Navigating the Quantum Resource Landscape of Entropy Vector Space Using Machine Learning and Optimization

TL;DR

This work develops a hybrid framework combining reinforcement learning and derivative-free optimization to map how entropy vectors and quantum resources evolve under unitary circuits and to identify states that violate Ingleton's entropy inequality. By training a Q-learning agent on entropy-constrained reward signals and by using CMA-ES/COBYLA to maximize the Ingleton gap, the authors generate arbitrarily many Ingleton-violating states, establish a maximal violation bound at $g_{Min.}\approx-0.1699$ (achieved with $6$ qubits), and analyze the accompanying resource signatures. They reveal that Ingleton-violating states occupy isolated, rare regions in Hilbert space and exhibit a characteristic resource profile: high entanglement and high total magic but low non-local magic, with a notable anti-correlation between entanglement entropy and entanglement capacity. The work further links resource evolution to the boundary structure of entropy cones, including the holographic and stabilizer cones, and offers a robust toolkit for engineering circuits with controlled information-theoretic features. These insights advance understanding of how entanglement and magic together shape information-theoretic constraints in quantum systems and provide practical methods for exploring entropy-cone geometry in high-dimensional quantum state spaces.

Abstract

We present a machine learning framework to study the dynamics of entropy vectors and quantum resources, including entanglement and magic, focusing on violations of entropy inequalities. Using a reinforcement learning agent formulated as a Markov decision process, we identify quantum circuits that optimally navigate the entropy vector space to generate violations of Ingleton's inequality. We complement this approach with a classical optimization algorithm to produce arbitrary numbers of Ingleton-violating states, with tunable degrees of violation, and empirically determine the maximal attainable violation for Ingleton's inequality. Our analysis reveals characteristic patterns of quantum resources that accompany Ingleton violation. A comprehensive statistical analysis shows that Ingleton-violating states occupy sharply-defined, isolated regions of the Hilbert space, and are extremely rare. Together, these results establish a unified computational toolkit for studying entropy vector dynamics, tracking quantum resource evolution, and engineering circuits with controlled information-theoretic features.

Figures (23)

• Figure 1: Workflow illustrating reinforcement learning protocol for generating states that fail a chosen entropy inequality, using a select gate set, and analyzing the entropy vector dynamics across violating circuits. After initialization, the circuit is updated with a gate chosen from the Q-learning. The agent calculates the entropy vector after each gate, and checks for violation of the inequality. A reward is computed based on the inequality evaluation, the Q-table is updated, and the process repeats until violation is reached.
• Figure 2: Left image shows the evolution of the left-hand and right-hand sides of MMI, in Eq. \ref{['MMI']}, as the system evolves from satisfaction to failure. After the $4$ gates in Figure \ref{['fig:mmi-violation-circuit']} are applied to ${|{0}\rangle}^{\otimes4}$, the system is in state ${|{GHZ_4}\rangle}$, and violates MMI. The right image shows the MMI difference, computed by Eq. \ref{['MMIDiff']}, which becomes positive at violation.
• Figure 3: Circuit which prepares the state ${|{GHZ_4}\rangle}$ starting from ${|{0}\rangle}^{\otimes 4}$. This circuit is constructed using the $H$ and $CNOT$ gates, according to the reinforcement learning protocol described in Figure \ref{['fig:ql_workflow']}. The final state ${|{GHZ_4}\rangle}$ violates MMI, as in Eq. \ref{['MMI']}.
• Figure 4: Example of an undirected graph that realizes all possible entropy vectors for $3$-qubit quantum states, using a min-cut prescription. Each vertex corresponds to an individual qubit, and each edge carries a weight $w_i$. The entanglement entropy $S_i$, of the $i^{th}$ qubit, corresponds to the weight of cutting both edges connected to it, e.g. $S_A = w_1+w_2$.
• Figure 5: Hypergraph possessing a min-cut protocol that realizes any $5$-qubit pure state entropy vector. The full graph is shown as the union of two subgraphs, with all $4$-edges shown to the left and all $3$-edges to the right. Eq. \ref{['HypergraphLinear']} demonstrates how these $15$ edges weights combine to produce any $5$-qubit pure state entropy vector.

Theorems & Definitions (2)

• Theorem 1
• Corollary 1