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Indefinite causal key distribution

Hector Spencer-Wood

TL;DR

The paper addresses secure quantum key distribution under indefinite causal order (ICO) realized by a quantum switch. It shows that eavesdroppers can be privately detected without publicly revealing key subsets, and provides security analysis against a class of individual attacks using a process-matrix framework, yielding a detection threshold $d_{thr} \,\approx\,0.096$ and tolerable error $P_{error} \le 2d_{thr} \approx 0.192$. The authors compare ICO with definite causal order, showing private detection can persist in ICO and can be realized with additional operations in definite order, but some ICO-specific strategies are not available otherwise. They discuss two-way protocol parallels, practical implementation considerations (coherence requirements, two-qubit-per-bit burden), and outline avenues for extending security proofs to correlated and coherent attacks. Overall, this work presents ICO as a prospective, albeit challenging, route to private eavesdropping detection in quantum cryptography and motivates further foundational and experimental exploration.

Abstract

We propose a quantum key distribution (QKD) protocol that is carried out in an indefinite causal order (ICO). In QKD, one considers a setup in which two parties, Alice and Bob, share a key with one another in such a way that they can detect whether an eavesdropper, Eve, has learnt anything about the key. To our knowledge, in all QKD protocols proposed until now, Eve is detected by publicly comparing a subset of Alice and Bob's key and checking for errors. We find that a consequence of our protocol is that it is possible to detect eavesdroppers without publicly comparing any information about the key. Indeed, we prove that it is not possible for eavesdroppers, performing any individual attack, to extract useful information about the shared key without inducing a nonzero probability of being detected. We also prove the security of this protocol against a class of individual eavesdropping attacks. The role ICO plays in causing unusual phenomena in quantum technologies is an important question. By considering it we find a two-way QKD protocol that exhibits a similar private detection feature, albeit with some interesting differences. After noting some implications of these differences and discussing some of the practicalities of our protocol, we conclude that this work is best considered as a first step in applying quantum cryptographic ideas in an ICO.

Indefinite causal key distribution

TL;DR

The paper addresses secure quantum key distribution under indefinite causal order (ICO) realized by a quantum switch. It shows that eavesdroppers can be privately detected without publicly revealing key subsets, and provides security analysis against a class of individual attacks using a process-matrix framework, yielding a detection threshold and tolerable error . The authors compare ICO with definite causal order, showing private detection can persist in ICO and can be realized with additional operations in definite order, but some ICO-specific strategies are not available otherwise. They discuss two-way protocol parallels, practical implementation considerations (coherence requirements, two-qubit-per-bit burden), and outline avenues for extending security proofs to correlated and coherent attacks. Overall, this work presents ICO as a prospective, albeit challenging, route to private eavesdropping detection in quantum cryptography and motivates further foundational and experimental exploration.

Abstract

We propose a quantum key distribution (QKD) protocol that is carried out in an indefinite causal order (ICO). In QKD, one considers a setup in which two parties, Alice and Bob, share a key with one another in such a way that they can detect whether an eavesdropper, Eve, has learnt anything about the key. To our knowledge, in all QKD protocols proposed until now, Eve is detected by publicly comparing a subset of Alice and Bob's key and checking for errors. We find that a consequence of our protocol is that it is possible to detect eavesdroppers without publicly comparing any information about the key. Indeed, we prove that it is not possible for eavesdroppers, performing any individual attack, to extract useful information about the shared key without inducing a nonzero probability of being detected. We also prove the security of this protocol against a class of individual eavesdropping attacks. The role ICO plays in causing unusual phenomena in quantum technologies is an important question. By considering it we find a two-way QKD protocol that exhibits a similar private detection feature, albeit with some interesting differences. After noting some implications of these differences and discussing some of the practicalities of our protocol, we conclude that this work is best considered as a first step in applying quantum cryptographic ideas in an ICO.
Paper Structure (23 sections, 1 theorem, 98 equations, 9 figures)

This paper contains 23 sections, 1 theorem, 98 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Background theory
  3. Indefinite causal order
  4. Quantum key distribution
  5. Quantum key distribution in an indefinite causal order
  6. Indefinite causal key distribution with no eavesdroppers
  7. Introducing an eavesdropper
  8. Security against individual attacks
  9. Problem setup
  10. Minimum probability of detection
  11. Eavesdroppers - Alice mutual information
  12. Alice - Bob mutual information
  13. Error rates
  14. Correlated individual attacks and beyond
  15. Private detection in a definite causal order
  16. ...and 8 more sections

Key Result

Theorem C.1

For any input state $|\psi\rangle$, $P(-^{C_c}) = 0$ if and only if where $r_{\mathbf{x}}^{\mu} \in \mathbb{C}~\forall \mu\in \{0,1,2,3\}, \mathbf{x} \in \mathbf{X}$ and $(\sigma_0, \sigma_1, \sigma_2, \sigma_3)$$= (\mathbbm{1}, \sigma_x, \sigma_y, \sigma_z)$.

Figures (9)

  • Figure 1: Quantum mechanics allows for more freedom in the ordering of events: (a) Alice can act on a state $\rho$ before Bob, (b) Bob before Alice, or (c) in a superposition of both orders, controlled on some quantum state $\omega$.
  • Figure 2: Indefinite causal key distribution with no eavesdroppers. A key is shared between Alice and Bob by sending a state $\rho$ to them in a superposition of orders controlled by the state $\omega$. Alice and Bob perform projective measurements randomly in either the Pauli $x$ or $z$-basis. After discarding cases in which Alice and Bob measured in different bases, they are left with identical keys. Regardless of the initial state $\omega$ of the control qubit, $\omega$ never changes when there are no eavesdroppers, a phenomenon we see not to be true when an eavesdropper is introduced.
  • Figure 3: Indefinite causal key distribution with a single eavesdropper, Eve, between Alice and Bob.
  • Figure 4: Summary of the proposed indefinite causal key distribution protocol.
  • Figure 5: Indefinite causal quantum key distribution with two eavesdroppers Eve and Yves. The fact there are now two eavesdroppers means correlated attacks are possible. All possible correlations are described mathematically by a process matrix $W^{\tilde{E}\tilde{Y}}$ shared by Eve and Yves.
  • ...and 4 more figures

Theorems & Definitions (4)

  • Definition B.1
  • Definition B.2
  • Theorem C.1
  • proof