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arXiv:2610.13579

Correlation-Based Measurement of Entanglement in Rank-2 Mixed States

M. I. Samar, V. M. Tkachuk

quant-phSubmitted Oct 9, 2026• Announced Oct 12, 2026• 0 citations
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Abstract

We study correlation-based entanglement evaluation in rank-2 mixed states of two- and multi-qubit systems. The relevant two-dimensional subspace is treated as a logical qubit embedded in the physical Hilbert space, and factorizable realizations of its logical Pauli operators are classified. For two qubits, this yields nontrivial canonical encodings related by local Clifford transformations. The relation between logical and physical Bloch vectors admits a geometric interpretation as a transversely deformed Bloch sphere, leading to closed-form expressions for two-qubit concurrence and generalized concurrence across $N$-qubit bipartitions. The entanglement scaling factor is obtained directly from complementary-subsystem correlations, without prior calibration of state-preparation parameters. Noisy simulations using the IBM Quantum FakeKingston backend model, with readout-error mitigation and zero-noise extrapolation, confirm the analytical results up to $N=10$ qubits. The protocol therefore enables entanglement estimation from only a small number of measurement settings for this class of rank-2 logical states.

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We study correlation-based entanglement evaluation in rank-2 mixed states of two- and multi-qubit systems. The relevant two-dimensional subspace is treated as a logical qubit embedded in the physical Hilbert space, and factorizable realizations of its logical Pauli operators are classified. For two qubits, this yields nontrivial canonical encodings related by local Clifford transformations. The relation between logical and physical Bloch vectors admits a geometric interpretation as a transversely deformed Bloch sphere, leading to closed-form expressions for two-qubit concurrence and generalized concurrence across NNN-qubit bipartitions. The entanglement scaling factor is obtained directly from complementary-subsystem correlations, without prior calibration of state-preparation parameters. Noisy simulations using the IBM Quantum FakeKingston backend model, with readout-error mitigation and zero-noise extrapolation, confirm the analytical results up to N=10N=10N=10 qubits. The protocol therefore enables entanglement estimation from only a small number of measurement settings for this class of rank-2 logical states.
The geometric measure of entanglement of mixed states is investigated based on its relation to spin correlations. We derive an analytical expression for the entanglement of rank-2 mixed states composed of arbitrary multiqubit pure states within a two-dimensional Hilbert subspace. The dependence of the entanglement of rank-2 mixed multiqubit states on the parameters characterizing arbitrary pure states in the mixture is studied both analytically and using quantum computing. For a particular class of mixed states, namely, equal-probability mixtures of an arbitrary multiqubit state and the Schrödinger cat state, we construct quantum protocols for detecting the geometric measure of entanglement based on its relation to spin correlations. We quantify the entanglement using both the AerSimulator and the ibm_marrakesh quantum computer. The effects of quantum-computing errors are also analyzed. The simulation results reproduce the analytical dependencies, while measurements on the quantum device demonstrate good agreement with theoretical predictions within estimated error bounds.
Quantum error mitigation (QEM) provides a practical route for estimating reliable observables on noisy intermediate-scale quantum (NISQ) devices. Traditional QEM strategies, including zero-noise extrapolation (ZNE) and Clifford data regression (CDR), rely on noise scaling or global regression, and their performance is constrained by the exponential growth of the system degrees of freedom. We construct a graph-enhanced mitigation (GEM) framework, which incorporates physical information into the model representation. In this work, quantum circuits are encoded as attributed graphs. Hardware-level physical information is mapped to node and edge features: local noise parameters such as calibration parameters T1T_1T1​, T2T_2T2​, and readout errors are encoded at nodes, while coupling-related information such as two-qubit gate errors is encoded as edge features. Graph neural networks are used to model how errors propagate along the physical coupling structure and build up into non-local correlations. This allows the model to capture local interactions and part of the resulting non-local correlations across qubits. A dual-branch affine correction is applied to maintain consistency with physical constraints. Experiments on 10-qubit and 16-qubit random circuits executed on superconducting quantum processors show that GEM provides a level of accuracy comparable to CDR at small scales, while yielding lower mean absolute error and improved stability in zero-shot transfer to larger systems. Results of the traditional QEM strategy indicate that global regression methods remain effective in low-dimensional settings but become less reliable as system degrees of freedom grow. In contrast, GEM makes use of local physical structures to show better scalability and generalization, while preserving the overall error propagation patterns. This work provides a practical scalable approach to QEM for NISQ devices.
Developing space- and time-efficient logical magic state preparation protocols will likely be an essential step towards building a large-scale fault-tolerant quantum computer. Motivated by this need, we introduce a scalable method for simulating logical magic state preparation protocols under the standard circuit-level noise model. When applied to protocols based on code switching, magic state cultivation, and magic state distillation, our method yields a complexity polynomial in (i) the number of qubits and (ii) the non-stabilizerness, e.g., stabilizer rank or Pauli rank, of the target encoded magic state. The efficiency of our simulation method is rooted in a curious fact: every circuit-level Pauli error in these protocols propagates to a Clifford error at the end. This property is satisfied by a large family of protocols, including those that repeatedly measure a transversal Clifford that squares to a Pauli. We provide a proof-of-principle numerical simulation that prepares a magic state using such logical Clifford measurements. Our work enables practical simulation of logical magic state preparation protocols without resorting to approximations or resource-intensive state-vector simulations.
While the structure of entangled quantum states is relatively well understood, the characterization of entangled measurements, especially in multipartite and high-dimensional settings, remains far less developed. In this work, we introduce a general approach to construct highly symmetric, locally encodable orthonormal measurement bases, as orbits of a single fiducial state under tensor-product actions of Pauli subgroups. This framework recovers the Elegant Joint Measurement-a two-qubit measurement whose local marginals form a regular tetrahedron on the Bloch sphere-as a special case, and we extend the construction to both more systems and higher dimensions. We analyze the entanglement cost required to implement these measurements locally via the Clifford hierarchy and use this criterion to classify them. We show how the symmetry of our constructions allows us to characterize their localizability, which is generally a challenging problem, and to identify certain classes of measurement bases that are efficiently localizable. Our approach offers a systematic toolkit for designing entangled measurements with rich symmetry and implementability properties.
Entanglement, and quantum correlation, are precious resources for quantum technologies implementation based on quantum information science, such as quantum communication, quantum computing, and quantum interferometry. Nevertheless, to our best knowledge, a directly or numerically computable measure for the entanglement of multipartite mixed states is still lacking. In this work, (i) we derive a measure of the degree of quantum correlation for mixed multipartite states. The latter possesses a closed-form expression valid in the general case unlike, to our best knowledge, all other known measures of quantum correlation. (ii) We further propose an entanglement measure, derived from this quantum correlation measure using a novel regularization procedure for the density matrix. Therefore, a comparison of the proposed measures, of quantum correlation and entanglement, allows one to distinguish between quantum correlation detached from entanglement and the one induced by entanglement and, hence, to identify separable but non-classical states. We have tested our quantum correlation and entanglement measures, on states well-known in literature: a general Bell diagonal state and the Werner states, which are easily tractable with our regularization procedure, and we have verified the accordance between our measures and the expected results for these states. Finally, we validate the two measures in two cases of multipartite states. The first is a generalization to three qubits of the Werner state, the second is a one-parameter three qubits mixed state interpolating between a bi-separable state and a genuine multipartite state, passing through a fully separable state.