Quantum ramp secret sharing from Haar scrambling

TL;DR

This work establishes a precise equivalence between Haar scrambling and quantum ramp secret sharing, showing that random Haar unitaries implement approximate ramp schemes rather than threshold schemes, with tunable parameters via the purity of the initial state. The authors introduce $l$-scrambling to unify scrambling notions and prove that Haar scrambling realizes ramp secret sharing with parameters $b=\frac{N}{2}-\epsilon$ and $g=\frac{N}{2}+\epsilon$ (for pure states), extendable to mixed states by purification and discarding shares, yielding $b=\frac{N+s}{2}-\epsilon$ and $g=\frac{N+s}{2}+\epsilon$. They further demonstrate that the protocol can be implemented efficiently using $t$-design circuits (e.g., Clifford group) with polynomial gate counts, addressing the apparent exponential cost of Haar randomness. The work discusses practical applications in secure quantum networks and NISQ-era cryptography, and provides conceptual insights into many-body dynamics and black-hole information through the secret-sharing lens. Overall, the results bridge scrambling, quantum secret sharing, and quantum information processing, offering a versatile framework for information localization, security, and fundamental physics.

Abstract

Quantum information scrambling has emerged as a powerful tool for studying the dynamics of chaotic quantum many-body systems, assessing benchmarking protocols, and even investigating exotic black hole models. During quantum information scrambling, localized quantum information disperses across the entire system, hiding from the observers who can only access part of it. On the other side, we have a fundamental cryptographic primitive called secret-sharing schemes, where a dealer shares a quantum secret with a group of parties, such that any subset of parties above a specific threshold size can reconstruct it. In this paper, we demonstrate that two protocols, Haar scrambling, which serves as a baseline for other scrambling techniques, and quantum secret sharing, are indeed equivalent. The scheme one gets out of this is of a ramp secret-sharing nature rather than a threshold scheme. We expect this because of the inherent randomness present in Haar unitaries. Moreover, by varying the purity of the initial states, we demonstrate that it is possible to obtain all possible types of ramp secret-sharing schemes. Furthermore, utilizing complexity theoretic arguments, we argue that the protocol can be implemented efficiently, i.e., in polynomial time. Finally, we explore potential applications, ranging from security implications for distributed quantum networks to cryptography in the Noisy Intermediate-Scale Quantum (NISQ) era, and draw insights for fundamental physics, such as many-body physics and quantum black holes.

Figures (9)

• Figure 1: $((k,n))$ Quantum Secret Sharing(QSS) threshold schemes; $k$ players out of $n$ players have to come together to reconstruct the original secret. Any arbitrary collection of players larger than size $k$ is called authorized players, while one with fewer than $k$ is called unauthorized players.
• Figure 2: Alice and Bob try to communicate via a scrambling unitary $U$ acting on N qubits. Alice has full control of the first qubit $A$, which is entangled with the reference system $R$. The rest of the qubits, marked green, can be in a mixed state $\rho$ purified by an external system called Memory. This pure state is denoted by $\ket{\sqrt{\rho}}$. The scrambling unitary acts only on Alice's qubit and the system's qubit, which is marked green. After the unitary evolution, Bob then has access to the system's set of qubits. If the scrambling unitary has a local structure, it is also possible to have an information light cone indicated by a yellow cone. The initial state of the system, including the reference, the system, and the memory, is: $\ket{\boldsymbol{\psi}} = \frac{1}{\sqrt{2}}(\ket{0}_R\ket{0}_{q_A} +\ket{1}_R\ket{1}_{q_A}) \ket{\sqrt{\rho}}$ Since the time evolution unitary doesn't act on reference and memory, the total dynamics is then given by: $\ket{\boldsymbol{\psi}(t)} = \frac{1}{\sqrt{2}} \left(\ket{0}_R U_S \otimes I_\text{MEM} \ket{0,\sqrt{\rho}} + \ket{1}_R U_S \otimes I_\text{MEM} \ket{1,\sqrt{\rho}} \right)$. More detail about this figure is in Appendix \ref{['app:toymodel']}.
• Figure 3: Suppose Alice decides to encode her quantum information in $A$, which is perfectly entangled with external reference state $R$. Alice's qubit then interacts with system $B$, which is also purified with external system $B'$. Thus the total initial state is $\ket{RA} \ket{BB'}$ while final state after scrambling unitary $U_{AB}$ is $\ket{\psi}_{RB'CD} = U_{AB} \otimes I_{RB'}\ket{RA} \ket{BB'}$.
• Figure 4: A qubit secret shared among $N$ parties in different scenarios. For the $((k,N))$ scheme, the mutual information between the reference $R$ and the parties goes through a sharp step function-like transition from $0$ to $2$ as the number of parties crosses the threshold $k$. In $((b,g))$ ramp scheme, the mutual information is zero for parties less than the size $b$ and two when it is greater than $g$ while some mutual information in the grey area i.e. when the player's size is greater than $b$ but less than $g$. Fig b: Includes also approximate $((k,N))$ scheme as for all $|A| \geq k$, $I(R:A) \geq 2 - \delta$ for some small $\delta$ and for all $|A| \geq k$, $I(R:A) \leq \epsilon$ for some small $\epsilon$ and the approximate $((b,g))$ scheme as $I(R:A) \geq 2 - \delta$ for all $|A| \geq g$ and $I(R:A) \leq \epsilon$ for all $|A| \leq b$.
• Figure 5: Mutual information between quantum secret and subsystem/parties of size $l$ for Haar scrambling and what we expect from the ramp scheme when the parties are in a pure state. Haar scrambling approximately behaves like a quantum ramp secret sharing scheme. In figure \ref{['fig:haarPure1']}, we have plotted the mutual information between the reference qubit $R$ and $l$ arbitrary qubits in the system of size $12$ after applying Haar scrambling. The curve for Haar scrambling approximates the $((3,9))$ ramp scheme with a grey area when $l$ is between $3$ and $9$, which gives the gap $G = 6$ and rampiness $R= 0.5$.

Theorems & Definitions (18)

• Definition 1: r9
• Definition 2: r8
• Definition 3
• Definition 4: r11
• Definition 5: yoshida2017efficient
• Definition 6: $l$- scrambling
• Definition 7: $l$- tripartite information
• Remark 1
• proof
• Definition 8