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Recovery from an eavesdropping attack on a qubit of a graph state

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Ankur Raina at Indian Institute of Science Education and Research, Bhopal
Ankur Raina
Shayan Srinivasa Garani
Shayan Srinivasa Garani
Shayan Srinivasa Garani
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Abstract and Figures

Graph states are multipartite entangled pure states that can describe distributed quantum information in a formal setting via the notion of nodes and edges. One qubit is present at each node, and the entangling interactions are represented via the edges. We investigate eavesdropping on one of the qubits of the graph state. The eavesdropper uses an ancilla qubit to unitarily interact with one of the qubits followed by a measurement on the ancilla qubit. We study the effect of eavesdropping on the graph state and its connections to the graph topology. We propose the use of a modified graph state by performing certain controlled unitary operations on the existing graph state. This improves the error correcting ability of the graph state, and the modified graph state can correct any error on one qubit.
An example of 5 nodes used for describing a graph state. The source is at node 1. CZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {CZ}$$\end{document} operations capture all the entangling interactions between qubits connected by undirected edges. Qubits at nodes 2, 3, 4 and 5 are initially prepared in |+⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert +\rangle $$\end{document} state. CZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {CZ}$$\end{document} gate is applied between every pair of qubits located at nodes connected by edges. The unitary interaction that entangles all the qubits is CZ12CZ14CZ15CZ23CZ24CZ34CZ35CZ45\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {CZ}_{12}\mathrm {CZ}_{14}\mathrm {CZ}_{15}\mathrm {CZ}_{23}\mathrm {CZ}_{24}\mathrm {CZ}_{34}\mathrm {CZ}_{35}\mathrm {CZ}_{45}$$\end{document}. The graph encodes the state |ϕ⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert \phi \rangle $$\end{document} into |ψ⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert \psi \rangle $$\end{document}
An example of 5 nodes used for describing a graph state. The source is at node 1. CZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {CZ}$$\end{document} operations capture all the entangling interactions between qubits connected by undirected edges. Qubits at nodes 2, 3, 4 and 5 are initially prepared in |+⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert +\rangle $$\end{document} state. CZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {CZ}$$\end{document} gate is applied between every pair of qubits located at nodes connected by edges. The unitary interaction that entangles all the qubits is CZ12CZ14CZ15CZ23CZ24CZ34CZ35CZ45\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {CZ}_{12}\mathrm {CZ}_{14}\mathrm {CZ}_{15}\mathrm {CZ}_{23}\mathrm {CZ}_{24}\mathrm {CZ}_{34}\mathrm {CZ}_{35}\mathrm {CZ}_{45}$$\end{document}. The graph encodes the state |ϕ⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert \phi \rangle $$\end{document} into |ψ⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert \psi \rangle $$\end{document}
… 
Block diagram depicting the encoding via U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {U}}$$\end{document}, eavesdropping operation via U and decoding after measurement in the {|+⟩,|-⟩}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\vert +\rangle ,\vert -\rangle \}$$\end{document} basis. Random variables X, Y, Z exist at the source, the destination and the eavesdropper. The quantum state encoded via U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {U}}$$\end{document} in the graph state undergoes unitary interaction U with the probe qubit of the eavesdropper. This is followed by measurement of the probe qubit by the eavesdropper. The graph state becomes mixed and undergoes the usual inversion via U†\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {U}}^{\dagger }$$\end{document}. The corresponding quantum states at various stages are also shown
Block diagram depicting the encoding via U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {U}}$$\end{document}, eavesdropping operation via U and decoding after measurement in the {|+⟩,|-⟩}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\vert +\rangle ,\vert -\rangle \}$$\end{document} basis. Random variables X, Y, Z exist at the source, the destination and the eavesdropper. The quantum state encoded via U\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {U}}$$\end{document} in the graph state undergoes unitary interaction U with the probe qubit of the eavesdropper. This is followed by measurement of the probe qubit by the eavesdropper. The graph state becomes mixed and undergoes the usual inversion via U†\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {U}}^{\dagger }$$\end{document}. The corresponding quantum states at various stages are also shown
… 
Comparison of results for attack on various nodes of a graph with N nodes. We used g=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g=1$$\end{document} and t=1.6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=1.6$$\end{document} s a variation of I(X; Y) as a function of t. b Variation of I(X; Z) as a function of t. c Variation of I(X; Y) as a function of t. dI(X; Z) as a function of t
Comparison of results for attack on various nodes of a graph with N nodes. We used g=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g=1$$\end{document} and t=1.6\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t=1.6$$\end{document} s a variation of I(X; Y) as a function of t. b Variation of I(X; Z) as a function of t. c Variation of I(X; Y) as a function of t. dI(X; Z) as a function of t
… 
The circuit detects four types of error, namely I,X,Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {I,X,Z}$$\end{document} and XZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {XZ}$$\end{document}. Sx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {S}}_x$$\end{document} detects X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {X}}$$\end{document} error, and Sz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {S}}_z$$\end{document} detects Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {Z}}$$\end{document} error. Measurement on the ancilla qubits aids the detection of the type of error. The state at various stages is denoted by |ψ1′⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert \psi _1'\rangle $$\end{document}, |ψ2′⟩,|ψ3′⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert \psi _2'\rangle ,\vert \psi _3'\rangle $$\end{document}
The circuit detects four types of error, namely I,X,Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {I,X,Z}$$\end{document} and XZ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {XZ}$$\end{document}. Sx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {S}}_x$$\end{document} detects X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {X}}$$\end{document} error, and Sz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {S}}_z$$\end{document} detects Z\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbf {Z}}$$\end{document} error. Measurement on the ancilla qubits aids the detection of the type of error. The state at various stages is denoted by |ψ1′⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert \psi _1'\rangle $$\end{document}, |ψ2′⟩,|ψ3′⟩\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vert \psi _2'\rangle ,\vert \psi _3'\rangle $$\end{document}
… 
An example of 5 nodes which requires some CX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {CX}$$\end{document} operations to improve error correction ability
+4
An example of 5 nodes which requires some CX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {CX}$$\end{document} operations to improve error correction ability
… 
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Quantum Information Processing (2019) 18:274
https://doi.org/10.1007/s11128-019-2387-2
Recovery from an eavesdropping attack on a qubit of a
graph state
Ankur Raina1·Shayan Srinivasa Garani1
Received: 18 May 2019 / Accepted: 18 July 2019 / Published online: 26 July 2019
© Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract
Graph states are multipartite entangled pure states that can describe distributed quan-
tum information in a formal setting via the notion of nodes and edges. One qubit is
present at each node, and the entangling interactions are represented via the edges. We
investigate eavesdropping on one of the qubits of the graph state. The eavesdropper
uses an ancilla qubit to unitarily interact with one of the qubits followed by a mea-
surement on the ancilla qubit. We study the effect of eavesdropping on the graph state
and its connections to the graph topology. We propose the use of a modified graph
state by performing certain controlled unitary operations on the existing graph state.
This improves the error correcting ability of the graph state, and the modified graph
state can correct any error on one qubit.
Keywords Modified graph state ·Probe qubit ·Schmidt decomposition ·Quantum
error correction ·Target qubit
1 Introduction
One of the principal advantages of quantum communication using quantum states is the
ability to know whether the data are eavesdropped upon. Unlike classical data storage
where it is difficult to know the presence of an eavesdropper, quantum information
storage has advantages stemming from the basic principle that measurement destroys
superposition of quantum states. In this light, quantum key distribution (QKD) has
received importance for the generation of a classical key that can be used for ensuring
perfect secrecy [1]. BB84 [2] and Ekert [3] protocols are two well-known protocols
that gave birth to the secrecy of information in the quantum realm. Both protocols
BShayan Srinivasa Garani
shayangs@iisc.ac.in
Ankur Raina
ankur@iisc.ac.in
1Department of Electronic Systems Engineering, Indian institute of Science, Bengaluru, India
123
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... In Eq. (31), CZ i j refers to the CZ gate between nodes i and j. The expressions are similar to what we used in [23] except the assumption of |φ k ∈ {|+ , |− }. In this paper, we allow for a general |φ k to accommodate mixed states. ...
... In this paper, we allow for a general |φ k to accommodate mixed states. Much of what follows in this section is taken from [23] expressing the Schmidt decomposition for the sake of completeness and readability. We reiterate the Schmidt decomposition for the above state with the two cases labeled by a = 1 and a = 1. ...
... The idea is to represent the encoding of basis state with respect to the ath node. Consider the encoded state labeled by a as [23] |ψ k (a) = ...
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