional resources and highdimensional receivers are naturally embodied by coin and walker degrees of freedom respectively. While characterizing the performance of the entanglement transfer scheme, we have been able to design schemes for entanglement accumulation and retrieval, thus drawing a complete picture for the manipulation of entanglement through a hetero-dimensional interface of great experimental potential. Indeed, the QWbased protocols addressed and studies in this paper are fully amenable to an implementation making use of polarization and OAM encoding. e scenario set by our schemes sets a promising framework for the use of low-dimensional entanglement as a resource to achieve otherwise complex entangled structures and states that can be experimentally synthesised and exploited. Let us show that if ũ j |ũ k = δ jk then the Schmidt coecients must change upon projection. Indeed, in this case, |Ψ γ has the form |Ψ γ = k √p k |ũ k |v k where v j |v k = δ jk and kp k = 1. Denoting with Ψ γ the matrix whose vectorization is |Ψ γ , this Schmidt decomposition amounts to the singular value decomposition Ψ γ = U √ DV † , with D = diag(p 1 , ...,p n ), V the unitary matrix whose columns are |v k , and U the (non-unitary) matrix with columns |ũ k . en where Pũ k = |ũ k ũ k | are in general non-orthogonal rank-1 projectors. Let us then prove that if a matrix is a convex combination of rank-1 projections, then it always majorizes the vector of coe cients of the convex combination. In our case, this translates to kp k Pũ k p. Let P k be rank-1 projections, p k ≥ 0 coe cients such that n k=1 p k = 1, and A ≡ n k=1 p k P k . We want to prove that A p, where p = (p k ) n k=1 is the vector of coe cients, and the majorization relation is de ned on Hermitian matrices via the corresponding vector of eigenvalues, that is, A p ⇐⇒ σ(A) p where σ(A) is the vector of eigenvalues of A. If A has dimension larger than n, we de ne λ(A) as the vector of the n largest eigenvalues, in order to make the majorization relation well-de ned. Without loss of generality, let us assume that the p k are in decreasing order: p 1 ≥ p 2 ≥ ... ≥ p n . De ne the partial sums A ≡ k=1 p k P k , so that A = A n . Observe that A ≥ A r whenever ≥ r. Because rank(P k ) = 1 for all k, we must also have rank(A ) ≤ . Denoting with λ ↓ j (A) the j-th largest eigenvalue of A, this implies that Using A = A n ≥ A for all 1 ≤ < n, we thus conclude that that is, λ(A) p, which is the de nition of A p.
Assuming |ũ k are orthogonal, then the Schmidt coecients of |Ψ γ are √p k = p −1/2 proj √ p k q k . We prove in this section that, for any state of the form |Ψ = √ p 1 |u, u + √ p 2 |v, v , with u|v = u |v = 0, there is some |γ such that the post-projected states are orthogonal, i.e. such that ũ|ṽ = 0 where √ p u |ũ = γ|u and √ p v |ṽ = γ|v .
De ne M ≡ tr W (|u v|) ∈ Lin(H (1) C ). Note that this is a 2 × 2 traceless matrix, as follows from u|v = 0. Our objective is then to nd |γ such that γ|M |γ = 0. For the purpose, we consider di erent scenarios: 1. If M is normal, then for some λ ∈ C and v i |v j = δ ij . en, are all suitable projections such that γ φ |M |γ φ = 0. Note that this also implies that we can nd orthogonal states that both correspond to valid projections.
If D = d 1 P 1 +d 2 P 2 and V † U = |1 w 1 |+|2 w 2 |, then where det(M ) = |det(M )|e iφ and we observed that ere are then two possibilities: either d 1 = d 2 , which implies M is normal, and this case was covered above, or d 1 = d 2 , which implies by the uniqueness of the singular value decomposition that |w 1 = |2 and |w 2 = |1 up to phases. Consequently, we would have where u i |u j = v i |v j = δ ij and u 1 |v 2 = u 2 |v 1 = 0. We can then use |γ = |v i as suitable projections, as v i |M |v i = 0.