(2,m)-threshold quantum data hiding

Technical Summary: (2, m)-Threshold Quantum Data Hiding

Problem Statement
The paper addresses the challenge of quantum data hiding, a protocol where information is encoded into quantum states shared among multiple parties such that the data remains concealed unless a sufficient number of parties collaborate. While classical data hiding relies on restricting communication, quantum data hiding offers stronger concealment even when classical communication (LOCC) is permitted, requiring global quantum operations for recovery.

Prior research established two extremes:

1. Two-party schemes: Concealing a bit between two parties where LOCC fails but joint measurement succeeds.
2. (m, m)-threshold schemes: Concealing data among $m$ parties where all $m$ parties must collaborate for recovery.

A gap existed in the literature regarding intermediate collaboration requirements. Specifically, no $(k, m)$-threshold scheme had been proposed for $2 \le k < m$. The authors aim to construct a $(2, m)$-threshold scheme where any pair of parties can perfectly recover a hidden bit, yet all parties performing only Local Operations and Classical Communication (LOCC) gain negligible information.

Methodology
The authors approach the problem through the lens of multiparty quantum state discrimination, specifically analyzing the gap between global discrimination and LOCC discrimination.

1. Theoretical Framework:

   • The paper defines the discrimination of an ensemble $\mathcal{E} = \{\eta_i, \rho_i\}$ using Positive Operator-Valued Measures (POVMs).
   • It introduces the concept of PPT (Positive Partial Transpose) measurements. Since every LOCC measurement is a PPT measurement, the optimal success probability of PPT discrimination ($p_{\text{PPT}}$) serves as an upper bound for LOCC discrimination ($p_L$).
   • The authors derive a dual characterization for $p_{\text{PPT}}$ (Theorem 1) and a trace-norm minimization formulation for two-state ensembles (Theorem 2).
   • Theorem 3 is a critical theoretical contribution, establishing that the optimal PPT discrimination probability of a multiparty state ensemble constructed from tensor products of two-party states is bounded by the sum of the PPT discrimination probabilities of the underlying two-party ensembles.

2. Scheme Construction:

   • System Architecture: For $m$ parties ($A_1, \dots, A_m$), the system is constructed such that every pair of parties $(A_k, A_{k'})$ shares a specific two-party subsystem. Each party $A_k$ holds $m-1$ subsystems, one for each other party.
   • Building Block: The scheme utilizes a specific two-party orthogonal PPT state ensemble $\mathcal{T} = \{ \frac{1}{2}, \tau_0; \frac{1}{2}, \tau_1 \}$. Crucially, this ensemble is chosen such that its optimal PPT discrimination probability is arbitrarily close to random guessing ($1/2 + \epsilon$), despite the states being orthogonal.
   • Encoding Process:
     • The hider prepares a global state $\tau_{\vec{b}}$ by tensoring independent instances of $\tau_{b_{k,k'}}$ for every pair $(k, k')$, where $b_{k,k'} \in \{0, 1\}$ are random bits.
     • To hide a target bit $x$, the hider broadcasts a classical bit string $\vec{c}$ derived from the random bits $b_{k,k'}$ and the target bit $x$ (using modulo-2 addition).
     • Specifically, for a fixed pair $(\alpha, \alpha')$, the bit $c_{\alpha, \alpha'} = b_{\alpha, \alpha'} \oplus x$, while other bits in $\vec{c}$ mask the other random bits.

Key Results

1. Perfect Recovery by Pairs: Because the states $\tau_0$ and $\tau_1$ are orthogonal, any pair of parties $(A_k, A_{k'})$ can perform a joint measurement on their shared subsystem to perfectly distinguish the state and recover the bit $x$.
2. LOCC Security: The paper proves that the maximum average probability of correctly guessing the hidden bit $x$ using any LOCC measurement is bounded by:
   $$\frac{1}{2} \le p_L(\mathcal{E}) \le \frac{1}{2} + \binom{m}{2} \left( p_{\text{PPT}}(\mathcal{T}) - \frac{1}{2} \right)$$
   Since $p_{\text{PPT}}(\mathcal{T})$ can be made arbitrarily close to $1/2$ (by choosing $\epsilon$ sufficiently small), the information gain for any LOCC strategy (even by all $m$ parties) can be made arbitrarily small.
3. Separable States: The proposed scheme can be implemented using only separable states of low-dimensional quantum systems (specifically, two-qutrit states in the example provided, though two-qudit states for $d \ge 2$ are noted as possible).

Significance and Claims
The paper claims to be the first to propose a $(2, m)$-threshold quantum data-hiding scheme, filling the gap between two-party and full-collaboration schemes. Its significance lies in three main areas:

• Intermediate Thresholds: It demonstrates that quantum data hiding is achievable with intermediate collaboration requirements ($k=2$), rather than requiring the maximal collaboration of all parties.
• Resource Efficiency: By utilizing separable states rather than entangled states, the scheme enhances practical feasibility. The authors explicitly state that entanglement is not a necessary resource for realizing multiparty quantum data hiding.
• Theoretical Bounds: The work provides new upper bounds on the optimal LOCC discrimination of multiparty quantum states, derived via the relationship between global PPT discrimination and local two-party discrimination.

The authors remain modest regarding future extensions, noting that while their scheme hides a single bit, extending this to multiple bits or higher thresholds ($k > 2$) remains an open question for future investigation.