PaperlayerView exact revision on arXiv

Source-linked AI summary

Quantum secret sharing with qudit graph states

Adrian Keet, Ben Fortescue, Damian Markham, Barry C. Sanders

arXiv:1004.4619v3quant-ph

TL;DR

Some access structures cannot be realized by qubit graph-state schemes, motivating secret sharing with higher-dimensional systems. The paper develops prime-dimensional graph-state protocols for classical and quantum secrets over classical and quantum channels, including a (2, 3) case unavailable in the qubit formalism.

  • Problem

    Some access structures cannot be realized by existing qubit graph-state schemes, while higher-dimensional systems can be useful for direct use of larger Hilbert spaces and for rigorous general results.

    Source: §I · p. 1

  • Method

    The paper extends graph-state secret-sharing formalism to prime-dimensional systems, constructing classical- and quantum-secret protocols over classical and quantum channels.

    Source: §I · pp. 1–2

  • Results

    The framework covers CC, CQ, and QQ protocols and includes (2, 3) sharing, which the qubit formalism could not achieve.

    Source: §I · p. 2, §IV · p. 12

  • Takeaways & Limitations

    The framework is a step toward describing schemes for classical and quantum secrets of any size using graph states for general physical systems.

    Source: §IV · p. 12

    Although arbitrary-access quantum schemes can in principle use the formalism, no explicit conversion procedure to graph states is provided, and its use for classical secrets is unclear.

    Source: §IV · p. 13

Abstract

from arXiv · show

We present a unified formalism for threshold quantum secret sharing using graph states of systems with prime dimension. We construct protocols for three varieties of secret sharing: with classical and quantum secrets shared between parties over both classical and quantum channels.

I. INTRODUCTION

The paper extends graph-state secret sharing from qubits to prime-dimensional systems, covering classical and quantum secrets and classical and quantum channels. It focuses on threshold schemes and motivates higher dimensions as useful for access structures unavailable to qubit schemes.

I. INTRODUCTION

  • Secret sharing distributes a classical string or quantum state so only certain collaborating player subsets can recover it, with access and adversary structures identifying the relevant subsets.The introduction notes applications including money transfer and voting.Source: §I · p. 1
  • Higher-dimensional shares can use their larger Hilbert space directly, and some access structures cannot be realized by existing qubit graph-state schemes or any qubit-share scheme.A qutrit implementation is cited as an example of an access structure unavailable with qubit shares.Source: §I · p. 1
  • The formalism extends qubit graph-state protocols to prime-dimensional systems and covers classical and quantum secrets shared over classical and quantum channels.It also demonstrates three-party classical-secret sharing over quantum channels, where at least two parties are required to recover the secret.Source: §I · pp. 1–2
  • A (k, n) threshold scheme lets any k or more players recover the secret, while fewer than k players obtain no knowledge; concatenating threshold protocols can construct arbitrary access structures.The paper considers three varieties: CC, CQ, and QQ schemes, distinguished by the secret type and the channels between dealer and players and among players.Source: §I · pp. 1–2

II. GRAPH STATES FOR QUDITS

  • Graph states are multipartite entangled states with graphical and stabilizer descriptions, used in information processing including computation and error correction.The paper adopts a framework previously considered for higher-dimensional graph states, with notation adapted to secret sharing.Source: §II · p. 2

A. Graph states

The paper represents qudit graph states using weighted undirected graphs over a finite field and summarizes their connections in an adjacency matrix. Its example shows a five-qudit graph in dimension 7.

A. Graph states

  • For prime d > 2, the graph has n vertices joined by weighted edges, with each edge weight drawn from F_d.A zero weight represents no edge, and vertices have no self-edges.Source: A. Graph states · p. 2
  • The n × n adjacency matrix A records edge weights, is symmetric, and has zero diagonal entries.The figure depicts a five-qudit graph in d = 7 alongside its adjacency matrix.Source: A. Graph states · p. 2, Figure 1 · p. 2
  • The graph-state construction is introduced using the computational basis and a two-qudit controlled-Z operator.The supplied passages do not state the construction formula.Source: A. Graph states · p. 2

B. Local operators and labelling

The paper labels qudit graph states by applying local generalized Pauli and S operators to individual vertices. It defines encoded graph states as labelled states with x = 0.

B. Local operators and labelling

  • Generalized Pauli operators use ω = e^2πi/d and satisfy X^d = Z^d = I; the listed operators provide the subset used for graph-state stabilizers and secret-sharing protocols.The operators Z, X, XZ, through XZ^(d−1) have mutually unbiased eigenbases.Source: B. Local operators and labelling · pp. 2–3
  • The S operator commutes with Z and transforms X by conjugation to XZ.The paper specifies that SXS^−1 = XZ.Source: B. Local operators and labelling · p. 3
  • Each vertex receives a label ℓ_i = (z_i, x_i, m_i), represented by applying S^m_i and X^x_i and Z^z_i to its graph-state subsystem.The paper combines the labels across vertices into vectors z, x, and m.Source: B. Local operators and labelling · p. 3
  • The paper applies S operators after X operators, unlike the cited qubit definition, and calls a labelled graph state with x = 0 an encoded graph state.The ordering distinction is stated relative to the earlier qubit definition.Source: B. Local operators and labelling · p. 3

C. Stabilizers

The paper characterizes encoded qudit graph states through stabilizers and shows how applying a stabilizer changes labels without changing the physical state. This equivalence supports reasoning about dependence on selected qudit labels.

C. Stabilizers

  • The encoded graph state satisfies a stabilizer for each vertex, and the full stabilizer set uniquely specifies the state up to a global phase.The stabilizers are tensor products of local operators, so appropriate parties can measure them using local operations and classical communication.Source: C. Stabilizers · p. 3
  • Applying stabilizer K_i changes x_i by 1 and shifts each neighboring z_j by A_ij modulo d, while leaving the physical state equivalent up to global phase.The exponents of the local operators need only be considered modulo d.Source: C. Stabilizers · p. 3
  • Equivalent labelings can therefore be used to determine how the state depends on particular qudit labels.The paper identifies this as an exploitable property, analogous to the cited qubit case.Source: C. Stabilizers · p. 3

D. Local measurements

Local Pauli measurements transform an encoded qudit graph into a reduced labelled graph, providing rules used for the dealer’s measurements in CQ and QQ protocols. The transformation generalizes local complementation, but the paper does not establish the full local-Clifford equivalence property for these rules.

D. Local measurements

  • The dealer’s local measurement leaves a reduced graph state shared by the other parties, with its access structure and labelling determined by the outcome.The section develops these rules for dealer measurements in CQ and QQ protocols.Source: D. Local measurements · pp. 3–4
  • Measuring XZ^m at a vertex removes that vertex and updates edges between its neighbors and the remaining graph labels.The edge update is A_jk → A_jk + mA_ijA_ik.Source: D. Local measurements · pp. 3–4
  • For a Z_i measurement, the reduced graph has vertex i and its incident edges removed.The remaining graph state is denoted |G_ℓ\v_i⟩.Source: D. Local measurements · p. 4
  • In the d = 5 square example, measuring XZ^2 at vertex 1 removes it and gives vertices 2 and 3 an edge of weight 1.The resulting labels include z = 0 and m = 3.Source: Figure 2 · p. 4
  • The XZ^m measurement transformation is an analogue of qubit local complementation, but full local-Clifford equivalence is not established here.The paper notes that this equivalence has been proven elsewhere [14].Source: D. Local measurements · p. 4

E. Dependence and access

The section distinguishes dependence of a reduced state on a secret from the ability to recover it with certainty. It gives an LOCC criterion for access to linear combinations of labels and uses graph structure to identify inaccessible labels.

E. Dependence and access

  • A subset accesses a secret only if it can use a measurement protocol on its reduced state to reveal the secret with certainty.Dependence of that reduced state on the secret alone is insufficient.Source: E. Dependence and access · p. 5
  • A subset can access a weighted sum of labels by LOCC when its supported weights cancel the adjacency contributions from every vertex outside the subset.The accessible value is the corresponding weighted sum of z labels.Source: E. Dependence and access · p. 5
  • A vertex label z_k is accessible by LOCC to a set containing vertex k and all its neighbors.The parties can retrieve it by measuring the corresponding stabilizer K_k.Source: E. Dependence and access · p. 5
  • Tracing out a vertex makes the remaining reduced graph state independent of that vertex’s labels, a property used to establish threshold access requirements.The labels are applied by local Pauli operations on the vertex being traced out.Source: E. Dependence and access · p. 5

1. Label “shuffling”

Label shuffling provides an equivalent description of an encoded graph state in which a label’s dependence is shifted between neighboring vertices. It is a relabelling, not a physical operation on the state.

1. Label “shuffling”

  • In the d = 5 square example, shuffling from vertex 1 sets its z label to 0 and changes the x and z labels at vertices 2 and 3.The displayed values are x_2 = −3/2 = 1 and z_3 = −6/2 = 2 modulo 5.Source: Figure 3 · p. 5, §1 · p. 5
  • Shuffling a label changes its assignment in the graph description without physically changing the encoded state.The relabelled graph state is physically equivalent to the original.Source: §1 · p. 5
  • For neighboring vertices i and j, the paper gives an equivalent relabelling that shifts z dependence from i toward j and its neighbors.This is the neighboring-vertex rule used to demonstrate label shuffling.Source: §1 · p. 5

A. CC scheme

The CC constructions encode classical secrets in labelled qudit graph states and use stabilizer measurements for recovery and label shuffling to establish which coalitions lack access. The section gives threshold examples from (n,n) through (3,5).

A. CC scheme

  • The CC approach encodes a secret dit in graph-state labels and uses joint stabilizer measurements for recovery; shuffling is used to show smaller coalitions lack access.For threshold schemes, the demonstration need only exclude coalitions of size k − 1.Source: A. CC scheme · pp. 5–6
  • Within this formalism, CC protocols provide a basis for the CQ and QQ cases, and the authors extend previously demonstrated qubit protocols to prime-dimensional systems.The classical protocols are described as a useful basis for CQ and QQ protocols.Source: A. CC scheme · p. 6
  • The (n,n) tree protocol lets all n parties recover s from a stabilizer measurement, and shuffling shows that all n are necessary.The secret is encoded on the first vertex’s z label.Source: §1 · p. 6
  • For the (2,3) protocol, each pair can recover s by measuring a stabilizer, while shuffling establishes that no single player can access it.The encoding sets z_2 = 2s and z_3 = s.Source: §2 · p. 6
  • In the (3,4) ring protocol, any three parties can measure s, while sequential shuffles make both adjacent and nonadjacent pairs independent of it.The shuffle for adjacent parties follows the nonadjacent-pair shuffle.Source: §3 · p. 6, Figure 6 · p. 7
  • In the (3,5) ring protocol, any three parties can recover s, while two parties can shuffle its dependence away and are insufficient.The argument covers triples with either three adjacent parties or only two adjacent parties.Source: §4 · p. 7

B. The CQ scheme

The CQ schemes use a dealer’s local qudit measurement to produce a labelled reduced graph state on which players can perform a classical secret-sharing protocol. The section gives constructions for (2, 3), (3, 5), and (n, n) thresholds, with measurement-based recovery and security checks.

B. The CQ scheme

  • Across the CQ protocols, the dealer’s measurement produces a reduced labelled graph state whose labels are correlated with the outcome and support a players’ CC protocol.The dealer and authorized players can also use measurement statistics to verify stabilizers of the original graph state.Source: B. The CQ scheme · p. 7, §1 · p. 8
  • The protocols are secure against certain attacks, including intercept-resend, but not all attacks by dishonest participants; noisy channels are not considered.The authors note that existing noisy-channel protocols could potentially be adapted to this model.Source: B. The CQ scheme · p. 7
  • For an (n, n) CQ protocol, an extended tree state yields a reduced n-party graph state with the dealer’s outcome encoded in player 1’s label.Players recover the secret when their measurement parameters satisfy t1 = tD + t2 + ... + tn, which occurs with probability 1/d.Source: §1 · pp. 7–8
  • In the (2, 3) CQ protocol, the dealer’s qudit is attached to players 2 and 3 with weights two and one, and any authorized pair can sometimes recover the dealer’s result.Authorized pairs randomly choose among possible measurements; the selected bases match the dealer’s with probability 1/d.Source: §2 · pp. 8–9
  • In the (3, 5) CQ protocol, no pair alone can access the dealer’s result, while any three players can recover it.Security checks show the eavesdropper’s state is uncorrelated with the dealer’s measurement result.Source: §3 · p. 10
  • For CQ security checks, the dealer and an authorized set combine measurement results to simulate measurements of stabilizers of the original graph state.For the (3, 5) protocol, successful checks imply that the eavesdropper is uncorrelated with the dealer’s measurement result.Source: §3 · p. 10

C. The QQ scheme

The QQ scheme extends graph-state secret sharing to quantum secrets by encoding them across players’ qudits. The paper gives perfect threshold examples for (2,3) and (3,5), while the (n,n) protocol permits some subthreshold information.

C. The QQ scheme

  • The dealer generalizes the QQ scheme by teleporting a secret qudit into a shared graph state, then communicating the generalized Bell measurement result for player corrections.For suitable graphs, the encoded secret is a superposition of labelled graph states whose labels have the access structure of corresponding CC protocols.Source: C. The QQ scheme · pp. 10–11
  • The (n,n) protocol requires all n players for perfect reconstruction, but fewer than n can still obtain incomplete information about the secret.The reduced states of fewer than n players depend on the secret amplitudes α_j.Source: §1 · p. 11
  • Any pair can recover the secret in the (2,3) protocol, and any single player gets no information, making it a perfect threshold scheme.For d = 3, the state is the qutrit code presented in.Source: §2 · pp. 11–12
  • In the (3,5) construction, each vector z occurs once in the expansion, letting players 3, 4 and 5 measure to project onto a subspace associated with some |xy⟩_1,2.Each such outcome occurs with equal probability 1/d^2.Source: §3 · p. 12
  • In the (3,5) protocol, any three players can recover the secret and complementary pairs obtain no information, yielding a perfect threshold scheme.The construction also describes how players can isolate the encoded secret to a chosen subsystem.Source: §3 · p. 12

IV. CONCLUSION

The paper develops a unified framework for three types of secret sharing using prime-dimensional graph states, including the (2, 3) threshold scheme unavailable in the qubit formalism. Arbitrary threshold schemes and generalizations beyond prime dimensions remain open directions.

IV. CONCLUSION

  • The framework covers three types of secret sharing with prime-dimensional graph states and includes (2, 3) sharing, unavailable in the qubit formalism.It supports classical dits and prime-dimensional quantum states as secrets.Source: §IV · p. 12
  • Whether every achievable threshold scheme can be built in this formalism remains open, as does generalization to arbitrary dimension d.The authors note that arbitrary-dimensional states might allow a richer array of protocols.Source: §IV · p. 12
  • Quantum-secret schemes for arbitrary access structures are possible in principle, but no explicit procedure converts them to graph states, and their use for classical secrets is unclear.The authors connect this possibility to arbitrary-access schemes using high-dimensional stabilizer states and local equivalence between stabilizer states and qudit graph states.Source: §IV · p. 13
  • Further CQ-protocol analysis could consider dishonest-participant attacks, channel noise, and overall efficiency.Source: §IV · p. 13
Loading 1004.4619v3…