Delocalized information in quantum networks

Given a quantum state containing some information, one aims to store and protect that information against noise or external interference. In our scenario, this is accomplished by encoding the state into a larger network consisting of multiple network nodes or parties, such that the information is delocalized or diluted over the network. This delocalization process is called uploading. Two different uploading procedures should be distinguished.

First, we consider the case where an arbitrary state is directly encoded into a larger system of n particles. This task involves global operations (at the logical level) and can be accomplished by constructing an entangled state between a logical block and an auxiliary system aux (see figure 4). Consider the state:

$$\begin{eqnarray}&&\left|\psi \right\rangle =\displaystyle \frac{1}{\sqrt{2}}\left({\left|0\right\rangle }_{{aux}}\left|{0}_{L}\right\rangle +{\left|1\right\rangle }_{{aux}}\left|{1}_{L}\right\rangle \right),\end{eqnarray} \tag{ 5 }$$

where the logical qubits are defined by some orthogonal codewords $\left|{0}_{L}\right\rangle ,\left|{1}_{L}\right\rangle$. This means that the system aux is maximally entangled with the whole logical block of qubits. Consider now an arbitrary state $\left|\phi \right\rangle =\alpha \left|0\right\rangle +\beta \left|1\right\rangle$ that we want to upload into the logical level. Hence, by performing a Bell measurement between the aux system and $\left|\phi \right\rangle$, the uploading into the logical level is deterministically achieved, i.e. the global state of the remaining particles is now $\alpha \left|{0}_{L}\right\rangle +\beta \left|{1}_{L}\right\rangle$ (up to unitary Pauli correction operations at the logical level). We remark that the logical Pauli correction operation can in general not be implemented locally, however are irrelevant in the sense that they actually do not need to be implemented as the logical subspace is unchanged. It suffices to relabel the basis states, and adapt the information processing and subsequent downloading processes accordingly.

Zoom In Zoom Out Reset image size

On the other hand, the uploading process can also be done from a single node within the network without using an entangled state. In this case, only the network state (e.g. a computational wire) is used, and a single party (or possibly also several parties) aim to upload quantum information that is given in the form of an auxiliary quantum state to the logical level, thereby delocalizing the quantum information.

The reverse process to the uploading of information is the downloading, i.e. once information is spread out among the nodes of the network, we require the possibility of localizing this information in one particular physical system (again, by LOCC). In general, this is achieved by suitable measurements on the rest of the particles. However, crucial differences exist when localizing information with standard encoded states (Dicke states) or in the correlation space framework. Encoding properties defines the characteristics of the localization process. In particular, two features are specially relevant. The ideal situation is the one where the localization is performed in a deterministic way, and the information is downloaded in an a-priori chosen place. However, due to the properties of each encoding state, one can be restricted to the case where only probabilistic, or open-destination downloading process is possible. Probabilistic downloading means that information can only be localized probabilistically in a heralded way. This may be unsatisfactory in many cases. Open-destination downloading in turn is still deterministic, however the precise location can not be determined a priori. This corresponds to a probabilistic download attempt to a given site, which however can be repeated if not successful. In this way quantum information can be downloaded quasi-determistically to individual sites within a given region of small size.

We remark that for standard encodings, there are examples where the probability to localize information from two logical blocks is arbitrarily small, even for the open-destination scenario [33]. For correlation space encodings, this problem does not occure as we show below. The main difference is that for correlation space encodings, only a fraction of the qubits need to be measured, and unsuccessful downloaded can be repeated. This leads to a success probability approaching one to download information to an unspecified site within a region of fixed size (see section 3.4).

The effect of local errors can affect the global state of the resource, jeopardizing the stored information. In particular, we are interested in errors occurring in single particles when the information is delocalized, and how robust is the encoded state under that noise. The effect of local noise should have limited influence on the global state, in order to consider an encoded state robust for storage. We find effective noise maps for this process. Besides, we study the stability of the resource states under loss of particles.

In order to analyze the effective noise when errors affect particles at the physical level, one can study the following scenario. Consider the maximally entangled state

$$\begin{eqnarray}&&\left|{{\rm{\Phi }}}^{+}\right\rangle =\displaystyle \frac{1}{\sqrt{2}}\left({\left|0\right\rangle }_{{aux}}\left|{0}_{L}\right\rangle +{\left|1\right\rangle }_{{aux}}\left|{1}_{L}\right\rangle \right),\end{eqnarray} \tag{ 6 }$$

where an auxiliary single particle is defined on ${{ \mathcal H }}_{{aux}}={{\mathbb{C}}}^{2}$, entangled with a logical encoded block which lives on ${{ \mathcal H }}_{L}={{\mathbb{C}}}^{{2}^{\otimes n}}$. The state resulting from the introduction of local noise on the physical particles of a logical block is defined as the Choi–Jamiolkowski state [52, 53] i.e.

$$\begin{eqnarray}&&{\rm{\Gamma }}={{\rm{i}}{d}}_{{{ \mathcal H }}_{{aux}}}\otimes {\xi }^{\otimes j}\left(\left|{{\rm{\Phi }}}^{+}\right\rangle \left\langle {{\rm{\Phi }}}^{+}\right|\right),\end{eqnarray} \tag{ 7 }$$

where the noisy channel ${\xi }^{\otimes j}$ affects j particles of the logical qubit. If one then aims to localize (by local operations) the information of the block into one physical system q, it is relevant to study how close this final state is from a perfect Bell pair $\left|{\rm{\Phi }}\right\rangle =\tfrac{1}{\sqrt{2}}\left({\left|0\right\rangle }_{{aux}}{\left|0\right\rangle }_{q}+{\left|1\right\rangle }_{{aux}}{\left|1\right\rangle }_{q}\right),$ which would be the resulting state of a perfect localization. Notice that Γ contains the full information about the effective noise map, which is in principle accessible. We remark that we use the effect of the performed processes on part of a maximally entangled state only as a tool to analyze the resulting effective noise maps or the resulting process fidelity.

Similarly, the stability under losses can be analyzed in the following way. We consider an initial state of the form $\left|{{\rm{\Phi }}}^{+}\right\rangle \left\langle {{\rm{\Phi }}}^{+}\right|$ (6) shared by the auxiliary particle and the logical qubit, and we assume that some particles in the logical block are lost. The effect of a lost particle is described by tracing out that particle, i.e. ${{\rm{\Gamma }}}_{f}={{Tr}}_{i}\left(\left|{{\rm{\Phi }}}^{+}\right\rangle \left\langle {{\rm{\Phi }}}^{+}\right|\right)$, where i are the lost particles. Tracing out is equivalent to performing an average over all the possibles outcomes of a measurement of the system i in any basis.

Finally, we compute in both cases (errors and losses) the fidelity of the final state with respect to the initial one (6), denoted as Choi–Jamiolkowski fidelity (CJ fidelity):

$$\begin{eqnarray}&&{F}_{{\rm{CJ}}}=\left\langle {{\rm{\Phi }}}^{+}\right|{{\rm{\Gamma }}}_{f}\left|{{\rm{\Phi }}}^{+}\right\rangle .\end{eqnarray} \tag{ 8 }$$

This parameter is a suitable measure to analyze how stable is a resource encoded state against loss of particles [53]. Note that in order to compute the CJ fidelity, we replace the state of each lost particle by an identity operator, i.e. consider a completely depolarizing map acting on each of these qubits.

An essential property of the encoded states is the local entanglement that each physical particle shares with respect to the rest. We study this characteristic, which determines the behavior, strength and suitability of the resource states. Consider a maximally entangled state of logical qubits (between two blocks of particles):

$$\begin{eqnarray}&&\left|{{\rm{\Phi }}}_{L}^{+}\right\rangle =\displaystyle \frac{1}{\sqrt{2}}\left(\left|{0}_{L}\right\rangle \left|{0}_{L}\right\rangle +\left|{1}_{L}\right\rangle \left|{1}_{L}\right\rangle \right).\end{eqnarray} \tag{ 9 }$$

We study the entropy of entanglement of a single qubit i within a block by computing the Von Neumann entropy of its reduced density operator, i.e.

$$\begin{eqnarray}&&S({\rho }_{i})=-{Tr}\left({\rho }_{i}\mathrm{log}{\rho }_{i}\right).\end{eqnarray} \tag{ 10 }$$

Even though the entanglement between blocks of logical qubits is maximal, in general this is not the case for the local entanglement of a single party. In fact, encoded states with low local entanglement are of particular importance, since this leads to advantages in order to construct and maintain the states.

Other encoding codewords can be explored based on the ones studied here. We consider some additional examples in appendix A.

Once the information is stored in the resources in a delocalized way, it is important to analyze the security of the storage, i.e. how much influence a party has about the stored state. At this point, we can make a distinction between trusted and untrusted parties. The first case analyzes how errors or losses of the physical qubits can affect the global state, and has been studied in previous sections. We focus here in the case the parties are not trusted and we hence study how much information a single party (or a group of them) can access from a encoded state, i.e. how secure is the storage against attack of parties that aim to access the information even though not authorized to do so. In order to study this, we perform a similar analysis as was done in [54] in the context of quantum secret sharing. As in [54] we use the mutual information as figure of merit. We only consider the situation where the initial state is the desired one, i.e. pure and unaltered. We remark that this does not correspond to full security analysis, but rather illustrates how much information an untrusted party can access in the network given that the resource states were distributed as they should. Given the scenario of section 3.1.1, where the q system can upload or encode any state, we analyze the mutual information between any party of the resource and the q system. The mutual information determines how much information a particle t (or a set of them) of the resource can obtain about the system q, and therefore, about the global encoded state once the party q applies the Bell measurement (to achieve uploading), local in its system.

The quantum mutual information between two systems is defined as

$$\begin{eqnarray}&&I\left(q;t\right)=S\left({\rho }_{q}\right)+S\left({\rho }_{t}\right)-S\left({\rho }_{q,t}\right),\end{eqnarray} \tag{ 11 }$$

where S represents the Von Neumann entropy (10) of the reduced density operator of system q, system t, and both systems, respectively. We will use the mutual information to assess information accessible by individual parties, and to determine the corresponding security features.

A general uploading process can be accomplished from a state of the form (5) by proceeding as explained in section 3.1.1.

In order to analyze how information can be localized from the logical level into one physical constituent, we consider a downloading process (with LOCC) from two entangled blocks (9) where information is eventually localized into one of the physical qubits of each block (sites k and $k^{\prime}$), such that we end up with a Bell pair between the two final qubits (see figure 5). Essentially, this procedure is accomplished by performing suitable measurements in all except one particle in each block. This process can only succeed deterministically for an a-priori fixed site if the local entanglement of the logical qubit is maximal.

Zoom In Zoom Out Reset image size

Consider an initial logical Bell state of the form (9) and two fixed sites k and $k^{\prime}$. The initial entanglement at the logical level is maximal, but the local entropy of entanglement for a single site k is ${E}_{{\rm{inital}}}\left({\rho }_{k}\right)\leqslant 1.$ If downloading succeeds deterministically, information is localized in the sites k and $k^{\prime}$, and the final state is a perfect Bell pair. Therefore, the entropy of entanglement of particle k at the end is ${E}_{{\rm{final}}}\left({\rho }_{k}\right)=1.$ Since the process only involves LOCC and single-site measurements, one can directly conclude that localization only succeeds deterministically when the local entanglement of the particle in the initial state is maximal.

If the local entanglement is not maximal, one has to face with a probabilistic localization (in the sense that the final place where the information is localized cannot fixed a priori), or a deterministic but imperfect (in terms of the fidelity of the final state) downloading.

In case the local entropy of entanglement is maximal, consider e.g. $\left|{0}_{L}\right\rangle ={\left|0\right\rangle }^{\otimes n},\left|{1}_{L}\right\rangle ={\left|1\right\rangle }^{\otimes n}$, localization is simply accomplished by performing X measurements on all except one particle of each logical block. This is in fact also the case for any graph state encoding, where Z measurements are enough to localize information.

In order to compute the CJ fidelity (8) we make use of some properties of Dicke states. Any Dicke state (1) of n particles and k excitations can be j-times decomposed as [38]:

$$\begin{eqnarray}&&\left|n,k\right\rangle ={\left[\displaystyle \frac{n!}{\left(\genfrac{}{}{0em}{}{n}{k}\right)\left(\genfrac{}{}{0em}{}{n}{j}\right)}\right]}^{\tfrac{1}{2}}\sum _{q={q}^{{\prime} }}^{{q}^{{\prime\prime} }}\displaystyle \frac{\left|j,q\right\rangle \left|(n-j),(k-q)\right\rangle }{\sqrt{q!(j-q)!(k-q)!(n-k-j+q)!}},\end{eqnarray} \tag{ 13 }$$

where $q^{\prime} =\max \left(0,(j+k-n\right)$ and $q^{\prime\prime} =\min \left(j,k\right)$, for $q^{\prime\prime} \gt q^{\prime} .$ On the other hand, given a state ${\rho }_{n,k}=\left|n,k\right\rangle \left\langle n,k\right|$, tracing out m number of particles leads to:

$$\begin{eqnarray}&&{{Tr}}_{m}\left({\rho }_{n,k}\right)=\sum _{i=0}^{k}\displaystyle \frac{\left(\genfrac{}{}{0em}{}{n-{n}_{f}}{i}\right)\left(\genfrac{}{}{0em}{}{{n}_{f}}{k-i}\right)}{\left(\genfrac{}{}{0em}{}{n}{k}\right)}{\rho }_{{n}_{f},k-i},\end{eqnarray} \tag{ 14 }$$

where n_f is the final number of particles, i.e. $m=n-{n}_{f}$. In particular, for any element $\left|n,{k}_{1}\right\rangle \left\langle n,{k}_{2}\right|$ we find:

$$\begin{eqnarray}&&{{Tr}}_{m}\left(\left|n,{k}_{1}\right\rangle \left\langle n,{k}_{2}\right|\right)=\sum _{i=0}^{\min ({k}_{1},{k}_{2})}\displaystyle \frac{\left(\genfrac{}{}{0em}{}{n-{n}_{f}}{i}\right)\sqrt{\left(\genfrac{}{}{0em}{}{{n}_{f}}{{k}_{1}-i}\right)\left(\genfrac{}{}{0em}{}{{n}_{f}}{{k}_{2}-i}\right)}}{\sqrt{\left(\genfrac{}{}{0em}{}{n}{{k}_{1}}\right)\left(\genfrac{}{}{0em}{}{n}{{k}_{2}}\right)}}\left|{n}_{f},\left({k}_{1}-i\right)\right\rangle \left\langle {n}_{f},\left({k}_{2}-i\right)\right|.\end{eqnarray} \tag{ 15 }$$

Computing the CJ fidelity (8) is hence straightforward for any pair of codewords $\left|{0}_{L}\right\rangle =\left|n,{k}_{1}\right\rangle ,\left|{1}_{L}\right\rangle =\left|n,{k}_{2}\right\rangle$ and for any number of losses:

$$\begin{eqnarray}&&\begin{array}{l}{F}_{{\rm{CJ}}}=\sum _{i,j=1}^{2}\left({ \mathcal C }\sum _{q=q^{\prime} }^{q^{\prime\prime} }\left(\displaystyle \genfrac{}{}{0em}{}{n-{n}_{f}}{q}\right)\displaystyle \frac{1}{q!(n-{n}_{f}-q)!}\right.\\ \quad \left.\times \,{\left[\displaystyle \frac{\left(\genfrac{}{}{0em}{}{{n}_{f}}{{k}_{i}-q}\right)\left(\genfrac{}{}{0em}{}{{n}_{f}}{{k}_{j}-q}\right)}{\left(\genfrac{}{}{0em}{}{n}{{k}_{i}}\right)\left(\genfrac{}{}{0em}{}{n}{{k}_{j}}\right)({k}_{i}-q)!({k}_{j}-q)!({n}_{f}-{k}_{i}+q)!({n}_{f}-{k}_{j}+q)!}\right]}^{\tfrac{1}{2}}\right),\end{array}\end{eqnarray} \tag{ 16 }$$

where

$$\begin{eqnarray}&&{ \mathcal C }=\displaystyle \frac{n!}{4\left(\genfrac{}{}{0em}{}{n}{n-{n}_{f}}\right)\sqrt{\left(\genfrac{}{}{0em}{}{n}{{k}_{i}}\right)\left(\genfrac{}{}{0em}{}{n}{{k}_{j}}\right)}},\end{eqnarray} \tag{ 17 }$$

and with $q^{\prime} =\max \left[0,(\max \left({k}_{1},{k}_{2}\right)-{n}_{f})\right]$ and $q^{\prime\prime} =\min \left[\left(n-{n}_{f}\right),\min \left({k}_{1},{k}_{2}\right)\right]$. In figure 6 one can see an illustrative analysis of the CJ fidelity for different codewords configurations. Two main conclusions can be extracted. On the one hand, one obtains a higher CJ fidelity when the codewords are close to each other in terms of number of excitations. Moreover, if both Dicke states are close to the limit of zero or maximal number of excitations, the CJ fidelity is again increased. Therefore, it turns out that the best choice of Dicke state codewords, if one is only interested on the robustness under losses, is $\left|{0}_{L}\right\rangle =\left|n,0\right\rangle ,\left|{1}_{L}\right\rangle =\left|n,1\right\rangle$, or the symmetric choice: $\left|{0}_{L}\right\rangle =\left|n,n\right\rangle ,\left|{1}_{L}\right\rangle =\left|n,n-1\right\rangle$.

Zoom In Zoom Out Reset image size

However, we find that a conflict of interest exists if one requires other properties more than just robustness under losses. For instance, as shown above, the success probability of localizing information increases with the local entanglement.

We analyze now the entanglement properties of the Dicke-type encoding. Consider a logical maximally entangled state of the form (9). We study the local entanglement that a physical particle shares with the rest of the system by computing the Von Neumann entropy (10) of its reduced density matrix. For the encoding codewords $\left|{0}_{L}\right\rangle =\left|n,{k}_{1}\right\rangle ,\left|{1}_{L}\right\rangle =\left|n,{k}_{2}\right\rangle$, and from the properties introduced before, it is easy to see that the entropy of entanglement of a single particle is:

$$\begin{eqnarray}&&S\left({\rho }_{A}\right)=-\left(\sum _{i=1}^{2}\displaystyle \frac{n-{k}_{i}}{2n}\right)\mathrm{log}\left(\sum _{i=1}^{2}\displaystyle \frac{n-{k}_{i}}{2n}\right)-\left(\sum _{j=1}^{2}\displaystyle \frac{{k}_{j}}{2n}\right)\mathrm{log}\left(\sum _{i=1}^{2}\displaystyle \frac{{k}_{j}}{2n}\right).\end{eqnarray} \tag{ 18 }$$

The entropy of entanglement is minimal (see figure 7) for encoding codewords which are, again, close to each other in terms of excitations and close to the boundaries with small or large (due symmetry) number of excitations. This result is consistent with the analysis of the robustness against losses, i.e. the less local entanglement, the more delocalized the information is and the more stability against losses.

Zoom In Zoom Out Reset image size

An explanatory example is the extreme case with codewords $\left|{0}_{L}\right\rangle ={\left|0\right\rangle }^{\otimes n},\left|{1}_{L}\right\rangle ={\left|1\right\rangle }^{\otimes n}$, where the local entanglement is maximal and the stability is minimal, i.e. when a single particle is lost, the information and entanglement is completely destroyed.

Following section 3.1.5, we study the amount of information a single party can access once the quantum information has been delocalized over the network. The results are shown in figure 8. One finds again a relation with the local entanglement. We see that the entanglement is a key property that defines how the information is spread around the system, i.e. the less local entanglement, the more distributed. Hence, a single particle within a encoding configuration with low local entanglement has a low information about the global state of the logical system.

Zoom In Zoom Out Reset image size

The general uploading procedure of section 3.1.1 is applicable for our wire configuration. In this case, a wire plays the role of the logical block, with codewords $\left|{0}_{L}\right\rangle ={\rm{\Phi }}\left(\left|0\right\rangle \right),\left|{1}_{L}\right\rangle ={\rm{\Phi }}\left(\left|1\right\rangle \right)$. Note that orthogonality of correlation space states does not always imply orthogonality of the corresponding wire states [51]. However, orthogonality is fulfilled for the period wires (4) we make use in this paper.

An alternative interpretation of uploading is possible [51], where any physical state is uploaded into the correlation space. We review this process in the appendix B.

In contrast to the Dicke-type encoding, it is known how to localize information of the correlation space into some physical site of a wire [39]. This process does not imply the measurement of all the parties of the wire and we show that it can be made quasi deterministic. Based on the protocol developed in [39], we analyze how information is delocalized within the correlation space, and which properties influence this spreading of information.

We briefly review the downloading procedure of [39], and point out some additional features. Given a quantum wire of the form (4), one can always find a basis such that

$$\begin{eqnarray}&&A[{m}_{0}]={r}_{0}\left|{\varphi }_{0}\right\rangle \left\langle 0\right|,A[{m}_{1}]={r}_{1}\left|{\varphi }_{0}\right\rangle \left\langle 0\right|+\left|{\varphi }_{1}\right\rangle \left\langle 1\right|,\end{eqnarray} \tag{ 19 }$$

with ${r}_{0}\gt 0$, ${r}_{1}\geqslant 0$, ${r}_{0}^{2}+{r}_{1}^{2}=1$ and $\left\langle {\varphi }_{i}| {\varphi }_{j}\right\rangle ={\delta }_{{ij}}$. In particular, if ${r}_{1}=0$ the operators are of the form:

$$\begin{eqnarray}&&A[{m}_{0}]=\left|{\varphi }_{0}\right\rangle \left\langle 0\right|,A[{m}_{1}]=\left|{\varphi }_{1}\right\rangle \left\langle 1\right|,\end{eqnarray} \tag{ 20 }$$

with $\left|{\varphi }_{0}\right\rangle =\cos \tfrac{\theta }{2}\left|0\right\rangle +{{\rm{e}}}^{{\rm{i}}\alpha }\sin \tfrac{\theta }{2}\left|1\right\rangle$ and $\left|{\varphi }_{1}\right\rangle =\sin \tfrac{\theta }{2}\left|0\right\rangle -{{\rm{e}}}^{{\rm{i}}\alpha }\cos \tfrac{\theta }{2}\left|1\right\rangle .$ In fact, we show that if one can find a basis $\left\{{m}_{0},{m}_{1}\right\}$ such that matrices of the form (20) exists, the local entropy of entanglement of any particle of the wire is maximal for any θ (see appendix C). Therefore, we can relate the local entanglement of the wire with the probabilistic behavior of the localization process (see below).

Consider the case of a wire which admits a basis where the matrices are of the form (20). The wire is prepared in the state ${\rm{\Phi }}{\left(\left|\psi \right\rangle \right)}_{k}^{n}$ by appropriate measurements of the first k particles. As before, by expanding the $k+1$ site in the $\left\{\left|{m}_{0}\right\rangle ,\left|{m}_{1}\right\rangle \right\}$ basis and mapping the state of the correlation space into $\left\{\left|{\varphi }_{0}\right\rangle ,\left|{\varphi }_{1}\right\rangle \right\}\to \left\{\left|+\right\rangle ,\left|-\right\rangle \right\}$ by single measurements of the next physical qubits (from site $k+2$ to $k^{\prime}$), we end up with a state of the form: $\sum _{s=0,1}{Z}_{m}^{s}{\left|{\psi }_{m}\right\rangle }_{k+1}{\left|{m}_{s}\right\rangle }_{k^{\prime} +1}{\rm{\Phi }}{\left(\left|{\psi }_{s}\right\rangle \right)}_{k^{\prime} +2}^{n}$. In order to accomplish the mapping, around τ measurements are needed (up to by-products), where τ is the period factor of the wire. With a measure on the site $k^{\prime} +1$ one achieves the downloading of the state $\left|\psi \right\rangle$ at position $k+1$. For a more general wire configuration (19), a filter operation [39] is needed in order to recover orthogonality, leading to a probabilistic process for an a priori fixed site downloading.

In particular, we show that the success probability of localizing the information in the period wire (4) is directly related with its local entanglement. The period wires are defined by the matrices (4), and the parameter ϕ. As we discuss below (see section 3.4.4) ϕ determines the local entanglement of the wire. For this wire, there exists a basis $\left|{m}_{i}\right\rangle$ such that the MPS matrices are expressed in the form (19), with

$$\begin{eqnarray}&&{r}_{1}=\displaystyle \frac{1}{2}\left(1+{{\rm{e}}}^{-{\rm{i}}\phi }\right).\end{eqnarray} \tag{ 21 }$$

Following [39], one concludes that the success probability of an a-priori fixed site downloading is

$$\begin{eqnarray}&&p=1-\left|{r}_{1}\right|=1-\cos \left(\displaystyle \frac{\phi }{2}\right).\end{eqnarray} \tag{ 22 }$$

This shows that the probability depends exclusively on the entanglement factor, and the process is deterministic (p = 1) for a maximally entangled wire $(\phi =\pi )$. Henceforth, the entanglement of the wire is one crucial property that defines how the information is spread over the particles.

If we do not restrict ourselves to the case where the localization has to be achieved in a specidfic, pre-defined site, and we allow for open destination downloading, the success probability can be made arbitrarily close to 1 using larger wires. By repeating the filtering process l times, the success probability becomes $p=1-{\left|{r}_{1}\right|}^{l}$. This is an important difference between the Dicke-type encoding and the correlation space,since here there is no need of measuring all the qubits to localize information and, therefore, if the localization procedure fails, one can keep repeating it until succeeding. Besides, once the process succeeds, information is localized in a qubit that is detached from the unmeasured part of the wire. The remaining part of the wire remains functional and can still be used to store and process information.

In an encoding scenario, each wire plays the role of a logical qubit with codewords given by the states of the correlation space, typically $\left|{0}_{L}\right\rangle ={\rm{\Phi }}\left(\left|0\right\rangle \right),\left|{1}_{L}\right\rangle ={\rm{\Phi }}\left(\left|1\right\rangle \right)$.

Consider again an auxiliary qubit entangled with a wire in the state

$$\begin{eqnarray}&&\left|{{\rm{\Phi }}}^{+}\right\rangle =\displaystyle \frac{1}{\sqrt{2}}\left({\left|0\right\rangle }_{{aux}}{\rm{\Phi }}{\left(\left|0\right\rangle \right)}_{1}^{n}+{\left|1\right\rangle }_{{aux}}{\rm{\Phi }}{\left(\left|1\right\rangle \right)}_{1}^{n}\right),\end{eqnarray} \tag{ 23 }$$

with $n\to \infty .$ Note this state can be obtained from a single wire and a Bell pair, by performing a Bell measurement between one qubit of the Bell state, and the first particle of the wire, measuring the next k particles to implement the rotation $\left\{A[{m}_{0}]\left|R\right\rangle ,A[{m}_{1}]\left|R\right\rangle \right\}\to \left\{\left|0\right\rangle ,\left|1\right\rangle \right\}$ in the correlation space.

We study the robustness of this wire encoding under loss of particles, for different configurations. For that purpose, we analyze again the CJ fidelity (8) with respect to the state (23) when several particles of the wire are lost (see figure 9). The CJ fidelity is computed following the steps specified in section 3.1.3. Similarly to the Dicke-type encoding, the stability of this encoding under loss of particles depends on the local entanglement of the wire, where smaller entanglement leads to higher robustness. This confirms the relation between the entanglement and the spreading of information that we showed above.

Zoom In Zoom Out Reset image size

In analogy to section 3.3.4, we analyze the local entanglement of a single particle within a wire in the logical state $\left|{{\rm{\Phi }}}_{L}^{+}\right\rangle =\tfrac{1}{\sqrt{2}}\left(\left|{0}_{L}\right\rangle \left|{0}_{L}\right\rangle +\left|{1}_{L}\right\rangle \left|{1}_{L}\right\rangle \right)$, with $\left|{0}_{L}\right\rangle ={\rm{\Phi }}\left(\left|0\right\rangle \right),\left|{1}_{L}\right\rangle ={\rm{\Phi }}\left(\left|1\right\rangle \right)$(see figure 10). The parameter ϕ is the factor that determines the local entanglement of the wire, a result consistent with the studies of previous sections. Therefore, denoting ϕ as entanglement factor is justified. One can see the influence of the parameter ϕ on the local entanglement from the analysis of section 3.4.2. From equation (21), one can infer that the only case where MPS matrices (20) can be written in the form (19) is for $\phi =\pi$. In appendix C it is shown that if a basis exists such that the matrices can be expressed as in equation (19), the local entanglement of the wire is maximal (which is the case for ϕ = π). For other values of the entanglement parameter, MPS matrices are of the general form (20) and the local entanglement depends on ϕ in the way shown in figure 10. Note that the local entanglement of the wire can be made arbitrarily low.

Zoom In Zoom Out Reset image size

We now analyze the amount of information available for a single party within the wire. Figure 11 shows that the mutual information of a single party depends both on the entanglement and the period factors of the wire. In order to understand these results we need to discern how the information of the correlation space is distributed throughout all the particles. We evaluate is the correlation length of the wire, which characterize the length scale on which two-point correlations vanish. Therefore, the correlation length can be seen as a value that determines the distance over which information is spread in the wire, and we study the relation of ϕ and τ with it. The correlation length is defined as [48]:

$$\begin{eqnarray}&&\xi =\displaystyle \frac{-1}{\mathrm{log}\left(\tfrac{| {\lambda }_{2}| }{| {\lambda }_{1}| }\right)},\end{eqnarray} \tag{ 24 }$$

where $| {\lambda }_{1}|$ and $| {\lambda }_{2}|$ are the two largest eigenvalues of the transfer matrix ${E}_{{\mathbb{I}}}=\sum A[i]\otimes \overline{A[i]}$. Figure 12 shows the dependence of the correlation length with the period and the entanglement factor separately. It is clear that both parameters have an important influence on the distribution of the information over the wire. This can be seen from the downloading process studied in previous sections, when one aims to localize information deterministically. There we showed that the entanglement factor defines the success probability of the localization procedure. Moreover, in case one succeeds, a number of particles of the order of the period factor has to be measured in order to accomplish the localization.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

We conclude that both, the local entanglement and the period factor, are essential parameters that determine how the information of the correlation space is distributed through the qubits of the wire, and how many particles are needed to localize that information. This is hence related to the amount of information accessible for each party, as we show in our analysis (figure 11) i.e. the more distributed the information is, the less information a party can access. Besides, our analysis reveals that the position of the particle also matters. Particles close to the left boundary, where the state of the correlation space is defined (section 2.4), have a higher access to information than qubits sited away. The correlation length also determines how the amount of accessible information decreases with the distance to that left boundary.

Consider an initial Bell state and a period quantum wire. A generalized Bell measurement is performed between the second qubit of the Bell pair and the first one of the wire. After by-product corrections (assuming successful uploading), the remaining state is of the form:

$$\begin{eqnarray}&&\left|{{\rm{\Phi }}}^{+}\right\rangle =\displaystyle \frac{1}{\sqrt{2}}\left({\left|0\right\rangle }_{{aux}}{\rm{\Phi }}{\left(\left|0\right\rangle \right)}_{k}^{n}+{\left|1\right\rangle }_{{aux}}{\rm{\Phi }}{\left(\left|1\right\rangle \right)}_{k}^{n}\right).\end{eqnarray} \tag{ 25 }$$

We transport the state two periods (2τpositions) in the wire before starting a localization process. In the ideal case (no error), the state of the correlation space ${\rm{\Phi }}\left(\left|i\right\rangle \right)$ is first mapped into itself by the transport. Assuming the localization process is successful (otherwise it could be repeated until succeeding), the remaining state is again a Bell pair between the first qubit of the initial Bell state (aux) and the qubit of the wire where downloading succeeds, decoupled from the rest of the wire. In order to examine the stability of the wire under errors in the transport, we study the fidelity of this final state in case some error (or loss) occurs during the transport with respect to the ideal case (Bell pair obtained), with a similar spirit as for the CJ fidelity (8).

When an error affects a physical particle, the corresponding error on its matrix operator of the correlation space is in general non-trivial. For instance, for a maximally entangled period wire with ϕ = π, if a Pauli error affect any particle, the effect on the correlation space is (see appendix D for proof):

$$\begin{eqnarray}&&Z\left|j\right\rangle \longrightarrow A[j]Z,\quad X\left|j\right\rangle \longrightarrow {XA}[j]X,\quad Y\left|j\right\rangle \longrightarrow {\rm{i}}{X}{A}[j]{ZX}.\end{eqnarray} \tag{ 26 }$$

Note that, when one transports information by measuring on the X basis, a X error in the measured qubit has no influence on the correlation space state.

For other entanglement configuration of the wire, these relations between the physical errors and they correspondence in the correlation space become cumbersome. However, if the qubit where any CPTP error occurs is measured in any basis (which is the case we typically come up against), it is known [56] which is the effect on the correlation space.

Consider now, for instance, a Pauli Z error affecting the kth particle of the wire with probability one during a transport process. The state ${\rm{\Phi }}\left(\left|i\right\rangle \right)$ for $i=\left\{0,1\right\}$ is now mapped into some state ${\rm{\Phi }}\left(\left|{\chi }_{i}\right\rangle \right)$ after the transport. Assume also that the subsequent downloading process is successful. If the local entanglement of the wire is maximal, i.e. T = Z, the error shows up completely in the end, i.e. the final state is the Bell state $\left|{{\rm{\Phi }}}^{-}\right\rangle =\tfrac{1}{\sqrt{2}}\left(\left|00\right\rangle -\left|11\right\rangle \right)$. However, if one reduces the local entanglement of the wire (the parameter ϕ) , the error appears diluted at the end of the process. For instance, for $\phi =\tfrac{\pi }{3}$, the final state expressed in the Bell basis is: $\tfrac{\sqrt{3}}{2}\left|{{\rm{\Phi }}}^{+}\right\rangle +\tfrac{1}{2}\left|{{\rm{\Phi }}}^{-}\right\rangle$. Figure 13 shows the fidelity of the final state $\left|\varphi \right\rangle$ after the transport-localization process with some Z error affecting one particle in the transport, i.e. $F={\left|\left\langle \varphi \left|{{\rm{\Phi }}}^{+}\right\rangle \right.\right|}^{2}$, for different entanglement parameters. The remarkable result is that this fidelity does not depend on the position where the error happens or the period parameter of the wire. Therefore, the stability of the wire depends exclusively on the entanglement, as one can conclude due to the strong similarity between figures 13 and 10.

Zoom In Zoom Out Reset image size

The price to pay for the protection against errors is a lower success probability of delocalizing and localizing information. However, when considering arbitrarily large wires, these processes can be made quasi-deterministic with open destination targets. One can always choose regions such that localization within this region has an arbitrary high probability of success. The size of this regions depends entirely on the local entanglement of the wire. Notice that for a given success propability p for the downloading process, the required number of repetitions to succeed with a probability $1-\epsilon$ is given by $n=\mathrm{log}(\epsilon )/\mathrm{log}(1-p)$.

The worst case of the previous scenario occurs when the particles are lost. We consider the overall process of uploading–processing–downloading, where we assume that one or more particles are lost (and hence traced out) during the transport. We study again the fidelity with respect to the ideal Bell state. Note that tracing out a particle is equivalent to averaging over the possible outcomes of a measurement in a given basis. By observation, one can easily check that, in our case, this map is analogous to a phase-flip channel $(\sigma \to q\sigma +(1-q)Z\sigma Z)$ with probability $q=\tfrac{1}{2}$ at the physical level. The Kraus decomposition of this map can be obtained by first expressing the final state in the Bell basis followed by a subsequent diagonalization

$$\begin{eqnarray}&&\left|{\psi }_{00}\right. \rangle \left. \langle {\psi }_{00}\right|\mathop{\longrightarrow }\limits^{\xi }\sum {\lambda }_{{i}_{1},{j}_{1},{i}_{2},{j}_{2}}\left|{\psi }_{{i}_{1}{j}_{1}}\right. \rangle \left. \langle {\psi }_{{i}_{2}{j}_{2}}\right|\mathop{\longrightarrow }\limits^{{\rm{d}}{\rm{i}}{\rm{a}}{\rm{g}}.}\displaystyle \sum _{i}{m}_{i}\left|{m}_{i}\right. \rangle \left. \langle {m}_{i}\right|,\end{eqnarray} \tag{ 27 }$$

where the Kraus operators K_i are identified from: ${K}_{i}\left|{\psi }_{00}\right\rangle =\sqrt{{m}_{i}}\left|{m}_{i}\right\rangle$. For the case of one lost particle, this is always a rank-2 channel, independently of the position of the lost qubit.

A remarkable effect is observed when more than one qubit is lost. We find a fidelity dependence with the relative position of the lost particles (see figure 14). The effect of the relative position of the particles is strongly related with the period factor τ of the wire. Given two losses, if the distance between them corresponds with one period τ, the fidelity is minimal. However, the fidelity is maximized when the relative distance coincides with half of a period $\tfrac{\tau }{2}$. In a more general scenario, with m lost qubits, one has to consider the $\left(\begin{array}{c}n\\ m\end{array}\right)$ relative distances of every pair of lost particles. The fidelity is maximal (minimal) when the maximal possible number of relative distances is equal to $\tfrac{\tau }{2}\left(\tau \right)$. We conjecture that this effect is due to the non-vanishing two-point correlation functions.

Zoom In Zoom Out Reset image size

It is possible to cut a chain from an arbitrary site of the wire in any direction, by using similar techniques than for localization (section 3.4.2). For simplicity, consider a wire with maximal local entanglement:

$$\begin{eqnarray}&&{\rm{\Phi }}{\left(\left|L\right\rangle \right)}_{1}^{n}=\sum \left\langle {s}_{n}\right|A\left[{s}_{n-1}\right]\cdots A\left[{s}_{1}\right]\left|L\right\rangle \left|{s}_{1}\ldots {s}_{n}\right\rangle .\end{eqnarray} \tag{ 28 }$$

Taking property (20) into account, one can expand the kth site in its basis, i.e.

$$\begin{eqnarray}&&{\rm{\Phi }}=\sum \left\langle {s}_{n}\right|A\left[{s}_{n-1}\right]\cdots {\left|{\varphi }_{i}\right\rangle }_{k}\left\langle i\right|\cdots A\left[{s}_{1}\right]\left|L\right\rangle \left|{s}_{1}\ldots {s}_{n}\right\rangle {\left|{m}_{i}\right\rangle }_{k}.\end{eqnarray} \tag{ 29 }$$

By appropriate measurements of the next q sites, one can perform the transformation $\left\{\left|{\varphi }_{0}\right\rangle ,\left|{\varphi }_{1}\right\rangle \to \left|+\right\rangle ,\left|-\right\rangle \right\}$ in the correlation space. Expanding the site $r=k+q+1$ in its basis:

$$\begin{eqnarray}&&{\rm{\Phi }}=\sum \left\langle {s}_{n}\right|A\left[{s}_{n-1}\right]\cdots {\left|{\varphi }_{j}\right\rangle }_{rk}\left\langle i\right|\cdots A\left[{s}_{1}\right]\left|L\right\rangle \left|{s}_{1}\ldots {s}_{n}\right\rangle {\left|{m}_{i}\right\rangle }_{k}{\left|{m}_{j}\right\rangle }_{r}.\end{eqnarray} \tag{ 30 }$$

Finally, by measuring site r and selecting the outcome ${\left|{m}_{0}\right\rangle }_{r}$

$$\begin{eqnarray}&&{\rm{\Phi }}=\sum \left\langle {s}_{n}\right|A\left[{s}_{n-1}\right]\cdots {\left|{\varphi }_{0}\right\rangle }_{rk}\left\langle i\right|\cdots A\left[{s}_{1}\right]\left|L\right\rangle \left|{s}_{1}\ldots {s}_{n}\right\rangle {\left|{m}_{i}\right\rangle }_{k}={\rm{\Phi }}{\left(\left|{\varphi }_{0}\right\rangle \right)}_{r^{\prime} }^{n}\otimes {\rm{\Phi }}{\left(\left|L\right\rangle \right)}_{1}^{k},\end{eqnarray} \tag{ 31 }$$

one succeeds in cutting the chain into two wires, both preserving their functionality, but separately. Note that the information of the initial wire remains completely in the right one. In case the entanglement is not maximal, one cannot select the position where the cutting is accomplished, but it eventually succeeds, similar as in the downloading process.

Given a network constructed from period wires (see below), it is important to be able to connect particles upon request, i.e. to establish entanglement between two (or more) parties. Consider again a maximally entangled wire, i.e. $\phi =\pi$. Given two, previously fixed, qubits of the wire, denoted as k₁ and k₂,

$$\begin{eqnarray}&&{\rm{\Phi }}{\left(\left|L\right\rangle \right)}_{1}^{n}=\sum \left\langle {s}_{n}\right|A\left[{s}_{n-1}\right]\cdots {\left|{\varphi }_{i}\right\rangle }_{{k}_{1}}\left\langle i\right|\cdots {\left|{\varphi }_{j}\right\rangle }_{{k}_{2}}\left\langle j\right|\cdots \cdots A\left[{s}_{1}\right]\left|L\right\rangle {\left|{m}_{i}\right\rangle }_{{k}_{1}}{\left|{m}_{j}\right\rangle }_{{k}_{2}}\left|{s}_{1}\ldots {s}_{n}\right\rangle ,\end{eqnarray} \tag{ 32 }$$

one can create a direct communication link between them by applying a double localization process (figure 15). This is achieved by measuring the particles toward the exterior direction in each case, mapping ${\left|{\varphi }_{i}\right\rangle }_{{k}_{1}}\to \left|{i}_{+}\right\rangle$ to the left, and ${\left|j\right\rangle }_{{k}_{2}}\to \tfrac{1}{\sqrt{2}}\left(\left|{\varphi }_{0}\right\rangle +{\left(-1\right)}^{j}\left|{\varphi }_{1}\right\rangle \right)$ to the right. If there are $\tau -1$ particles between the two qubits, and we measure the $\tau -1$ particles in the X basis, where τ is the period factor of the wire, a Bell pair is obtained between the selected qubits and the state is decoupled from the rest of the wire in both directions (see figure 15). In case the local entanglement of the wire is not maximal, this process becomes probabilistic and target sites cannot be fixed previously.

Zoom In Zoom Out Reset image size

Note that this process has two inconveniences. First, not any pair of particles can be chosen, since one is restricted to the property of leaving one period between them. Secondly, in case that we obtain unwanted outcomes (by-products) in the measurement of the intermediate nodes, the final state is not a perfect Bell pair.

Nevertheless, both problems can be solved by considering compensation stations placed in different positions of the network. This stations are simply constructed with a certain number of auxiliary wires of limited length. These auxiliary wires are coupled with the main wires in order to process information until period restrictions and measurement by-products are fixed. In a sense, these boxes serve to extend the distance between two sites to a full period. All possibilities are present, and one only needs to cut all wires except the one with the proper length. Notice that the length of the wire can be properly chosen such that cutting and correcting suitable byproducts is possible.

We conceive the 1D quantum wires as fundamental building blocks. In order to construct more complex structures, one has to be able to couple wires in order to route the information or create direct links between any constituents.

For simplicity, we restrict ourselves to the case of maximally entangled wires. However, these processes are valid also for lower entangled wires, by applying appropriate filtering operations and taking into account the impossibility of a priori fixed destination. We start by showing that two 1D wires can be coupled in the boundary qubits to merge them into a larger wire. Given two wires initialized in the $\left|+\right\rangle$ state,

$$\begin{eqnarray}&&{\rm{\Phi }}{\left(\left|+\right\rangle \right)}_{1}^{n}\otimes {\rm{\Phi }}^{\prime} {\left(\left|+\right\rangle \right)}_{n^{\prime} }^{k}=\sum \left\langle {s}_{k}\right|A\left[{s}_{k-1}\right]\cdots A\left[{s}_{n^{\prime} }\right]\left|+\right\rangle \left\langle {s}_{n}\right|A\left[{s}_{n-1}\right]\cdots A\left[{s}_{1}\right]\left|+\right\rangle \left|{s}_{1}\ldots {s}_{n}\right\rangle \left|{s}_{n^{\prime} }\ldots {s}_{k}\right\rangle ,\end{eqnarray} \tag{ 33 }$$

we perform the operation $P=\left|0\right\rangle \left\langle {m}_{0}0\right|+\left|1\right\rangle \left\langle {m}_{1}1\right|$ between the last qubit of the left chain and the first qubit of the right one, merging them into an intermediate node r. The basis $\left\{{m}_{0},{m}_{1}\right\}$ corresponds to the one where operators are expressed as equation (20). Note that for the wires we use this basis coincides with the computational basis. Therefore, one obtains

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Phi }} & = & \sum \left\langle {s}_{k}\right|A\left[{s}_{k-1}\right]\cdots A\left[i\right]\left|+\right\rangle \left\langle i\right|A\left[{s}_{n-1}\right]\cdots A\left[{s}_{1}\right]\left|+\right\rangle \left|{s}_{1}\ldots {s}_{n-1}\right\rangle \left|{s}_{n^{\prime} +1}\ldots {s}_{k}\right\rangle {\left|i\right\rangle }_{r}=\\ & = & \sum \left\langle {s}_{k}\right|A\left[{s}_{k-1}\right]\cdots \left|{\varphi }_{i}\right\rangle \left\langle i\right|A\left[{s}_{n-1}\right]\cdots A\left[{s}_{1}\right]\left|+\right\rangle \left|{s}_{1}\ldots {s}_{n-1}\right\rangle {\left|i\right\rangle }_{r}\left|{s}_{n^{\prime} +1}\ldots {s}_{k}\right\rangle ={\rm{\Phi }}{\left(\left|+\right\rangle \right)}_{1}^{k},\end{array}\end{eqnarray} \tag{ 34 }$$

and the merging successes (see figure 16(a)).

Zoom In Zoom Out Reset image size

More than two wires can be coupled. One can consider, for instance, three wires where a projection $P=\left|0\right\rangle \left\langle 000\right|+\left|1\right\rangle \left\langle 111\right|$ is performed on their extremities (first site of each), i.e.

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Phi }} & = & \sum \left\langle {s}_{n^{\prime\prime} }\right|A\left[{s}_{n^{\prime\prime} -1}\right]\cdots A\left[i\right]\left|+\right\rangle \left\langle {s}_{n^{\prime} }\right|A\left[{s}_{n^{\prime} -1}\right]\cdots A\left[i\right]\left|+\right\rangle \left\langle {s}_{n}\right|A\left[{s}_{n-1}\right]\cdots A\left[i\right]\left|+\right\rangle \\ & & \times \,\left|{s}_{2}\ldots {s}_{n}\right\rangle \left|{s}_{2^{\prime} }\ldots {s}_{n^{\prime} }\right\rangle \left|{s}_{2^{\prime\prime} }\ldots {s}_{n^{\prime\prime} }\right\rangle {\left|i\right\rangle }_{r}.\end{array}\end{eqnarray} \tag{ 35 }$$

If now the states of the correlation space are all mapped $\left|{\varphi }_{i}\right\rangle \to \left|i\right\rangle$ by suitable local measurements and one expands the next site (denoted as q):

$$\begin{eqnarray}\begin{array}{rcl}{\rm{\Phi }} & = & \sum \left\langle {s}_{n^{\prime\prime} }\right|A\left[{s}_{n^{\prime\prime} -1}\right]\cdots A\left[j\right]\left|0\right\rangle \left\langle {s}_{n^{\prime} }\right|A\left[{s}_{n^{\prime} -1}\right]\cdots A\left[j\right]\left|0\right\rangle \left\langle {s}_{n}\right|A\left[{s}_{n-1}\right]\cdots A\left[j\right]\left|0\right\rangle \left|{s}_{q-1}\ldots {s}_{n}\right\rangle \\ & & \times \,\left|{s}_{q^{\prime} -1}\ldots s{}_{n^{\prime} }\right\rangle \left|{s}_{q^{\prime\prime} -1}\ldots {s}_{n^{\prime\prime} }\right\rangle {\left|{jjj}\right\rangle }_{{qq}^{\prime} q^{\prime\prime} }{\left|j\right\rangle }_{r}.\end{array}\end{eqnarray} \tag{ 36 }$$

If now localization on each wire is performed (see previous section), by mapping the correlation state to $\left|\pm \right\rangle$ and measuring the following site in the computational basis, a 4-party GHZ state $\sum {\left|{jjj}\right\rangle }_{{qq}^{\prime} q^{\prime\prime} }{\left|j\right\rangle }_{r}$ is obtained. By measuring the r qubit in the X basis, a GHZ of three qubits (one of each wire) is found and decoupled from the rest of the wires (figure 16(b)). Note that the remaining wires are still functional. Note also that by constructing router stations with auxiliary wires, any particle can be chosen on demand to be part of this GHZ state. This process can be expanded for an arbitrary large number of wires, therefore generating n-party GHZ states.

With analogous techniques, one can prove that two wires can be coupled in any middle point of them, such that a four-party GHZ state can be obtained by downloading information in the four diverging directions (see figure 16(b)). Given this merging configuration, one can easily check that a cutting can be performed in any of the directions, such that one can transport information through the remaining chain leafs (see figure 16(c)). That is, a network with a given structure (intersecting wires), one can either generate GHZ states between parties next to an intersection vertex, or use any of the intersecting vertices to route information. To this aim all but an incoming and outgoing wire are cut.

We emphasize that the merging of wires is a not process that takes place when processing information in the network. This should rather be understood as a tool that describes the required resource states in such an entanglement based network, and how they can (in principle) be generated from individual wire states.

Given the quantum period wires and the tools studied above, it is possible to construct a networks with arbitrary topology where information can be transport through in a protected way, and where direct communication links between any two (or more) particles can be established. These is accomplished by merging 1D wires and including compensation stations between regions as described above (see figure 17).

Zoom In Zoom Out Reset image size