Consecutive Measurement Tradeoffs in Quantum Cryptography

Abstract

Mistrustful quantum cryptographic protocols encode information in incompatible observables, so that any attempt by a dishonest party to access multiple pieces of information necessarily involves a tradeoff. A natural class of such strategies proceeds via consecutive measurements, where each measurement disturbs the state and influences subsequent information extraction. We introduce consecutive measurement theorems (CMTs) as a unified framework to quantify this tradeoff and argue that they capture the fundamental limitations underlying security in mistrustful quantum cryptography. Our main result is a tight characterization of the achievable region of success probabilities for single and consecutive measurements, strictly improving all previously known bounds. We further establish robust variants expressed in terms of fidelity and trace distance that remain valid under perturbations. These results yield strengthened and unified security guarantees for a range of primitives, including relativistic bit commitment, quantum oblivious transfer and quantum private query, and clarify the role of measurement disturbance as the key limitation on adversarial information extraction.

Figures (8)

• Figure 1: Comparison of different lower bounds with $n=2$. Our tight CMT provides a significantly tighter bound than all previous works for all $V \in [\frac{1}{n}, 1]$. Note that Ref. shi2024relativistic only applies to the special case $n=2$, while Refs. Unruh2012quantumchailloux2017relativistic and our work are applicable for all $n \geq 2$. Ref. Unruh2012quantum imposes an extra condition: $V \geq \frac{1}{\sqrt{n}}$.
• Figure 2: Schematic of $\mathbb{F}_p$ RBC. There are two agents of Alice (Bob), denoted by A1 and A2 (B1 and B2), which are spatially separated to satisfy the no-signaling principle. A1 sends $x\in \mathbb{F}_q$ to B1, and B1 responds with $a\in \mathbb{F}_q$, where $a=x\cdot y-b$. Here $y\in \mathbb{F}_p$ is the committed value and $b\in\mathbb{F}_q$ is a pre-shared uniformly random element held by B1 and B2. To reveal $y$, B2 sends the value $b$ to A2.
• Figure 3: A general protocol for QOT. Alice and Bob exchange quantum messages with each other to transfer $x_b$ out of two data bits $(x_0,x_1)$ to Bob upon Bob's request $b$. Alice remains oblivious to $b$ and Bob remains oblivious to $x_{\overline{b}}$.
• Figure 4: Comparison of different no-go theorems for QOT. The pink surface is the no-go theorem for QOT in Theorem \ref{['thm:bound_RCMT']} derived from consecutive measurement (CMT) while the blue surface is the theorem in Refs. Amiri_2021Hu_2023 derived from pretty good measurement (PGM).
• Figure 5: A general protocol for QPQ. Bob communicates quantum messages with Alice to retrieve an entry $x_b$ from Alice's database $x\in[k]^n$. Alice remains oblivious to $b$ and Bob remains oblivious to other entries except $x_b.$

Theorems & Definitions (57)

• Theorem 1: Tight CMT
• Proposition 2
• Proposition 3: Upper bound of $\omega^{*}$(${\rm CHSH}_q(p)$)
• Proposition 4: Upper bound of $\omega^{*}$(${\rm CHSH}_{q}(p)^{\otimes m}$)
• Definition 1: Sum-binding
• Definition 2: $\mathbb{F}_p$ RBC
• Theorem 5: Sum-binding of $\mathbb{F}_p$ RBC
• Definition 3: $\mathbb{F}_p^{\otimes m}$ RBC
• Theorem 6: Sum-binding of $\mathbb{F}_p^{\otimes m}$ RBC
• Theorem 7: Robust CMT in fidelity