Primus for Quantum Mechanics

Today, we are introducing Primus for Quantum Mechanics, a specialized release in our Primus for X suite of AI researchers designed to assist with domain-specific theoretical work.

Primus for Quantum Mechanics is a mode you can select inside Primus: choose the Quantum information specialization when you start a project. Every Primus user has access to it today at primus.lab.cloud.

It supports Physlib's QuantumInfo library for Lean 4, alongside toqito, cvxpy semidefinite programming, sympy and QuTiP.

As an initial demonstration, Primus for Quantum Mechanics independently derived a complete mathematical solution to an unresolved problem in quantum state discrimination: determining the optimal probability with which two separated parties can discriminate four orthogonal two-qubit states, given a shared partially entangled resource state, when their measurements are restricted.

The resulting derivation was reviewed by a domain expert in quantum information theory who had previously obtained the result privately. The reviewer confirmed the accuracy of the final closed-form result and validated the AI's overall dual semidefinite programming framework.

Read the full research →

See the Lean formalization →

Understanding the problem

To understand the core problem, consider two parties, Alice and Bob, located in separate laboratories:

• The Secret Message: Alice and Bob share one of four orthogonal quantum states. Neither party can identify the state individually through local inspection alone.
• The Scrambler ($\alpha$): The signal states are parameterized by a Schmidt coefficient $\alpha$. When $\alpha = 1/\sqrt{2}$, the states are maximally entangled Bell states. When $\alpha = 1$, they reduce to unentangled product states.
• The Resource Link ($\varepsilon$): Alice and Bob share an auxiliary pair of qubits in a partially entangled state $\tau_\varepsilon$ to help them discriminate the states. The parameter $\varepsilon$ measures noise or deviation from maximum entanglement, ranging from maximally entangled ($\varepsilon = 0$) to a product state ($\varepsilon = 1$).

The Task: Alice and Bob must determine which of the four states they share using Positive Partial Transpose (PPT) measurements across their systems.

The Result: Primus for Quantum Mechanics constructed matching primal and dual solutions to derive the exact closed-form formula for the optimal success probability:

$$P_{\text{succ}} = \frac{1}{2}\left(1 + \sqrt{1 - v_+^2}\right)$$

Where $v_+ = \max\{v, 0\}$ and $v = 2\alpha\beta\lambda_0 - \lambda_1$. Here $\beta = \sqrt{1 - \alpha^2}$, and $\lambda_0 = (1 + \varepsilon)/2$ and $\lambda_1 = (1 - \varepsilon)/2$ are the Schmidt weights of the resource state $\tau_\varepsilon$.

From this formula the condition for perfect discrimination follows: $2\alpha\beta(1 + \varepsilon) \le 1 - \varepsilon$. It is a trade-off rather than a threshold: the more entangled the four states are, the more entanglement the shared resource must carry before the parties can identify which one they hold. Outside that region no PPT measurement succeeds perfectly, and therefore no local protocol does either.

A specialized domain in quantum information theory

While quantum computing broadly receives significant public attention, the specialized subfield of quantum state discrimination via cone programming and Positive Partial Transpose (PPT) constraints is highly technical.

Globally, only a few hundred researchers actively work at this specific intersection of convex optimization and quantum information theory.

Rather than relying on numerical approximations, Primus for Quantum Mechanics constructed exact primal-feasible and dual-feasible solutions whose values coincide, which certifies the formula as the exact optimum rather than a close approximation. Demonstrating that an automated system can produce certificates of this kind is a promising benchmark for computational physics.

Theoretical utility and context

Finding exact closed-form solutions for state discrimination provides benchmarks within quantum information theory:

• Bounding Information Leakage: Exact PPT bounds give strict limits on how much two separated parties can learn about a shared state. This is the standard tool for certifying quantum data hiding, where information is encoded so that it stays inaccessible to anyone who cannot bring the parts together for a joint measurement.
• Resource Quantification: The derived threshold provides an explicit metric for the minimum entanglement required to execute specific bipartite state discrimination tasks without error.
• Analytic Benchmarks: Bipartite state discrimination problems with broken symmetries rarely yield simple radical expressions. Having exact solutions for specific families provides test cases for broader quantum channel discrimination algorithms.

The road ahead for Primus for X

Our goal with Primus for X is to expand access to powerful tools that accelerate research, so the frontier belongs to more people and not just a few closed corporations. We look forward to expanding the Primus for X product suite into additional areas of theoretical physics, mathematics, and computer science.

We also want to work directly with quantum companies and quantum research teams on the problems slowing their progress. If that is you, please contact us.

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