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Mathematical Analysis of Entanglement Measures and Maximized Quantum Fisher Information

Volkan Erol

Published Sep 18, 2025• 0 citations
Publisher

Abstract

Entanglement measures such as concurrence, negativity and REE are well-known tools for quantifying non-classical correlations in quantum systems. However, these measures can lead to different state orderings for non-maximally entangled states. On the other hand, Quantum Fisher Information (QFI), while not an entanglement measure, provides a framework for analyzing a state's metrological potential. In this study, we numerically analyze the relationship between these entanglement measures and QFI for a large ensemble of random two-qubit states. We specifically focus on the maximized QFI (MQFI) obtained through local unitary rotations. Our findings demonstrate a strong correlation between entanglement and a state's metrological capacity, confirming that entanglement is a valuable resource. We show that while a state's QFI with a fixed generator can vary widely for a given entanglement value, local optimization to find the MQFI leads to a tighter, more predictable relationship. Our results also reinforce the principle that the metrological performance of all mixed states is fundamentally bounded by the theoretical limit of pure states. The polynomial fit equations for the upper and lower bounds of our data provide a quantitative description of these complex relationships.

DOI
10.20944/preprints202509.0528.v2

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Entanglement measures such as concurrence, negativity and REE are well-known tools for quantifying non-classical correlations in quantum systems. However, these measures can lead to different state orderings for non-maximally entangled states. On the other hand, Quantum Fisher Information (QFI), while not an entanglement measure, provides a framework for analyzing a state's metrological potential. In this study, we numerically analyze the relationship between these entanglement measures and QFI for a large ensemble of random two-qubit states. We specifically focus on the maximized QFI (MQFI) obtained through local unitary rotations. Our findings demonstrate a strong correlation between entanglement and a state's metrological capacity, confirming that entanglement is a valuable resource. We show that while a state's QFI with a fixed generator can vary widely for a given entanglement value, local optimization to find the MQFI leads to a tighter, more predictable relationship. Our results also reinforce the principle that the metrological performance of all mixed states is fundamentally bounded by the theoretical limit of pure states. The polynomial fit equations for the upper and lower bounds of our data provide a quantitative description of these complex relationships.
Entanglement measures such as concurrence, negativity and REE are well-known tools for quantifying non-classical correlations in quantum systems. However, these measures can lead to different state orderings for non-maximally entangled states. On the other hand, Quantum Fisher Information (QFI), while not an entanglement measure, provides a framework for analyzing a state's metrological potential. In this study, we numerically analyze the relationship between these entanglement measures and QFI for a large ensemble of random two-qubit states. We specifically focus on the maximized QFI (MQFI) obtained through local unitary rotations. Our findings demonstrate a strong correlation between entanglement and a state's metrological capacity, confirming that entanglement is a valuable resource. We show that while a state's QFI with a fixed generator can vary widely for a given entanglement value, local optimization to find the MQFI leads to a tighter, more predictable relationship. Our results also reinforce the principle that the metrological performance of all mixed states is fundamentally bounded by the theoretical limit of pure states. The polynomial fit equations for the upper and lower bounds of our data provide a quantitative description of these complex relationships.
Abstract We provide a quantitative analysis of the connection between entanglement measures and Maximized Quantum Fisher Information (MQFI) using extensive numerical analysis of 20,000 randomly generated two-qubit states. Our systematic study reveals strong empirical relationships between the metrological capacity of quantum states and three entanglement measures: concurrence, negativity, and relative entropy of entanglement. Optimization over local unitary transformations produces substantially more predictable relationships than fixed-generator approaches, as confirmed by bootstrap resampling, systematic data binning, and multiple model comparisons. Exponential fits reach R 2 > 0.99 and polynomial models achieve R 2 = 0.999, providing thorough empirical support for saturation behavior in quantum metrological advantage. Even separable states possess robust metrological utility (MQFI/4 ≈ 0.19, p < 0.0001), achieving approximately 19% of maximal Bell-state sensing capacity—confirming the importance of quantum coherence and discord beyond entanglement. Entanglement–metrology relationships remain stable under realistic decoherence, particularly phase damping. Our findings empirically validate predictions from quantum resource theory, establish empirical bounds for quantum sensor optimization, and offer direct practical guidance for the design of quantum sensing protocols.
Entanglement has been studied extensively for unveiling the mysteries of non-classical correlations between quantum systems. In the bipartite case, there are well known measures for quantifying entanglement such as concurrence, relative entropy of entanglement (REE) and negativity, which cannot be increased via local operations. It was found that for sets of non-maximally entangled states of two qubits, comparing these entanglement measures may lead to different entanglement orderings of the states. On the other hand, although it is not an entanglement measure and not monotonic under local operations, due to its ability of detecting multipartite entanglement, quantum Fisher information (QFI) has recently received an intense attraction generally with entanglement in the focus. In this work, we revisit the state ordering problem of general two qubit states. Generating a thousand random quantum states and performing an optimization based on local general rotations of each qubit, we calculate the maximal QFI for each state. We analyze the maximized QFI in comparison with concurrence, REE and negativity and obtain new state orderings. We show that there are pairs of states having equal maximized QFI but different values for concurrence, REE and negativity and vice versa.
Using extensive numerical analysis of 20,000 randomly generated two-qubit states, we provide a quantitative analysis of the connection between entanglement measures and Maximized Quantum Fisher Information (MQFI). Our systematic study shows strong empirical relationships between the metrological capacity of quantum states and three different entanglement measures: concurrence, negativity, and relative entropy of entanglement. We show that optimization over local unitary transformations produces substantially more predictable relationships than fixed-generator quantum Fisher information approaches using sophisticated statistical analysis, such as bootstrap resampling, systematic data binning, and multiple model comparisons. With exponential fits reaching R2>0.99R^2 > 0.99R2>0.99 and polynomial models reaching R2=0.999R^2 = 0.999R2=0.999, we offer thorough empirical support for saturation behavior in quantum metrological advantage. With immediate applications to realworld quantum sensing protocols, our findings directly empirically validate important predictions from quantum resource theory and set fundamental bounds for quantum sensor optimization and resource allocation. These intricate relationships are quantitatively described by the polynomial and exponential fit equations, which offer crucial real-world direction for the design of quantum sensors.
We investigate the speed at which entanglement can be generated by an interaction parameter encoded in a two-qubit Hamiltonian, quantified by the derivative of concurrence with respect to the coupling parameter. For arbitrary pure two-qubit states evolving under a general nonlocal interaction, we derive a bound relating this entanglement speed to the quantum Fisher information (QFI). Specifically, we show that ∣∂gC∣≤FQ(g)1−C2|\partial_g C| \le \sqrt{F_Q^{(g)}} \sqrt{1-C^2}∣∂g​C∣≤FQ(g)​​1−C2​, where FQ(g)F_Q^{(g)}FQ(g)​ is the QFI associated with estimation of the parameter. This establishes FQ1−C2\sqrt{F_Q}\sqrt{1-C^2}FQ​​1−C2​ as an upper bound on the speed of entanglement generation in parameter space. We further derive the saturation conditions and identify the states and dynamical regimes for which equality is attained. At saturation, concurrence evolves at the maximum rate permitted by the distinguishability of the underlying quantum state. These results reveal a direct connection between quantum metrology and entanglement generation, showing that the same information-theoretic quantity that governs parameter-estimation precision also limits the speed at which entanglement resources can be created.
The violation of Bell's theorem is a very simple way to see that there is no underlying classical interpretation of quantum mechanics. The measurements made on the photons shows that light signal (information) could travel between them, hence completely eliminating any chance that the result was due to anything other than entanglement. Entanglement has been studied extensively for understanding the mysteries of non-classical correlations between quantum systems. It was found that violation of Bell's inequalities could be trivially calculated and for sets of nonmaximally entangled states of two qubits, comparing these entanglement measures may lead to different entanglement orderings of the states. On the other hand, although it is not an entanglement measure and not monotonic under local operations, due to its ability of detecting multipartite entanglement, quantum Fisher information (QFI) has recently received an intense attraction generally with entanglement in the focus. In this work, we visit violation of Bell's inequalities problem with a different approach. Generating a thousand random quantum states and performing an optimization based on local general rotations of each qubit, we calculate the maximal QFI for each state. We analyze the maximized QFI in comparison with violation in Bell's inequalities and we make similar comparison of this violation with commonly studied entanglement measures, negativity and relative entropy of entanglement. We show that there are interesting orderings for system states.