twirled-bell-witness

Agent Prompt: Exact Analytical Solution for PPT Discrimination of Two-Qubit Ensembles

Your objective is to derive and rigorously prove the closed-form analytical solution for the positive-partial-transpose (PPT) discrimination probability of a specific quantum state ensemble. Numerical approximation is insufficient; your final deliverable must be a rigorous mathematical proof consisting of exact, symbolic matching primal and dual SDP certificates.

1. Mathematical Setup

Let $\mathcal{X}_1, \mathcal{X}_2, \mathcal{Y}_1, \mathcal{Y}_2$ each denote $\mathbb{C}^2$, and write $\mathcal{X} = \mathcal{X}_1 \otimes \mathcal{X}_2$ and $\mathcal{Y} = \mathcal{Y}_1 \otimes \mathcal{Y}_2$. Alice holds the pair of systems $\mathcal{X}$ and Bob holds $\mathcal{Y}$. Fix real numbers $\alpha \geq \beta \geq 0$ with $\alpha^2 + \beta^2 = 1$, and define four orthonormal states in $\mathcal{X}_1 \otimes \mathcal{Y}_1$:

$\ket{\psi_0} = \alpha\ket{00} + \beta\ket{11}, \qquad \ket{\psi_1} = \beta\ket{00} - \alpha\ket{11}$

$\ket{\psi_2} = \alpha\ket{01} + \beta\ket{10}, \qquad \ket{\psi_3} = \beta\ket{01} - \alpha\ket{10}$

For $\varepsilon \in [0,1]$, define a resource state in $\mathcal{X}_2 \otimes \mathcal{Y}_2$:

$\ket{\tau_\varepsilon} = \sqrt{\frac{1+\varepsilon}{2}}\ket{00} + \sqrt{\frac{1-\varepsilon}{2}}\ket{11}$

At $\varepsilon = 0$ this is maximally entangled; at $\varepsilon = 1$ it is a product state. For an index set $S \subseteq \{0,1,2,3\}$, define the ensemble $\eta_\varepsilon^{S}$ where the states are drawn with uniform probability $1/\vert{}S\vert{}$:

$\eta_\varepsilon^{S} = \left\{ \left( \frac{1}{\vert{}S\vert{}}, \ket{\psi_i} \otimes \ket{\tau_\varepsilon} \right) : i \in S \right\}$

Let $\rho_i = \bigl(\ket{\psi_i}\otimes\ket{\tau_\varepsilon}\bigr) \bigl(\bra{\psi_i}\otimes\bra{\tau_\varepsilon}\bigr)$.

2. The PPT SDP Formulation

A measurement is a collection $\{P_i\}_{i \in S}$ of positive semidefinite operators on $\mathcal{X}\otimes\mathcal{Y}$ summing to $\mathbb{1}$. The PPT measurement class requires each $P_i$ to have a positive semidefinite partial transpose across the bipartition between Alice's systems ($\mathcal{X}$) and Bob's systems ($\mathcal{Y}$). The optimal PPT discrimination probability, $\mathrm{opt}_{\mathrm{PPT}}$, is given exactly by this Semidefinite Program (SDP):

$\begin{aligned} \text{maximize} \quad & \frac{1}{\vert{}S\vert{}}\sum_{i \in S} \langle P_i, \rho_i\rangle \\ \text{subject to} \quad & \sum_{i \in S} P_i = \mathbb{1}, \\ & P_i \succeq 0, \quad i \in S, \\ & \mathrm{T}_{\mathcal{Y}}(P_i) \succeq 0, \quad i \in S \end{aligned}$

where $\mathrm{T}_{\mathcal{Y}}$ is the partial transpose on Bob's systems.

3. The Target: Four-State Family Solution

Take $S = \{0,1,2,3\}$, meaning the ensemble is the full orthonormal basis with the resource attached. It is known that the solution takes the following closed form:

$\mathrm{opt}_{\mathrm{PPT}}\bigl(\eta_\varepsilon^{\{0,1,2,3\}}\bigr) = \frac{1}{2}\Bigl(1 + \sqrt{1-v^2}\Bigr)$

Your task is to independently discover the exact function $v(\alpha, \varepsilon)$ and rigorously prove it. As a calibration anchor, for the symmetric three-state family $S = \{0,1,2\}$ at $\alpha = \beta = 1/\sqrt{2}$, the optimal separable (and PPT) value is known to be $\frac{1}{3}\bigl(2 + \sqrt{1-\varepsilon^2}\bigr)$. Use this structure to guide your ansatz.

4. Required Execution Pipeline

1. High-Precision Sampling: Solve the SDP densely over a grid of $(\alpha, \varepsilon)$ values. Default solver tolerances are too loose; you must achieve roughly 10 digits of precision before fitting.
2. Symbolic Regression: Use the high-precision numerical data to recover the explicit algebraic form of $v(\alpha, \varepsilon)$ via symbolic regression (without looking it up).
3. Exact Exact Dual Construction (The Proof): Read the structure of the optimal dual solution off the numerical solver (eigenvalues, eigenvectors, tight constraints). Guess the exact algebraic form of the primal and dual operators. Verify the primal-dual matching in exact arithmetic using a computer algebra system (e.g., SymPy). A verified matching primal and dual pair is the only acceptable proof.

5. Known Pitfalls & Guidelines

• System Ordering (Critical): The states live in $\mathcal{X}_1 \otimes \mathcal{Y}_1 \otimes \mathcal{X}_2 \otimes \mathcal{Y}_2$, but the bipartition for the partial transpose is $(\mathcal{X}_1\mathcal{X}_2)$ against $(\mathcal{Y}_1\mathcal{Y}_2)$. A system permutation is required before transposing. Getting this wrong produces plausible numbers that silently answer a different question.
• Exploit Symmetry: Both ensembles carry a symmetry group. Twirling over it block-diagonalizes the SDP, reducing it to small invariant blocks. This is the most reliable route to a closed form.
• Start at Boundaries: Solve analytically for $\varepsilon = 0$, $\varepsilon = 1$, and $\alpha = \beta$ across all $\varepsilon$ to pin the edges of the surface and constrain your candidate formula.

6. Recommended Tools

• toqito (Python): State discrimination, PPT measurement helpers, partial transpose/trace with explicit system indexing.
• CVXPY: Use MOSEK (preferred) or SDPA as the backend for the required 10-digit accuracy.
• SymPy / mpmath: For exact-arithmetic verification of the conjectured certificate and high-precision sampling.
• PySR: For the symbolic regression step to recover $v(\alpha, \varepsilon)$.