Lossless Address Coding for Quantum Networks Dick Maryopi
arXiv:2607.23510v1 [cs.IT] 26 Jul 2026
University of Naples Federico II, Naples, 80125 Italy. [email protected]
Abstract—As quantum systems advance toward interconnected architectures, the ability to identify nodes, manage resources, and support network-level functions becomes increasingly critical. In this work, we propose a lossless source coding scheme for addressing in quantum networks that enables compact, hierarchical, and coherently processable quantum address states. Specifically, we introduce a prefix–suffix address space and develop an isometric hierarchical encoder-decoder that guarantees unique decodability. We further provide a practical Huffman-based procedure that embeds prefix-free, length-eigenstate codewords into the address space, thereby preserving isometry. The scheme is particularly designed for networks with hierarchical structure, heterogeneous cluster sizes, and configurable address assignment to accommodate dynamic network conditions. A numerical example on a 13-node network demonstrates that the proposed hierarchical encoding scheme is feasible and achieves perfect fidelity. This work establishes a rigorous connection between source coding theory and quantum network design, offering a practical framework towards scalable and coherent quantum addressing. Index Terms—Quantum Networks, Quantum Addressing, Quantum Compression, Lossless Source Coding.
I. I NTRODUCTION We are currently witnessing a crucial transition in the field of quantum technologies, in which the development of quantum computing is progressing from isolated systems toward interconnected systems that form quantum networks. This networking of quantum systems not only enhances the overall computational capabilities but also enables a wide range of applications, including secure quantum communication, high-precision sensing, and many other distributed quantum information processing tasks. Given this potential, it is envisioned that quantum networks will evolve into a largerscale, and globally interconnected architecture, referred to as the quantum internet [1]–[5]. However, extending quantum systems to large-scale networks, particularly at a global level, is far from trivial and has been a challenging research effort. This is mainly due to the fact that the networks must satisfy fundamentally different physical constraints, such as coherence preservation, restriction of measurement, and the non-clonability of unknown quantum information. Consequently, most informationprocessing techniques in classical networks cannot be directly applied in the quantum setting. In addition, entanglement, which will serve as the fundamental resource in quantum networking, still faces inherent technical challenges in its generation, storage, and distribution, despite its beneficial properties that enable flexible networking.
All of those factors lead to time-varying entanglement network topologies that are dependent on the availability of entanglement and its manipulation, thereby creating a highly dynamic condition. In addition, practical quantum networks should accommodate an increasing number of nodes and a wide range of network sizes. In such networks, identifying participating nodes, determining the source and destination of the communicating parties, allocating resources among nodes, and supporting many other functionalities become even more challenging. These tasks all rely on an addressing mechanism, where directing to an incorrect address may have a significant impact. Hence, addressing should operate in a lossless manner and remain valid as the network grows, despite the steady changes in the entanglement network’s topology. Until recently, only a limited number of studies have investigated the problem of addressing within quantum networks [6]– [10]. Among these works, the initial concepts of addressing were introduced in [6] in the context of the quantum internet. It proposed a quantum-native addressing functionality that shifts from a classical location-aware addressing scheme to one built on an entangled overlay network via link augmentation. In contrast, [7] proposed a classical–quantum hybrid frame structure that carries classical addresses for quantum data payloads. The work in [8] extends quantum-native addressing to enable scalable, compact routing, in which the matching-and-forward operation is performed via quantum address splitting using Schrödinger’s oracle. Unlike [8], which uses a superposition of node identifiers to achieve scalability, the work in [9] uses addressing to perform the superposition of tasks. Meanwhile, [10] explores quantum addresses that can be in entangled states to facilitate quantum routing. While these works establish important mechanisms for quantum addressing, they do not fundamentally resolve the question of how the address should be represented so that it is flexible enough to accommodate the structure and dynamics of the networks while satisfying the quantum-physical constraints. In this work, we approach this question from an information-theoretic coding perspective. In this context, naming or labeling a set of objects is essentially a natural instance of a source coding task [11]. Therefore, we use this coding-theoretic lens and show that the challenge of addressing in quantum networks can be viewed as a lossless coding problem, where address should be represented as decodable quantum states and processed coherently to remain meaningful under a dynamic environment. To that end, we propose a lossless source coding scheme
Tier -2
where V is the set of nodes with cardinality |V | = n and E ⊆ V ×V is the set of quantum links that physically connect the nodes. The node set V is bipartitioned into disjoint two-tier network layers as depicted in Fig. 1. Each layer is designed to serve different functions and has distinct characteristics [8]. In the tier-1 layer we have a set of user nodes that need to consume entanglement resources. It induces a subgraph given by
Lay er
Tier -1
Lay er
Gu = G[Vu ] = (Vu , Eu ), with |Vu | = nu .
Quantum physical link
ESP Node
User Node
In the tier-2 layer we have a set of Entanglement Service Provider (ESP) nodes, which is responsible for distributing entanglement to user nodes in the tier-1 layer. The tier-2 layer induces a subgraph given by Ge = G[Ve ] = (Ve , Ee ), with |Ve | = ne .
Fig. 1. A simple hierarchical quantum network model considered in this paper. Network nodes are partitioned into two functional layers: 1) the tier-2 layer serves as an entanglement service provider (ESP), and 2) the tier-1 layer consists of user nodes organized into clusters that consume the distributed entanglement.
II. S YSTEM M ODEL We consider a quantum network represented as a graph G = (V, E),
(1)
(3)
We require that all nodes be partitioned into these two layers in such a way that the partition satisfies V = Ve ∪ Vu , Ve ∩ Vu = ∅.
for quantum networks, particularly those with a hierarchical structures, to support scalability. The main contributions of this work can be summarized as follows: 1) We provide a framework for addressing in quantum networks by treating node identification as an informationtheoretic lossless source coding problem. 2) We design an isometric hierarchical encoder and its adjoint decoder, and prove unique decodability and coherent (measurement-free) reconstruction. 3) We develop a concrete encoding procedure that uses spectral diagonalization and Huffman-based prefix/suffix codes to produce codewords suitable for dynamical addressing. 4) We demonstrate the scheme numerically on a toy example of a 13-node network, showing negligible recovery and isometry errors, and validate a hierarchical lossless encoding in practice. The remainder of the paper is organized as follows. Section II introduces the system model considered in this paper. Section III formulates the addressing problem and states the design requirements. Section IV develops the core contribution, that is, a hierarchical lossless-coding construction, including the prefix–suffix structure and the isometric encoder/decoder. Section V presents a concrete, implementable encoding procedure (Huffman-based spectral coding) and discusses practical design choices. Section VI illustrates the approach with a numerical example that validates the construction on a small network. Finally, Section VII summarizes the results and outlines directions for future work.
(2)
(4)
To ensure the network functions are scalable as the network grows, we further partition all nodes in V into ne clusters. We denote the resulting j-th cluster by the subgraph Gj = (Vj , Ej ), for j = 1, . . . , ne ,
(5)
such that ne [ Vj , with Vj ∩ Vj ′ = ∅ for all j ̸= j ′ . V = j=1
Thus, every node vi ∈ V belongs to exactly one cluster Vj . To capture a more realistic modeling of a heterogeneous network scenario, we allow the clusters to have varying sizes |Vj |. In this work, we consider that each cluster Vj is served by exactly one ESP node ve,j ∈ Ve = {ve,1 , . . . , ve,ne }. Accordingly, each cluster j is composed of the ESP node ve,j and a (possibly empty) set of user nodes Vu,j ⊆ Vu , i.e., Vj = {ve,j } ∪ Vu,j . where Vu,j denotes the set of user node in cluster j. We require that each ESP node belongs to the cluster itself, i.e. ve,j ∈ Vj for all j. This allows cluster label and its representative coincide, thereby simplifying the constructions of the hierarchical encoder and decoder in Sections IV–V. The structure induces the node-cluster association for any node v ∈ V , given by µ : V → {1, . . . , ne }, µ(v) = j iff v ∈ Vj ,
(6)
which has a natural two-level hierarchy for addressing. Further, each user node vi ∈ V is assigned a unique identifier, referred to as an address, drawn from an address space A. The specific properties and structure of A will be presented in Section IV.
III. P ROBLEM F ORMULATION In the following, we aim to develop an addressing scheme that simultaneously satisfies the required quantum properties and the constraints imposed at the network level. From a quantum perspective, the addressing mechanism must support a representation that can be embedded into quantum states and coherently manipulated by quantum operations. This requirement arises because address information may accompany quantum data or entanglement in various quantum-native network functionalities and, therefore, must remain compatible with the laws of quantum mechanics. At the same time, the addressing scheme must remain efficient at the network level. In particular, the address representation should be compact to avoid excessive overhead, while also supporting hierarchical identification and accommodating heterogeneous cluster sizes. In some cases, only the cluster identity may be needed, rather than the individual identity. Moreover, because quantum network topologies may evolve as entanglement links are created, consumed, or reconfigured, address assignment must be configurable and remain valid under dynamic conditions. Hence, scalability and flexibility are essential properties of the addressing scheme. Specifically, we seek to construct an injective addressing function f : V → A, (7) where A denotes the quantum address space. The injectivity of f ensures that each node in the network can be uniquely identified from its associated quantum address. To satisfy the network constraint, it is required that for each node vi ∈ V , the corresponding quantum address |ci ⟩ ∈ A can be expressed as |ci ⟩ = |κj σij ⟩ ,
(8)
This source message |vi ⟩ is then encoded into a codeword |ci ⟩ by a lossless code c. In this manner, each node vi ∈ V , can be assigned a distinct codeword |ci ⟩ = |c(vi )⟩ ∈ A that represents its address, and the resulting set of codewords is uniquely decodable 1 . In a quantum setting, this assignment must be implemented as an isometric linear map satisfying ⟨v|v ′ ⟩ = ⟨c(v)|c(v ′ )⟩, for all v, v ′ ∈ V,
(9)
which ensures that the encoding is physically realizable [13], [14]. To this end, we use the quantum lossless coding framework introduced by Boström [11], adapting and extending it to suit the specific requirements of our setting. A. Quantum Message Space To represent the source message |vi ⟩ ∈ V and construct the corresponding codewords |ci ⟩ ∈ A, we work with a twodimensional symbol space, that is, a Hilbert space H with the computational basis B = {|x⟩ : x ∈ {0, 1}}.
(10)
Any quantum message |v⟩ of length ℓ has then a form of quantum bit string, which is composed of quantum bits via tensor multiplication |v⟩ = |x1 x2 . . . xℓ ⟩ :=
ℓ O
|xk ⟩ ,
(11)
k=1
where |xk ⟩ ∈ B. Such a quantum message lives in ℓ-fold tensor product of Hilbert space H⊗ℓ :=
ℓ O
H = span(B ℓ ),
(12)
k=1
where |κj ⟩ denotes the prefix address for identifying the jth cluster and |σij ⟩ denotes the suffix address for identifying the i-th user node in the j-th cluster. Taking into account the heterogeneity in cluster sizes, the lengths of the prefix ℓ(κj )and the length of the suffix ℓ(σij ) are both treated as variable parameters. IV. H IERARCHICAL L OSSLEES C ODING To tackle the problem of addressing, we propose to design a scheme based on a lossless source coding approach. We model the node set V as a discrete information source, which is described by an ensemble V = {p, V}. Thereby, each node vi ∈ V , indexed by i = 1, . . . , n, is represented by a quantum message |vi ⟩ ∈ V that is prepared by the source with probability p(vi ) > 0, for all vi ∈ V . In practical scenarios, the probability p(vi ) can be specified, for example, based on empirical knowledge or on measurement data from the traffic observed at node vi . Since each node corresponds to a distinct physical entity, the specific source message |vi ⟩ prepared for node vi is known a priori. A case that is referred to as visible coding [12].
equipped with the message basis ( ) ℓ O ℓ B = |ω⟩ := |ωk ⟩ : |ωk ⟩ ∈ B .
(13)
k=1
To support variable-length codewords, we need a larger space, for which [11] recommends using the Fock space. For practical deployment, we can restrict the Fock space to a bounded message space H
⊕ℓmax
:=
ℓM max
H⊗ℓ ,
(14)
ℓ=0
that is, the direct sum of all block message spaces up to length ℓmax . In our setting, this space serves as the bounded quantum bitstring space. For our purpose, we therefore choose our address space A as a subspace of H⊕ℓmax . 1 The source message |v ⟩ attached to node v can be seen as a MAC i i address, whereas the codeword |ci (vi )⟩ plays the role of an IP address in a classical network.
B. Encoder ⊗r
Let V ⊂ H be the source message space of fixed length r, spanned by the message basis BVr . In accordance with the scheme proposed by Boström [11], we define a lossless quantum code as an isometric embedding c : V −→ A ⊂ H
⊕ℓmax
,
(15)
which associates each source message |v⟩ with a distinct address codeword |c(v)⟩ such that |c(v)⟩ ̸= |c(v ′ )⟩
whenever
|v⟩ ̸= |v ′ ⟩ .
(16)
This encoding is realized by the isometric encoding operator X C := |c(ω)⟩ ⟨ω| . (17) r ω∈BV
Recall that the operator C from V to A is isometry if C † C = IV and CC † = ΠA , where ΠA is the projector to its range A [14]. Therefore, C maps the orthonormal basis BVr of source messages onto a corresponding set of mutually orthogonal codewords in the address subspace A. Although the encoder is built on basis BVr , linearity ensures the codeword address of any node v is obtained by applying the encoder to the source message for that node, given by |c(v)⟩ = C |v⟩ ,
for all |v⟩ ∈ V.
(18)
Consequently, once the encoder has been specified, it can be used to perform lossless encoding of any source message in V of an arbitrary node v ∈ V .2 However, to accommodate the hierarchical addressing structure defined in (8), the encoder (17) must be decomposed into encoders that together form a hierarchical structure. To that end, we require a hierarchical code ĉ so that the address codeword can be expressed as |ĉ(v)⟩ = |κ(v)σ(v)⟩ ,
(19)
where κ is a prefix code and σ is a suffix code. Additionally, it is essential to ensure that the hierarchical encoder maintains the isometric property. By recognizing that the source message space can be decomposed as V=
ne M
Vj ,
Vj := span{|ω⟩ : ω ∈ Vj },
(20)
j=0
we may characterize the prefix and suffix codes as follows. 1) Prefix Code : The prefix code κ assigns the source message of any node v ∈ V to a prefix address given by κ : Ve −→ Ae ⊂ H⊕ℓmax ,
(21)
where Ve ⊂ V denotes the source message space associated with the ESP set Ve , and Ae is the code space of the prefix addresses. Our objective is to employ this code in such a way that all nodes belonging to the same cluster Vj are assigned 2 This mechanism enables dynamic address assignment for a collection of
nodes in a manner analogous to the Dynamic Host Configuration Protocol (DHCP) in classical communication networks.
an identical prefix |κj ⟩, while nodes in distinct clusters are assigned mutually distinct prefixes. This prefix code κ induces an isometric prefix-encoding operator X |κ(ω)⟩ ⟨ω| , (22) K := ω∈BVe
which maps basis message |ω⟩ to its cluster prefix. Thus, we can express (23) κµ(v) = |κ(v)⟩ = K |v⟩ , where µ denotes the node–cluster mapping defined in (6), which returns the index j whenever the node v belongs to the cluster Vj . By assuming |v⟩ ∈ Ve is normalized, we have ∥K |v⟩ ∥ = ∥ |κ(v)⟩ ∥ = 1. 2) Suffix Code: We now turn to the characterization of the suffix code σ, which provides an intra-cluster identifier that distinguishes nodes within the same cluster. For each cluster Vj , we introduce a cluster-specific suffix code σj : Vj −→ Aj ⊂ H⊕ℓj ,
(24)
where Aj is the code space of the the suffix address in cluster j and ℓj denotes the maximum length of suffix address in cluster j. The suffix mapping is defined as ( σij , v ∈ Vu,j , (25) σj (v) := ∅, v = ve,j , such that a suffix address σij is assigned whenever the node v belongs to cluster j and is not the ESP node, whereas no address is assigned when v is the ESP node ve,j . Here, equation (25) defines the abstract intra-cluster suffix assignment, which is formulated without imposing any operational constraints. This suffix code σ induces an isometric suffix-encoding operator X Sj := |σj (ω)⟩ ⟨ω| , (26) ω∈BVj
such that we can write |σj (v)⟩ = Sj |v⟩
(27)
for all |v⟩ ∈ Vj , mapping each node to its intra-cluster suffix address. By hierarchically composing the prefix and suffix encoders, we obtain the complete hierarchical encoder, formulated as follows. Proposition 1 (Hierarchical Encoder). Let K and Sj are defined as in (22) and (26) respectively. Then, we have a unique isometry map ne M ĉ : V → Ae ⊗ Aj j=1
given by the hierarchical encoder b= C
ne M
(|κj ⟩ ⟨κj | ⊗ Sj ) ,
j=1
b |v⟩ = |ĉ(v)⟩ = |κj σij ⟩ for all v ∈ V . that satisfies C
Proof. Let fix an arbitrary cluster j. By construction, for any j and v ∈ Vj we want to have the codeword
Ve
|ĉ(v)⟩ = |κj σj (v)⟩ = |κj ⟩ ⊗ |σj (v)⟩ . On the subspace Vj , the suffix is implemented by any isometry Sj , such that |ĉ(v)⟩ = |κj ⟩ ⊗ Sj |v⟩ ,
Since K and Sj are isometries and |κj ⟩ is normalized, it bj is an isometry on Vj . follows that C Next, the source space decomposes as a direct sum of orthogonal subspaces as in (20), which in turn induces the following decomposition of the address space: Ab =
ne M j=1
Abj ,
ne M
(|κj ⟩ ⟨κj | ⊗ Sj ) .
j=1
Further, since each Sj is an isometry and |κj ⟩ ⟨κj | is an orthogonal projector, their direct sum defines an isometry. b |v⟩ = |ĉ(v)⟩ for all By construction this operator satisfies C v ∈V.
C. Decoder and Unique Decodability b is an isometry from V into Since the hierarchical encoder C ⊕ℓmax b A⊂H , its inverse is well defined on its range. Within the framework of quantum information theory, the inverse of an isometric encoding is given by the adjoint operator restricted to the corresponding code subspace [14]. Thus, the b can be formulated as follows. hierarchical decoder of C b : V → Ab be the Proposition 2 (Hierarchical Decoder). Let C hierarchical encoder of Proposition 1, with V=
ne M j=1
Vj ,
Ab =
ne M j=1
Abj ,
Abj := |κj ⟩ ⊗ Aj .
For each cluster j, define the local decoder b j : Abj → Vj , D
b j := |κj ⟩ ⟨κj | ⊗ S † D j
Then the global hierarchical decoder b : Ab → V, D
b := D
ne M j=1
bj = C b† D
bC b = IV , and the hierarchical code is uniquely satisfies D decodable both locally and globally.
V2 V=
L
. . . Vne
j
Ae
b Encoder C
b1 A
Vj
b2 . . . Abne A
b= A
L
j
bj A
Fig. 2. Hierarchical decomposition of the source space and its image under b which maps each subspace Vj into mutually orthogonal isometric encoder C, bj address subspace A
.
Proof. Fix a cluster j. On Vj , the encoder of Proposition 1 is given by bj : Vj → Abj C
bj := |κj ⟩ ⟨κj | ⊗ Sj , C
where Sj is an isometry. Consequently,
bj C bj = (|κj ⟩ ⟨κj | ⊗ S † )(|κj ⟩ ⟨κj | ⊗ Sj ) D j
with Abj := |κj ⟩ ⊗ Aj .
Consequently, the global hierarchical encoder must be blockdiagonal with respect to this decomposition, and hence it can be written as b= C
V1
v ∈ Vj ,
Thus, on Vj , a local encoder that implements this code is given by bj = |κj ⟩ ⟨κj | ⊗ Sj . C
Address / codeword space
Source space
= (|κj ⟩ ⟨κj |)2 ⊗ Sj† Sj = |κj ⟩ ⟨κj | ⊗ IVj .
Therefore, upon restriction to the subspace Vj we obtain bj C bj = IV . D j
In particular, for every |v⟩ ∈ Vj it follows that bj C bj |v⟩ = |v⟩ , D
which demonstrates the decodablity on each cluster subspace Vj . Using the direct-sum decompositions M M V= Vj , Ab = Abj , j
j
the global encoder and decoder decompose as M M b= bj , b= bj . C C D D j
Thus,
bC b= D
M j
j
bj C bj = D
M
IVj = IV ,
j
b =C b † is which proves global decodability and shows that D ′ b b b a left inverse of C on V. In particular, if C |v⟩ = C |v ⟩, then b yields applying D bC b |v⟩ = D bC b |v ′ ⟩ = |v ′ ⟩ , |v⟩ = D
so the encoder is injective, and hence the hierarchical code is uniquely decodable. These encoder and decoder appear to resemble the Ahlswede and Cai scheme [15] in that the source space is partitioned into subspaces, and each equipped with its own local isometry. In that scheme, the information regarding the subspace to which the source message belongs must be transmitted through a classical side channel. Our construction differs by embedding this information directly into the prefix
of codeword address, as illustrated in Fig 2. In addition, the hierarchical encoder and decoder described above naturally extends to multiple layers. The suffix code of one layer may be treated as the prefix code of the next, inducing a further direct-sum decomposition of the corresponding code subspaces. This recursive construction leads to a multi-layer hierarchical code, but a full treatment lies beyond the scope of the present paper. V. E NCODING P ROCEDURE Having presented the construction of the hierarchical encoder and decoder, we now proceed to describe a procedure that we propose for their implementation. The procedure is enabled by considering the following key observations. Observation 1 (Visible coding). Since each node is a distinct physical entity, we know which node v ∈ V corresponds to |v⟩ ∈ V when preparing the source message. Therefore, the encoder operates in the visible coding regime, where the output state of the source message to be encoded is known a priori [11], [12]. In our setting, visible coding allows the prefix and suffix code, κ and σj , to perform one-to-one mapping between the source message and the corresponding prefix and suffix addresses. For such a mapping, we use a prefix-free code in our encoding procedure, motivated by its desirable properties given in the following observation. Observation 2 (Prefix-free code). A prefix-free code assigns codewords in such a way that no codeword is a prefix of another. When embedded as basis in the quantum bitstring space H⊕ℓmax , these codewords form a prefix-free set in the sense of Definition 3.1 of Müller–Rogers [16]. In particular, they occupy mutually orthogonal subspaces and therefore span a prefix-free Hilbert subspace. Moreover, prefix-free codewords have an additional structural benefit for our hierarchical construction, as given in the follwoing observation. Observation 3 (Concatenation prefix-free codewords). Prefixfree codewords have definite length and are therefore length eigenstates in H⊕ℓmax . By Lemma 2.5 of Müller–Rogers [16], the concatenation of two variable-length quantum codewords is an isometry whenever the first component is a length eigenstate. This observation is consistent with Proposition 1, where b we proved that the structure of the hierarchical encoder C preserves the isometry. Hence, by implementing prefix-free codewords, Observation 3 confirms that for each cluster j, the concatenated map |κj ⟩ ⟨κj | ⊗ Sbj is an isometry. With those observations in place, we propose the following encoding procedure. It may be viewed as an extension of the Bostrom [11] and Ahlswede [15] scheme, in which we incorporate a hierarchical structure and, in addition, modify the underlying mapping by embedding a prefix-free code, such as Huffman code, into the basis codewords. In particular,
under the Huffman scheme, the source probability naturally corresponds to the eigenvalues of the source message matrix in the quantum setting. Once the eigenvalues are treated as classical probabilities, a standard Huffman algorithm can be used to generate classical codewords. This allows for a oneto-one mapping where each Huffman codeword is assigned to the associated eigenstate [17]. Procedure 1 (Huffman-based Hierarchical Encoding). Suppose the ensemble {p(v), |v⟩}v∈V represents a discrete quantum source. 1) Prefix stage. (a) Diagonalize the local ESP source matrix X ρe = p(v) |v⟩ ⟨v| v∈Ve
e and use its eigenvectors {|ωk ⟩}dk=1 as the orthonormal basis of Ve ⊂ V. e denote the corresponding eigenvalues of (b) Let {λk }dk=1 ρe . Use these eigenvalues as the canonical probabilities for coding:
p(ωk ) := λk = ⟨ωk | ρe |ωk ⟩ , where k = 1, . . . , de . (c) Construct a binary Huffman code h : {1, . . . , de } → {0, 1}ℓk of variable length ℓk for the probability distribution {λk }, and represent each outcome by the quantum state |h(ωk )⟩. (d) Define the prefix isometry K=
de X
|h(ωk )⟩ ⟨ωk | .
k=1
P For each prefix node, express |vej ⟩ = k αjk |ωk ⟩ so that |κj ⟩ = K |vej ⟩. 2) Suffix stage. (a) Choose vej whose corresponding |c(vej )⟩ is alength eigenstate. Then, set Vj = span {|vej ⟩} ∪ Vuj for a prescribed partition Vuj ⊆ V \Ve , where j = 1, . . . , ne . (b) For each P cluster j, form the local source matrix ρj = v∈Vj p(v) |v⟩ ⟨v| and diagonalize it. Use the d
j eigenvectors {|φj,k ⟩}k=1 as the orthonormal basis of Vj . dj (c) Let {λj,k }k=1 be the eigenvalues of ρj . Use these eigenvalues as the canonical probabilities:
p(φj,k ) := λj,k = ⟨φj,k | ρj |φj,k ⟩ , where k = 1, . . . , dj . (d) Construct a local binary Huffman code hj for the distribution {λj,k } and represent each outcome by the quantum length eigenstate |hj (φj,k )⟩.
VI. N UMERICAL E XAMPLE
(e) Define the suffix isometry Sj =
dj X
|hj (φj,k )⟩ ⟨φj,k | .
k=1
3) Hierarchical encoding. For any |v⟩ ∈ Vj , bj |v⟩ = |κj ⟩ ⊗ Sj |v⟩ . |b c(v)⟩ = C
4) Hierarchical decoding. Decoding is performed coherently by the adjoint isometry b† = C
ne M j=1
b† , C j
which reconstructs the state of original source message b † |b via |v⟩ = C c(v)⟩.
The procedure described above is only one of several possible approaches. In our case, when a local density operator ρ possesses degenerate eigenvalues, we may arbitrarily select any orthonormal basis within each degenerate eigenspace. Alternatively, for example, a Gram–Schmidt procedure can be employed when starting from a set of non-orthogonal source messages, as suggested in [11]. Furthermore, any other prefix-free code can also be employed effectively. Our choice of the Huffman code is motivated by its optimality in attaining the Shannon entropy in the classical case [18]. Depending on the statistical distribution of the source messages (traffic), alternative prefix codes may be more advantageous. For instance, the Golomb code is particularly well suited when the source messages follow a geometric distribution. Several implementation strategies are available for realizing the encoders K and Sj , particularly to accommodate the space H⊕ℓmax , whose elements are vectors of variable length. One approach is the zero-extended form (ZEF) of Schumacher and Westmoreland [19], which embeds indeterminate-length strings into fixed-length registers by padding with zeros. An alternative is to embed states into the tape Hilbert space of a quantum Turing machine (QTM) and implement coherent write/read operations (prefix parsing or delimiters) on the tape [16]. One of useful features in our procedure, the source message can be coherently reconstructed without the need for any projective measurements due to the isometry. Moreover, in contrast to the schemes proposed by Boström [11] and Ahlswede [15], our construction does not require any classical communication of codeword lengths or subspace indices for decoding the address codewords. This follows from the fact that the prefix, which is appended to each suffix, already encodes the subspace indices and is chosen to be an orthogonal length eigenstate. As investigated by Müller and Rogers [16], the concatenation for arbitrary non-length-eigenstate codewords remains an open problem. Thus, for the time being, we leave the cluster empty of user nodes if the prefix is not a length eigenstate to avoid operational complexity. The expense of this construction is a likely increase in the total codeword length due to the appending of the prefix.
To demonstrate the effectiveness of hierarchical coding, we apply Procedure 1 to a small network as a toy example. We consider a network comprising 13 nodes. Among these, 3 nodes function as ESP nodes within the tier-2 layer. All nodes are partitioned into 3 clusters, each managed by a single ESP. In the tier-1 layer, we specify that each cluster includes 6, 4, and 0 user nodes, respectively. We model the network nodes as a discrete information source specified by the ensemble V = {p, V}. Each cluster is modeled as an independent source, so the joint distribution factorizes as P (V ) =
3 Y
P (Vj ),
j=1
which is valid because the cluster preparations are independent and no cross-cluster quantum correlations are present. Consequently, the ensemble decomposes into independent cluster components and separate prefix and local suffix codes are justified. a) Prefix stage: Suppose the ESP node probabilities are p(ve1 ) = 0.5,
p(ve2 ) = 0.3,
p(ve3 ) = 0.2.
We work in a two-dimensional Hilbert space Ve span{|0⟩ , |1⟩} and have prepared the ESP states |ve1 ⟩ = |0⟩ ,
|ve2 ⟩ = |1⟩ ,
=
|ve3 ⟩ = √12 (|0⟩ + |1⟩).
The local ESP source matrix is ρe =
3 X j=1
0.6 p(vej ) |vej ⟩ ⟨vej | = 0.1
0.1 . 0.4
Following Procedure 1, we diagonalize ρe , and obtain the eigenpairs 0.924 λ1 ≈ 0.641, |ω1 ⟩ ≈ , 0.383 λ2 ≈ 0.358,
|ω2 ⟩ ≈
−0.383 , 0.924
with ⟨ω1 |ω2 ⟩ = δ12 . We then use the eigenvalue distribution {λ1 , λ2 } as the Huffman probabilities. For two symbols the natural prefix assignment is |h(ω1 )⟩ = |0⟩ ,
|h(ω2 )⟩ = |1⟩ ,
and the prefix isometry is K = |0⟩ ⟨ω1 | + |1⟩ ⟨ω2 | . The address of ve3 is obtained by |k(ve3 )⟩ = K |ve3 ⟩.
TABLE I T HE NODES UNDER CONSIDERATION GROUPED BY CLUSTER , WITH THEIR ADDRESS ASSIGNMENT. Node p(v)
p(v|j) |v⟩
1
v11
0.15
0.30
|00⟩
1
v21
0.07
0.14
|01⟩
1
v31
0.06
0.12
|10⟩
1
v41
0.06
0.12
|11⟩
1
v51
0.05
0.10
0.71 |01⟩ + 0.71 |11⟩
1
v61
0.05
0.10
0.71 |10⟩ + 0.71 |11⟩
1
v71
0.06
0.12
2 2 2 2 2
v12 v22 v32 v42 v52
0.08 0.06 0.05 0.06 0.05
3
v13
0.2
Cluster
|ĉ(v)⟩
1
0.50 |00⟩ + 0.50 |01⟩ + 0.50 |10⟩ + 0.50 |11⟩
0.267 0.2 0.167 0.2 0.167
|00⟩ |01⟩ |10⟩ |11⟩ 0.71 |01⟩ + 0.71 |11⟩
−0.27 |0010⟩ − 0.27 |0110⟩ + 0.65 |1010⟩ + 0.65 |1110⟩ 0.27 |0011⟩ − 0.27 |0100⟩ − 0.65 |1011⟩ + 0.65 |1100⟩ −0.27 |0010⟩ + 0.27 |0110⟩ + 0.65 |1010⟩ − 0.65 |1110⟩ −0.27 |0011⟩ − 0.27 |0100⟩ + 0.65 |1011⟩ + 0.65 |1100⟩ −0.38 |0100⟩ + 0.92 |1100⟩
1 1 1 1 1
1
0.71 |0⟩ + 0.71 |1⟩
0.71 |0⟩ + 0.71 |1⟩
1
b) Suffix stage (cluster 1): Assume cluster 1 has conditional probabilities as specified in Table I. We use a four-dimensional Hilbert space V1 = span{|00⟩ , |01⟩ , |10⟩ , |11⟩} and prepare the node states (including the ESP) as source messages |v⟩ in Table I. The local source matrix is computed as 7 X p(vi1 | ve1 ) |vi1 ⟩ ⟨vi1 | ρ1 = i=1 0.33 0.03 ≈ 0.03 0.03
Fidelity
0.56 |0010⟩ + 0.37 |0011⟩ − 0.64 |0100⟩ + 0.06 |0110⟩ + 0.23 |1010⟩ + 0.15 |1011⟩ − 0.26 |1100⟩ + 0.02 |1110⟩ −0.71 |0010⟩+0.08 |0011⟩−0.58 |0100⟩−0.13 |0110⟩− 0.29 |1010⟩ + 0.03 |1011⟩ − 0.24 |1100⟩ − 0.05 |1110⟩ 0.14 |0010⟩ − 0.82 |0011⟩ − 0.33 |0100⟩ + 0.22 |0110⟩ + 0.06 |1010⟩ − 0.34 |1011⟩ − 0.14 |1100⟩ + 0.09 |1110⟩ −0.17 |0010⟩+0.19 |0011⟩+0.04 |0100⟩+0.89 |0110⟩− 0.07 |1010⟩ + 0.08 |1011⟩ + 0.02 |1100⟩ + 0.37 |1110⟩ −0.62 |0010⟩+0.19 |0011⟩−0.38 |0100⟩+0.54 |0110⟩− 0.26 |1010⟩ + 0.08 |1011⟩ − 0.16 |1100⟩ + 0.22 |1110⟩ −0.03 |0010⟩−0.45 |0011⟩−0.21 |0100⟩+0.78 |0110⟩− 0.01 |1010⟩ − 0.18 |1011⟩ − 0.09 |1100⟩ + 0.32 |1110⟩ −0.09 |0010⟩−0.09 |0011⟩−0.75 |0100⟩+0.52 |0110⟩− 0.04 |1010⟩ − 0.04 |1011⟩ − 0.31 |1100⟩ + 0.21 |1110⟩
0.03 0.22 0.03 0.08
0.03 0.03 0.20 0.08
0.03 0.08 . 0.08 0.25
1 We diagonalize ρ1 and let {λk,1 , |φk,1 ⟩}dk=1 denote its eigenpairs. Using the spectral rule we take the eigenvalues as the Huffman probabilities. For this instance the diagonalization yields (rounded)
λ1,1 ≈ 0.40, λ2,1 ≈ 0.29, λ3,1 ≈ 0.18, λ4,1 ≈ 0.13, which sum to unity up to rounding, and the orthonormal basis vectors |φ1k ⟩ for V1 stacked as the columns of matrix 0.606 −0.794 −0.004 0.053 0.419 0.296 −0.754 −0.410 . Φ1 = 0.382 0.247 0.643 −0.616 0.558 0.470 0.130 0.670
Next, we construct a local Huffman code h1 that assigns a shorter codeword to the larger eigenvalue. One valid assignment is |h1 (φ1,1 )⟩ = |1⟩ ,
|h1 (φ1,2 )⟩ = |01⟩ ,
|h1 (φ1,3 )⟩ = |000⟩ ,
|h1 (φ1,4 )⟩ = |001⟩ ,
1 1 1 1 1 1
and the suffix isometry is S1 = |0⟩ ⟨φ1,1 | + |01⟩ ⟨φ1,2 | + |000⟩ ⟨φ1,3 | + |001⟩ ⟨φ1,4 | . For the implementation of the computation of the encoder S1 , we employ the ZEF scheme [11], [19], which is selected due to its relatively straightforward and practically tractable realization. It provides a finite-dimensional Hilbert space and simple matrix representations of the encoders. In addition, we may introduce a row-selection (permutation) matrix P to aggregate the utilized padded indices while preserving the isometric property. In this setting, we can express S1 = P Γ1 Φ†1 , where Γ1 is constructed such that its columns are given by the ZEF of the basis codewords |h(φ1,k )⟩. c) Suffix stage (cluster 2): In an analogous manner, we can compute the hierarchical encoder and decoder for cluster 2. Here, we choose the same basis as in cluster 1 to describe the source messages in cluster 2. With the probabilities listed in Table I, we obtain the corresponding source matrix ρ2 =
5 X
p(vi2 | ve2 ) |vi2 ⟩ ⟨vi2 |
i=1
0.27 0 ≈ 0 0
0 0.28 0 0.08
0 0 0.17 0
0 0.08 . 0 0.28
Accordingly, the spectral decomposition of the source matrix ρ2 yields its associated eigenvalues, given by λ1,2 ≈ 0.37, λ2,2 ≈ 0.27, λ3,2 ≈ 0.20, λ4,2 ≈ 0.17,
and its eigenvectors, stacked as columns in the matrix Φ2 , given by 0 1 0 0 0.70 0 0.70 0 . Φ2 = 0 0 0 −1 0.70 0 −0.70 0 As in cluster 1, we assign a Huffman codeword to each basis vector based on its eigenvalue distribution to obtain the suffix encoder S2 = |0⟩ ⟨φ2,1 | + |01⟩ ⟨φ2,2 | + |000⟩ ⟨φ2,3 | + |001⟩ ⟨φ2,4 | . d) Hierarchical encoding: For any |v⟩ ∈ Vj , with j = 1, 2 , we obtain the address codeword |b c(v)⟩ = |κj ⟩ ⊗ Sj |v⟩ , as provided in Table I. As shown in the table, the encoded addresses are represented as a superposition of 4-qubit strings that are constructed from the tensor product of the basis codewords assigned to the prefix and suffix. The encoded prefix ensures that codewords originating from different clusters lie in orthogonal subspaces, thereby preventing overlaps. This is shown by the evidence that even identical source messages appearing in both clusters are mapped to distinct address codewords and can be uniquely decoded. Meanwhile, within each cluster, the source messages and their corresponding codewords do not need to be orthogonal. The only requirement is an isometric encoder on the entire cluster subspace. e) Decoding and Evaluation: The source message |v⟩ can be recovered from the address codeword by applying the b † , computed as decoder D = C b † |b |v⟩ = C c(v)⟩ .
For each source message |v⟩, we evaluate its fidelity as b † |b F (v) = | ⟨v| C c(v)⟩ |2 ,
which serves as a quantitative measure of the accuracy of the proposed address coding scheme. As shown in Table I, it is possible to attain fidelity 1, which confirms the degree to which the representation is lossless. However, it is important to note that we examined this address coding in an ideal, noise-free environment. An extension to a noisy setting would be an interesting direction for future work, for example, by equipping the address with an appropriate quantum errorcorrecting scheme. In addition, we also asses the isometry error of any encoder X by computing ϵiso (X) := ∥X † X − I∥2 . As shown in Table II, each encoder preserves its isometry with only a negligible error due to finite numerical precision. Altogether, the numerical evaluation confirms the isometry property stated in Proposition 1, as well as the decodability guarantee established in Proposition 2.
TABLE II N UMERICAL E VALUATION OF THE HIERARCHICAL ENCODERS . Encoder K S1 S2 b1 C b2 C b C
Size 2×2 8×4 8×4 16 × 4 16 × 4 16 × 8
Isometry error 8.48 × 10−8 5.87 × 10−16 6.14 × 10−16 2.54 × 10−7 2.54 × 10−7 1.70 × 10−7
VII. C ONCLUSION We have shown that addressing in quantum networks is naturally framed as a lossless coding problem. By introducing a hierarchical prefix–suffix address space and implementing isometric, prefix-free encoders and their adjoints, we ensure that addresses are uniquely decodable and compatible with coherent processing. Our Huffman-based procedure demonstrates how the address structure can be exploited to produce length-eigenstate codewords, thereby avoiding classical side channels for subspace indices. The design explicitly accommodates heterogeneous cluster sizes and dynamic reconfiguration, allowing address assignment to be adjusted as the entanglement topology evolves. Future work will analyze performance under realistic noise models, explore multi-layer recursive hierarchies, and investigate their use for different network functionalities. Overall, lossless address coding provides a principled, scalable foundation for addressing in next-generation quantum networks. U SE OF AI D ISCLOSURES Some portions of this manuscript were refined using AIassisted tools for grammar, clarity, and organization. All AI outputs were reviewed, edited, and verified by the authors. All concepts and technical content are solely those of the authors. R EFERENCES [1] H. J. Kimble, “The quantum internet,” Nature, vol. 453, no. 7198, pp. 1023–1030, 2008. [2] W. Dür, R. Lamprecht, and S. Heusler, “Towards a quantum internet,” European Journal of Physics, vol. 38, no. 4, p. 043001, 2017. [3] S. Wehner, D. Elkouss, and R. Hanson, “Quantum internet: A vision for the road ahead,” Science, vol. 362, no. 6412, p. eaam9288, 2018. [4] A. S. Cacciapuoti, M. Caleffi, F. Tafuri, F. S. Cataliotti, S. Gherardini, and G. Bianchi, “Quantum internet: Networking challenges in distributed quantum computing,” IEEE Network, vol. 34, no. 1, pp. 137–143, 2020. [5] W. Kozlowski, S. Wehner, R. V. Meter, B. Rijsman, A. S. Cacciapuoti, M. Caleffi, and S. Nagayama, “Architectural principles for a quantum internet,” Internet Engineering Task Force, RFC 9340, March 2023. [Online]. Available: https://www.rfc-editor.org/rfc/rfc9340 [6] A. S. Cacciapuoti, J. Illiano, and M. Caleffi, “Quantum internet addressing,” IEEE Network, vol. 38, no. 1, pp. 104–111, 2024. [7] S. DiAdamo, B. Qi, G. Miller, R. Kompella, and A. Shabani, “Packet switching in quantum networks: A path to the quantum internet,” Phys. Rev. Res., vol. 4, p. 043064, Oct 2022. [Online]. Available: https://link.aps.org/doi/10.1103/PhysRevResearch.4.043064 [8] M. Caleffi and A. S. Cacciapuoti, “Quantum internet architecture: Unlocking quantum-native routing via quantum addressing,” IEEE Transactions on Communications, vol. 74, pp. 3577–3599, 2026. [9] J. Miguel-Ramiro, A. Pirker, and W. Dür, “Genuine quantum networks with superposed tasks and addressing,” npj Quantum Information, vol. 7, no. 135, 2021. [Online]. Available: https: //doi.org/10.1038/s41534-021-00472-5
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