The quantum internet: an efficient stabilizer states distribution scheme

In the present paper, we consider quantum intermediate relay nodes where the contents of outgoing states are causal functions of the contents of received states, referring to this mapping as Local Quantum Coding. First, we show that Local Quantum Coding –LQC–in intermediate nodes, can be used to distribute the particular class of graph states in a distributed way among remote client nodes. This feature will be proved to be important for the following reasons:

• It is useful to beat the scalability issues of the previous protocols.
• Providing a distributed architecture for multipartite entanglement distirbution

Second, we show that the scheme of using LQC, is nothing but a particular network coding strategy [22–24]. This allows to achieve the single-use capacity of the appropriate quantum network topologies, which normal entanglement routing fails to fulfill. To this aim:

• Different bounds for stabilizer states distribution capabilities of the underlying network would be obtained in terms of min-cuts capacities corresponding to different bipartitions of the remote client nodes requiring the graph state.

By considering the quantum network as a tensor network with a particular instance of tensors being specific coding isometries, the advantages of the advantages of our scheme are given in terms of the following figure of merits and applications:

• We show that by allowing the intermediate nodes in the quantum network to use LQC, an exponential latency reduction in the entanglement distribution time is achieved.
• We show that the use of LQC reduces drastically the memory qubits overhead of the entanglement distribution cost.
• We discuss how by allowing intermediate use of quantum error correcting codes, distributed storage can be achieved in quantum networks.

The paper is structured as follows. In section. 2 some preliminaries are be given. In particular, graph states and their entanglement structure will be highlighted. Additionally, the concept of tensor networks along with the notion of min-cut will be presented. In section. 3, our model of the distribution quantum network will be elaborated. An equivalence between our model and a network of stabilizer states contracted by Bell pairs is given in section. 4. In section. 5 our results comprising the different bounds on the consumed resources to single-use distribute a target graph state will be established. Section 6 illustrates the obtained results and bounds for some specific network topologies. In particular, the protocol will be benchmarked to Bell pair based protocols with the time complexity of the graph states distribution and the number of memory qubits taken as figure of merits. We also study the application of choosing the LQC schemes to be valid stabilizer quantum error correcting schemes in order to achieve distributed quantum storage within quantum networks. We finish the paper with conclusions and future work in section 7.

Graph states admit a simple description in terms of mathematical graphs [12]. A graph is a pair (G, V) of a finite set V = {1....,N} and a set E ⊂ [V]². The elements of V are called the vertices of the graph and the elements of E are called its edges. Additionally, we denote by N_a the set of neighboring vertices of the vertex a.

The graph state ∣G〉 that corresponds to the graph G is the pure state given as

$$\begin{eqnarray}&&| G\rangle =\displaystyle \prod _{\{a,b\}\in E}{U}_{{ab}}| +{\rangle }^{V}\end{eqnarray} \tag{ 1 }$$

where

$$\begin{eqnarray}&&| +{\rangle }^{V}={\otimes }_{a\in V}\displaystyle \frac{1}{\sqrt{2}}{\left(| 0\rangle +| 1\rangle \right)}^{a}\end{eqnarray} \tag{ 2 }$$

Similarly, U_ab is the controlled Z gate between the vertices a and b.

It is known that any graph state is equivalent to a stabilizer state under a certain class of local operations, the so-called local Clifford operations [12], which map the set of stabilizer states onto itself. Accordingly, graph states can be defined uniquely as the common eigenvector with eigenvalues equal to one in ${{\mathbb{C}}}^{V}$ to the set of independent commuting operators

$$\begin{eqnarray}&&{K}_{a}={\sigma }_{x}^{a}{\sigma }_{z}^{{N}_{a}}\end{eqnarray} \tag{ 3 }$$

for all a ∈ V.

Generally, the many interesting entanglement structure properties of any multipartite state are captured by the reduced state after tracing out parts of the system. Specifically, for a given pure multipartite state ∣Ψ〉. For a specific bipartition of the state, namely {A, B}, the reduced state on A

$$\begin{eqnarray}&&{\rho }_{{\rm{\Psi }}}^{A}={\mathrm{Tr}}_{B}(\rho )\end{eqnarray} \tag{ 4 }$$

where ρ = ∣Ψ〉〈Ψ∣_AB . The entanglement rank of the state ∣Ψ〉 in the bipartition {A, B}, quantifies the amount of Bell pairs needed to create such bipartition, and is given as [12]:

$$\begin{eqnarray}&&{r}_{A}(| {\rm{\Psi }}\rangle )={\mathrm{log}}_{2}[{rank}({\rho }_{{\rm{\Psi }}}^{A})]\end{eqnarray} \tag{ 5 }$$

In this work, we represent the quantum network by an undirected network. For simplicity, an undirected network is described by a tuple (G, V, E, d) where G is the underlying graph, V is the set of its nodes and E is the set of the point to point links in this network. To each link (u, v) ∈ E is assigned a value d_(u,v) representing the dimension of the quantum system that can be carried through this link. For two subsets of nodes S ⊂ V and T ⊂ V in the network is defined the notion of the cut $(\tilde{S},\tilde{T})$. This is a partition of the set of nodes $V=\tilde{S}\cup \tilde{T}$ such that $S\in \tilde{S}$ and $T\in \tilde{T}$. The value of a cut is given by the product:

$$\begin{eqnarray}&&\displaystyle \prod _{(u,v)\in E,u\in \tilde{S},v\in \tilde{T}}{d}_{(u,v)}\end{eqnarray} \tag{ 6 }$$

which is the product of the dimensions of independent links between the two subsets $\tilde{S}$ and $\tilde{T}$. In particular, the minimum cut between two sets of nodes S and T designated by MC_(S,T) is the cut which has the minimum value [22, 25], namely:

$$\begin{eqnarray}&&{\mathrm{MC}}_{(S,T)}=\mathop{\min }\limits_{{\left(\tilde{S},\tilde{T}\right)}_{S\in \tilde{S},T\in \tilde{T}}}\displaystyle \prod _{(u,v)\in E,u\in \tilde{S},v\in \tilde{T}}{d}_{(u,v)}\end{eqnarray} \tag{ 7 }$$

To better illustrate the notion of the min-cut, we consider the illustrating network in figure 1. It is a network with identical link dimensions for all the edges d_u,v in the graph. It is clear that the minimum cut separating the two sets of nodes S and T given in light blue and dark blue respectively, is given by the partition $(\tilde{S},\tilde{T})$ which are the respectively given by the set of nodes surrounded by the red and green regions. This min cut has value equals to four. It is much more convenient in the remaining of the paper, to work with the min-cut with respect to log₂ of the dimensions, hence the min-cut would be given by

$$\begin{eqnarray}&&{\mathrm{MC}}_{(S,T)}=\mathop{\min }\limits_{{\left(\tilde{S},\tilde{T}\right)}_{S\in \tilde{S},T\in \tilde{T}}}\displaystyle \sum _{(u,v)\in E,u\in \tilde{S},v\in \tilde{T}}{\mathrm{log}}_{2}({d}_{(u,v)})\end{eqnarray} \tag{ 8 }$$

Working with this definition of the min-cut would allow us to work later with the entanglement ranks instead of the dimensions of the states distributed. The reason we are considering undirected networks is that entanglement networks are symmetric on different bipartitions, in the sense that, a sender and a receiver might be interchanged. Similarly, the notions of the min-cut and the entanglement entropies of states, which will be needed later, are also independent of interchanging bipartitions.

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The distribution of entanglement in the central-node scheme hinges on establishing the most efficient star-shaped network configuration. In this setup, clients desiring a multipartite entangled state are positioned at the endpoints of this star, while the central node connects to each client through a chain of EPR pairs. Entanglement swapping is employed in nodes in the EPR chains to establish long-haul EPR pairs between the central node and the terminals. Thereafter, through local adjustments of these long-haul EPR pair connections, the central node can effectively transmit its multipartite entangled state to the remote clients [14, 15]. This central node approach is schematized in figure 4. This approach proves to be efficient as long as the connections between the central node and the terminals operate independently. To put it differently, when there are bottleneck nodes, which are nodes with a connectivity degree greater than two, the protocol demands additional resources in these particular nodes. This leads to sub-optimal performance, especially in scenarios featuring highly connected nodes, resulting in inefficiency concerning the available communication capacity and time.

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An alternative method for disseminating multipartite entanglement, particularly in the case of graph states, involves identifying the minimum-cost Steiner tree that connects all clients located at its terminals using point-to-point links [20]. This approach works perfectly for efficiently distributing GHZ states using local Clifford gates as a tool of transforming distributively the state. This is illustrated in figure 5. However, when dealing with general target graph states, an enhanced protocol is necessary. This entails the discovery and utilization of various Steiner trees that span different subsets of the terminals. These trees are employed to distribute a decorated edge graph state, which can then be transformed into any graph state having the same number of clients. For instance, when considering the distribution of a four qubit linear cluster state between the client nodes of the network in figure 3, this protocol does not use the quantum network infrastructure ones, but many usages are required.

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theorem  A graph state with a list of bipartite entanglement ranks $\{{r}_{{AB}}\}$ , corresponding to different possible bipartitions {A, B} of the end nodes, can be distributed in a quantum network by a single use of the latter if and only if:

$$\begin{eqnarray}&&{{MC}}_{\{A,B\}}\geqslant {r}_{{AB}}={{rank}}_{{{\mathbb{F}}}_{2}}({{\rm{\Gamma }}}_{{AB}})={\mathrm{log}}_{2}[{rank}({\rho }_{A}^{G})]\end{eqnarray} \tag{ 36 }$$

for every bipartition $\{A,B\}$ of the target graph state.

Proof. We prove first the sufficient part of the theorem. A bipartition of the state of the client nodes into $\{A,B\}$ would induce a bipartition of the network into a cut $(\tilde{S},\tilde{T})$ with the client nodes $A,B$ satisfying $A\subset \tilde{S}$ and $B\subset \tilde{T}$ respectively. We can regard A and B as source and sink nodes S and T respectively—in light of the discussion in section. 2 –and we can collect them in a single node as an effective network without affecting the communication capacities between the bipartitions. It is known that the entanglement that can be established between the source and sink nodes in a tensor network should satisfy [25]

$$\begin{eqnarray}&&Q(\{A,B\})\leqslant {{MC}}_{\{A,B\}}\end{eqnarray} \tag{ 37 }$$

For bi-partite pure entangled states the entanglement that can be extracted ${E}_{D}(A,B)$ (entanglement distillation) and the entanglement needed to create the state E_F (the entanglement of formation) are equal and they satisfy

$$\begin{eqnarray}&&\begin{array}{l}{E}_{D}(A,B)={E}_{F}(A,B)\\ =\,S(A)\\ =\,{rank}({\mathrm{Tr}}_{B}(\rho ))\end{array}\end{eqnarray} \tag{ 38 }$$

Accordingly, for graph states and by using equation (5), we have that

$$\begin{eqnarray}\begin{array}{lcc}Q(\{A,B\}) & = & {\mathrm{rank}}_{{{\mathbb{F}}}_{2}}({{\rm{\Gamma }}}_{{AB}})\\ & \leqslant & {{MC}}_{\{A,B\}}\end{array}\end{eqnarray} \tag{ 39 }$$

We prove the necessary part of the theorem by contradiction. Suppose that a target graph state with entanglement entropy r_AB satisfying:

$$\begin{eqnarray}&&{r}_{{AB}}\gt {{MC}}_{\{A,B\}}\end{eqnarray} \tag{ 40 }$$

in a given bipartition $\{A,B\}$ can be distributed in the underlying quantum network. Then we can necessarily find r_AB end-to-end independent links between the two sets A and B in the quantum network. In this case the min-cut between the two sets satisfies:

$$\begin{eqnarray}&&{{MC}}_{\{A,B\}}={r}_{{AB}}\end{eqnarray} \tag{ 41 }$$

which is in contradiction with the initial assumtion of ${r}_{{AB}}\gt {{MC}}_{\{A,B\}}$ □

This theorem underscores the presence of two distinct graph structures. One of these structures pertains to the physical network, ensuring connectivity among nodes, while the other represents the theoretical structure of the desired graph state. The pivotal aspect is the condition imposed by the theorem, emphasizing that these two graphs must meet specific criteria. To be precise, Theorem. 2 stipulates that the minimum cut capacities in various client bipartitions within the physical network must exceed the entanglement entropies in the corresponding bipartitions within the graph state, giving the possibility of the latter to be successfully distributed. We highlight that methods using entanglement entropies of entangled state have been used in a slightly different context in [26]. Although it is obvious, the fact that by distributing EPR pairs between a central node and clients on a star topology, any graph state can be distributed among the clients, can be derived from the previous theorem in network information theoretical settings.

corollary  The star network topology can distribute any graph state between the leafs of the network from the central node by a single use of the network.

Proof. On the one hand, we note that the entanglement rank r_AB for any bipartition $\{A,B\}$ of a graph state is bounded from above as:

$$\begin{eqnarray}&&{r}_{{AB}}\leqslant \min \{| A| ,| B| \}\end{eqnarray} \tag{ 42 }$$

On the other hand, the min-cut of any bipartition of the clients in the star network topology satisfies:

$$\begin{eqnarray}&&{{MC}}_{\{A,B\}}=\min \{| A| ,| B| \}\end{eqnarray} \tag{ 43 }$$

Therefore, we have that

$$\begin{eqnarray}&&{r}_{{AB}}\leqslant {{MC}}_{\{A,B\}}\end{eqnarray} \tag{ 44 }$$

and hence, by theorem 2 the star topology of the network allows for the distribution of any graph state. □

This corollary establishes the optimality of the central node protocol framework in a star network topology where there are no bottlenecks in the relay nodes. In fact, under these conditions, our protocol achieves identical performance to that of the central node protocol.

Similarly, we can use the results of Theorem. 2 to state the following corollary for GHZ states distribution in arbitrary quantum networks.

corollary  Any topology of a connected quantum network distributes, in a single use, a GHZ state between the clients

Proof. The proof is straightforward from Theorem 2 by noting that the entanglement ranks r_AB on any bipartition $\{A,B\}$ of the GHZ state satisfy

$$\begin{eqnarray}&&{r}_{{AB}}=1\end{eqnarray} \tag{ 45 }$$

On the other hand, the min-cut of any bipartition of the clients in any connected network satisfies:

$$\begin{eqnarray}&&{{MC}}_{\{A,B\}}\geqslant 1\end{eqnarray} \tag{ 46 }$$

Therefore, we have that

$$\begin{eqnarray}&&{r}_{{AB}}\leqslant {{MC}}_{\{A,B\}}\end{eqnarray} \tag{ 47 }$$

and hence, by Theorem 2, any connected topology of the quantum network can distribute a GHZ state between the clients by single use of the network. □

This corollary demonstrates that the results achieved in [20] regarding the Steiner tree protocol are indeed optimal. It is evident that the protocol's optimality is a direct consequence of the minimum Steiner tree, which connects the clients at its endpoints, effectively optimizing both communication overhead and communication time. Certainly, when it comes to distributing GHZ states exclusively, it's evident that our protocol, utilizing LQC, is indistinguishable from this particular protocol, and no advantage can be discerned at this stage. However, as we expand our consideration to encompass other categories of graph states, the distinctions between the two protocols become apparent. In pursuit of this objective, our protocol leverages local isometries to generate graph states with different entanglement entropies corresponding the target min-cut bounds imposed by the communication network structure. Conversely, the preceding protocol relies on a sequential process of GHZ state distribution to establish a highly connected edge graph state among the clients, granting them the capability to extract the desired graph state.

In order to better harness the capabilities of a given quantum network for entanglement distribution, the design of appropriate LQC strategies in relay nodes is needed. This is equivalent to finding the optimal coding strategies in relay nodes in network coding in order to achieve the capacity of a given network. Notably, our network architecture relying on LQC for entanglement distribution, might be regarded as a quantum network coding counterpart of the classical network coding paradigm.

Specifically, suppose we endow all the relay nodes in the quantum network with a Linear local quantum coding strategy (LLQC) given by:

$$\begin{eqnarray}&&\begin{array}{l}{S}_{x}={U}_{x}^{0}\otimes | 0\rangle \langle 0| +{U}_{x}^{1}\otimes | 1\rangle \langle 1| \\ ={I}^{\otimes n}\otimes | 0\rangle \langle 0| +{X}^{\otimes n}\otimes | 1\rangle \langle 1| \end{array}\end{eqnarray} \tag{ 48 }$$

This strategy is equivalent to a quantum repetition code, generating GHZ states in each node. It is easy to notice that this strategy is identical to the isometry in equation (21) up to a local unitary given by I ⨂ H^⨂n−1. Indeed this strategy can only generate entanglement ranks no higher than r_AB = 1, making it not a suitable candidate for achieving the network capacity bound in general. To illustrate this, we consider different sub-network structures appearing in the network of figure. 2. For the chain subnetwork (blue nodes), the isometry in equation (48) performed in the blue dashed node

$$\begin{eqnarray}&&{S}_{0}={I}_{0}\otimes | 0\rangle \langle 0{| }_{1}+{X}_{0}\otimes | 1\rangle \langle 1{| }_{1}\end{eqnarray} \tag{ 49 }$$

leads to the same performance of an entanglement swapping. Formally, the effect of the LQC 49 in the dashed node is equivalent, by Lemma 1, to

$$\begin{eqnarray}&&\langle {EPR}{| }_{0}| {EPR}{\rangle }_{01}| {EPR}{\rangle }_{02}=| {EPR}{\rangle }_{12}\end{eqnarray} \tag{ 50 }$$

which is exactly entanglement swapping performed at node 0. Differently, the performance of the same isometry, in the dashed nodes of sub-networks (orange nodes) and (end nodes), is not equivalent to entanglement swapping, and leads to better entanglement distribution rates as will be discussed in the next section. This suggests that our multiparty protocol provides an advantage over the Bell pair based approach when the entanglement distribution network has high connectivity.

To illustrate these advantages we consider regular networks with different degrees of connectivity. For instance, we will consider a tree network where each node has the same number of children nodes. We will consider also that the end relay nodes are connected to the same number of clients. In such a network the number of relay nodes is given by N ∼ O(n^p−1), where p is the depth of the network –the smallest distance from the central node and the clients–. Accordingly, the complexities of the single use protocol with the isometry 48 in each relay node, and the Bell pair based, are given respectively by

$$\begin{eqnarray}&&\begin{array}{l}\sim O(p)\\ \sim O({n}^{p})\end{array}\end{eqnarray} \tag{ 51 }$$

Clearly, we have an exponential time complexity reduction in the depth of the network for the distribution of GHZ like states. The results are illustrated in the plots of figure 6. We can easily see that the more the deeper the quantum network is the more clear the advantage of using a multipartite distribution strategy is.

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The similar investigation can be conducted for the advantage over the reduction of memory qubits in the network for GHZ. The figure of merit that we use in this case is the maximal number of memory qubits/per node used in the network to achieve the task of distributing a target graph state between the clients. For the LQC approach, the maximum number of memory qubits needed in each relay node is equal to the number of controlled qubits n^a _c required to generate locally the entangled state with an appropriate LQC strategy. For the case of the previous regular network, this is maximized by the number of connectivity n + 1 of the node according to lemma 1. Thus, the maximum number of memory qubits n_m satisfies

$$\begin{eqnarray}&&\mathop{\max }\limits_{a\in V/C}({n}_{m})={n}_{c}^{{a}^{* }}\leqslant n+1\end{eqnarray} \tag{ 52 }$$

Differently, in the Bell pair-based approach, the highest possible number of memory qubits n_m in each node is of the order of the number of clients in the network

$$\begin{eqnarray}&&\mathop{\max }\limits_{a\in V/C}({n}_{m})\sim {n}^{p}\end{eqnarray} \tag{ 53 }$$

The results are illustrated in figure. 7. We can notice how the entanglement distribution based on multipartite protocols is beneficial for networks with growing depth and connectivity.

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As a noise model, we assign to every entangled state distributed in the quantum network a probability of failure p. Namely, the noise model for the entanglement distribution is given by

$$\begin{eqnarray}&&{ \mathcal N }(\rho )=(1-p)\rho +{pe}\end{eqnarray} \tag{ 54 }$$

where e is the error flag which will abort the distribution protocol. Accordingly, the error propagation in the network for the distribution of n-EPR pairs would be given by

$$\begin{eqnarray}&&{{ \mathcal N }}^{\otimes n}({\rho }^{\otimes n})=\displaystyle \sum _{k=0}^{n}\displaystyle \left(\genfrac{}{}{0em}{}{n}{k}\right){\left(1-p\right)}^{n-k}{p}^{k}{\rho }^{\otimes n-k}\otimes {e}^{\otimes k}\end{eqnarray} \tag{ 55 }$$

Clearly, the probability that the protocol does not abort is when k = 0 which is given by

$$\begin{eqnarray}&&{p}_{{sucess}}={\left(1-p\right)}^{n}\end{eqnarray} \tag{ 56 }$$

As we can easily notice, the less qubit channels are used to establish the target state between the clients the less noisy is the protocol.

For the case of the tree network of depth p = 2, our protocol outperforms the protocol relying on central node and EPR pairs distribution. The performance of the two protocols is illustrated in figure. 8. It is clear that for different target graph states we have

$$\begin{eqnarray}&&{p}_{{sucess}}^{{LQC}}\gt {p}_{{sucess}}^{{EPR}}\end{eqnarray} \tag{ 57 }$$

showing that our protocol is more robust to noise than other protocols.

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A distributed storage system is an infrastructure that can split data across multiple physical servers, and often across more than one data center. It typically takes the form of a cluster of storage units. The storage system includes features that serves as a mechanism for data synchronization and coordination between cluster storage nodes that are geographically dispersed. Moreover, it has the intelligence to detect and respond to failures and cyber attacks. The objective is to achieve very low latency by storing data physically near the location it will be used.

Fundamentally, the transition from classical distributed storage to quantum distributed storage of quantum data, cannot be smooth due to the no-cloning theorem. Although, our distributed stabilizer states generation discussed in section 4, using intermediate quantum encoding, can be harnessed to enhance quantum storage of quantum data significantly.

Consider the network in figure. 9, where the bulk nodes (orange) denoted by {b_i } want to store their single qubit data $\{{q}_{{b}_{i}}\}$ into the neighboring cluster nodes (red) denoted as {c_i j}. The best that each node can do to store its qubit, independently of the other bulk nodes, is to encode it in a three qubit code given by a GHZ-like state

$$\begin{eqnarray}&&| {GHZ}{\rangle }^{{b}_{i}}=\displaystyle \frac{1}{\sqrt{2}}\left(| 000{\rangle }^{{c}_{i1}{c}_{i2}{c}_{i3}}+| 111{\rangle }^{{c}_{i1}{c}_{i2}{c}_{i3}}\right)\end{eqnarray} \tag{ 58 }$$

It can easily be noticed that by encoding in such codes, single qubit errors that might occur during the storage cannot be corrected by no means, making the encoding vulnerable to noise and adversarial attacks in the transmission. Differently, if the bulk nodes (orange) collaborates and look for local codes that harness the full capacity of the network as discussed in section 5, they might achieve better performance. to do so, let the bulk nodes choose to encode their single qubits in a five qubit code each. The five qubit code, denoted as [[5, 1, 3]], can be described by a cyclic five qubits graph state, which have maximal entanglement ranks among five qubit graph states. Equivalently, it might be described by the stabilizer generators given by [27, 28]:

$$\begin{eqnarray}&&{XZZXI},{IXZZX},{XIXZZ},{ZXIXZ}\end{eqnarray} \tag{ 59 }$$

This can encode at most a single logical qubit, and can correct single qubit errors and up to two erasure errors.

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corollary  By contracting the three $[[5,1,3]]$ codes with Bell pairs corresponding to edges according to equation ( 25 ), a nine qubit code $[[9,3,3]]$ is established between remote cluster nodes. This nine qubit code has as generators

$$\begin{eqnarray}&&\begin{array}{l}{XZZXIXZZX},{XIXXZZXZZ},{IXZZXIIXZ}\\ {ZXIIXZZXI},{ZYYXZZIII},{YXXYXXIII}\end{array}\end{eqnarray} \tag{ 60 }$$

It can be simply checked that these nine qubit Pauli operators are commuting. Moreover, it is shown in Appendix 7 that they are independent of each other. Therefore, they provide a stabilizer code storing three logical qubits -encoded locally in the three bulk nodes-, in nine physical qubits-shared between the remote cluster nodes-. Furthermore, it is shown in details in the appendix that the established nine qubit code can correct for single qubit errors and up to two erasure errors. As a result, the remotely established code between remote cluster nodes, and storing the three qubits from the bulk nodes is a [[9, 3, 3]] code. Not surprisingly, the obtained code satisfies, indeed, the quantum singleton bound given by [27, 28]:

$$\begin{eqnarray}&&k\leqslant n-2(d-1)\end{eqnarray} \tag{ 61 }$$

We can generalize the previous example of quantum storage to regular networks with a fixed number of neighbors. As a matter of fact we have the following proposition:

proposition  Let a quantum storage network where the boundary B, containing the quantum registers, and a bulk containing m nodes each of which encodes k logical qubits in a stabilizer code $[[l,k,d]]$ of l physical qubits and distance d. Then, by contracting the different codes, a stabilizer code $[[| B| ,{mk},D]]$ , encoding the mk qubits, can be established between the quantum registers $| B|$ , where its distance D satisfies:

$$\begin{eqnarray}&&D\leqslant \displaystyle \frac{| B| +{ml}-2m(d-1)-2{km}}{2}+1\end{eqnarray} \tag{ 62 }$$

Proof. Suppose we have a network with bulk nodes having k qubits encoded in a stabilizer quantum error correcting code $[[l,k,d]]$, where l is the number of neighboring nodes. The code-space of such code is given by

$$\begin{eqnarray}&&| \psi \rangle \langle \psi {| }_{i}=\displaystyle \frac{1}{{2}^{l-k}}\displaystyle \sum _{{b}_{i}}{u}_{{b}_{i}}\end{eqnarray} \tag{ 63 }$$

where $\{{u}_{{b}_{i}}\}$ are the elements of the corresponding stabilizer group of node i. Suppose also that we have m nodes in the bulk of the network and we have the storing registers sitting at the boundary B of the network. According to theorem 1 the state on the boundary would be given by the projector

$$\begin{eqnarray}&&\begin{array}{l}| {\rm{\Psi }}\rangle \langle {\rm{\Psi }}{| }_{B}=\mathrm{Tr}({\otimes }_{i=1}^{m}| \psi \rangle \langle \psi {| }_{i}| e\rangle \langle e{| }^{\otimes ({lm}+| B| )})\\ =\displaystyle \frac{1}{{2}^{(n-k)m+{lm}+| B| }}\displaystyle \sum _{{s}_{{AB}}}\displaystyle \sum _{{t}_{A}}\mathrm{Tr}({s}_{{AB}}{t}_{A})\\ =\displaystyle \frac{1}{{2}^{(| B| -{km})}}\displaystyle \sum _{{u}_{B}}{u}_{B}\end{array}\end{eqnarray} \tag{ 64 }$$

The last two equalities follow from the derivation of Theorem. 1. with km is the number of logical qubits stored into the $| B|$ quantum registers. Additionally, according to the quantum singleton bound, the best distance D, i.e., minimum Pauli error weight that cannot be corrected, that this stabilizer code can have should satisfy:

$$\begin{eqnarray}&&{km}\leqslant | B| -2(D-1)\end{eqnarray} \tag{ 65 }$$

therefore:

$$\begin{eqnarray}&&D\leqslant \displaystyle \frac{| B| -{km}}{2}+1\end{eqnarray} \tag{ 66 }$$

Differently, the single codes sitting in the bulk satisfy

$$\begin{eqnarray}&&{km}\leqslant {ml}-2m(d-1)\end{eqnarray} \tag{ 67 }$$

by adding the two inequalities (65) and (67) we get:

$$\begin{eqnarray}&&2{km}\leqslant | B| -2(D-1)+{ml}-2m(d-1)\end{eqnarray} \tag{ 68 }$$

and hence, D should satisfy:

$$\begin{eqnarray}&&D\leqslant \displaystyle \frac{| B| +{ml}-2m(d-1)-2{km}}{2}+1\end{eqnarray} \tag{ 69 }$$

□

Proposition 1 can be regarded as a straightforward result of known entropic inequalities. To better understand this, let each of the m nodes containing K EPR pairs where half of these pairs are encoded into a l-qubits quantum code with distance d and the other half is kept as a purifying system R_i . We can split the encoded l qubits part into three partitions A_i F_i C_i where $\dim ({A}_{i})=\dim ({F}_{i})={2}^{d-1}$ and $\dim ({C}_{i})={2}^{l-2(d-1)}$. The global system is thus given by ${RAFC}={R}_{i}^{{\otimes }_{i=1}^{m}}{A}_{i}^{{\otimes }_{i=1}^{m}}{F}_{i}^{{\otimes }_{i=1}^{m}}{C}_{i}^{{\otimes }_{i=1}^{m}}$. In the one hand, it is known that as long as the quantum code has distance d, any partition of d − 1 qubits of the code-space is independent of the encoded state [27, 28]. Formally, we should have

$$\begin{eqnarray}&&\begin{array}{l}S({R}_{i}{A}_{i})=S({R}_{i})+S({A}_{i})\\ S({R}_{i}{F}_{i})=S({R}_{i})+S({F}_{i})\\ S(R)=S({AFC})=\displaystyle \sum _{i=1}^{m}S({R}_{i})={mk}\end{array}\end{eqnarray} \tag{ 70 }$$

Accordingly, we can have

$$\begin{eqnarray}&&\begin{array}{l}S(R)=S({FC})-S(A)=\displaystyle \sum _{i=1}^{m}S({R}_{i})={mk}\\ S(R)=S({AC})-S(F)=\displaystyle \sum _{i=1}^{m}S({R}_{i})={mk}\end{array}\end{eqnarray} \tag{ 71 }$$

Applying the subbaditivity property of the entropy to the last two inequalities we get

$$\begin{eqnarray}&&\begin{array}{l}{mk}=S(R)\leqslant S(C)+S(F)-S(A)\\ {mk}=S(R)\leqslant S(C)+S(A)-S(F)\end{array}\end{eqnarray} \tag{ 72 }$$

Therefore

$$\begin{eqnarray}&&\begin{array}{l}{mk}\leqslant S(C)\\ =\displaystyle \sum _{i=1}^{m}S({C}_{i})\\ \leqslant m(l-2(d-1))\end{array}\end{eqnarray} \tag{ 73 }$$

On the other hand, we note that the contraction of the codes maps the entanglement in the bulk into the boundary B as RAFC → RB with

$$\begin{eqnarray}&&S(R)=\min \{{mk},| B| \}\end{eqnarray} \tag{ 74 }$$

Provided that ∣B∣ > km and that the state on the boundary is indeed a quantum code according to Theorem. 1, and in the same spirit as before we have:

$$\begin{eqnarray}\begin{array}{lcc}S(R) & = & {mk}\leqslant S({C}_{B})\\ & \leqslant & \dim ({C}_{B})\end{array}\end{eqnarray} \tag{ 75 }$$

where $\dim ({C}_{B})=| B| -2(D-1)$.

We can easily verify that the example of the Network in figure 9, with ∣B∣ = 9, l = 5, m = 3, d = 3 and k = 1 satisfy proposition 1 in the sense that we have the singleton bound D = 3 ≤ 4, which is the smallest distance code for ∣B∣ = 9 physical quantum registers and encoding k = 3 logical qubits.

Clearly, the collective efforts of the bulk nodes -by harnessing the full capacity of the network- to establish a common code for distributed storage is advantageous than encoding separately in a centralized way. This gives more resilience to errors that might occur during the storage time or during the transmission process, in the meanwhile, it provides more resilience to losses, or to dishonest participation, when the stored information is to be retrieved. Therefore, this shows the importance of choosing appropriate quantum network encoding for providing secure quantum cryptographical schemes for distributively storing quantum data. Indeed, this would not be possible if the network relies solely on bipartite entanglement distribution and bipartite entanglement swapping.