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Complete Factorization of Entangled Quantum States PART 2: Factorizing the GHZn and W States Using Semi-Structured Complex Numbers

Peter Jean‐Paul, Shanaz Wahid

Published Aug 25, 2026• 0 citations
Publisher

Abstract

This paper is a second in a three-part series providing an auxiliary parameter method for factorizing maximally entangled states. A major vulnerability of entangled states is they cannot be factored, leaving the whole composite system highly susceptible to decoherence. Factorizing these states creates a protective barrier that prevents local noise or measurements in one subsystem corrupting the rest of the system. This paper introduces an auxiliary parameter method that uses semi-structured complex numbers to create (1) an unphysical auxiliary parameter kn and (2) a unique j-conjugate framework to embed maximally entangled systems in semi-structured state space and factorize them into independent single-qubit product states. To demonstrate this method the maximally entangled GHZn and W States were completely factorized. These factorizations strictly satisfy classical local realism, passing the Mermin Inequality Test (for GHZ-type states) and the MARS inequality test (for W states) whilst in semi-structured state space. Finally, an inverse transformation maps these auxiliary terms back to zero, recovering the original standard Euclidean quantum states. This framework provides a novel algebraic tool for manipulating maximally entangled states as formal tensor product states.

DOI
10.20944/preprints202608.1721.v1

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This paper is a second in a three-part series providing an auxiliary parameter method for factorizing maximally entangled states. A major vulnerability of entangled states is they cannot be factored, leaving the whole composite system highly susceptible to decoherence. Factorizing these states creates a protective barrier that prevents local noise or measurements in one subsystem corrupting the rest of the system. This paper introduces an auxiliary parameter method that uses semi-structured complex numbers to create (1) an unphysical auxiliary parameter kn and (2) a unique j-conjugate framework to embed maximally entangled systems in semi-structured state space and factorize them into independent single-qubit product states. To demonstrate this method the maximally entangled GHZn and W States were completely factorized. These factorizations strictly satisfy classical local realism, passing the Mermin Inequality Test (for GHZ-type states) and the MARS inequality test (for W states) whilst in semi-structured state space. Finally, an inverse transformation maps these auxiliary terms back to zero, recovering the original standard Euclidean quantum states. This framework provides a novel algebraic tool for manipulating maximally entangled states as formal tensor product states.
Abstract Tracking non-local correlations through complex quantum circuits presents a massive computational bottleneck, as standard linear algebra forces multi-qubit state simulations to scale exponentially ( 2n\:{2}^{n}2n). This paper introduces a novel intermediate algebraic framework using semi-structured complex numbers to overcome non-separability constraints during intermediate tracking. By strategically mapping vanishing Euclidean coefficients to non-zero auxiliary forms ( −pk\:-pk−pk), an unphysical scaffolding layer is introduced that permits the factorization of maximally entangled multiqubit systems into independent, single-qubit tensor product states. Note:  p\:pp is the semi-structured complex unit and  k \:k\:kis an auxiliary variable. To validate the mathematical consistency and probability conservation of this extended state space, the bipartite Bell states are completely factorized and subjected to localized operator matrices, demonstrating strict compliance with statistical scalar product rules. Finally, an algebraic inverse transformation cleanly annihilates the auxiliary parameters. This process projects the system back to the physical domain, perfectly recovering standard Euclidean quantum states with zero loss of physical data. This framework serves as a practical analytical tool for quantum compiler optimization, allowing localized qubit tracking without altering fundamental quantum mechanics.
Abstract Self-testing was originally introduced as a device-independent method of certification of entangled quantum states and local measurements performed on them. Recently, in (Baccari et al 2020 Phys. Rev. Lett. 125 260507) the notion of state self-testing has been generalized to entangled subspaces and the first self-testing strategies for exemplary genuinely entangled subspaces have been given. The main aim of our work is to pursue this line of research and to address the question how ‘large’ (in terms of dimension) are genuinely entangled subspaces that can be self-tested, concentrating on the multiqubit stabilizer formalism. To this end, we first introduce a framework allowing to efficiently check whether a given stabilizer subspace is genuinely entangled. Building on it, we then determine the maximal dimension of genuinely entangled subspaces that can be constructed within the stabilizer subspaces and provide an exemplary construction of such maximally-dimensional subspaces for any number of qubits. Third, we construct Bell inequalities that are maximally violated by any entangled state from those subspaces and thus also any mixed states supported on them, and we show these inequalities to be useful for self-testing. Interestingly, our Bell inequalities allow for identification of higher-dimensional face structures in the boundaries of the sets of quantum correlations in the simplest multipartite Bell scenarios in which every observer performs two dichotomic measurements.
We analytically characterize the maximal quantum violation of the extended Mermin-Ardehali-Belinskii-Klyshko (EMABK) family of inequalities by nnn-qubit generalized Greenberger-Horne-Zeilinger (GHZ) states. We develop a correlation-tensor approach in which the Bell value is expressed as the Frobenius inner product of the quantum correlation tensor and an effective coefficient tensor. A rank constraint on the latter, together with the von Neumann trace inequality, yields an upper bound governed by the two largest singular values of the reshaped correlation-tensor. For generalized GHZ states, we determine the complete singular spectrum and obtain a piecewise analytic upper bound. We then construct two complementary measurement strategies that saturate the bound for EMABK throughout the entire parameter range. The first is a purely MABK strategy in which all measurement directions lie in the equatorial plane of the Bloch sphere. The second is a hybrid strategy that recursively combines lower-order MABK anti-diagonal operators with the fully-σzσ_zσz​ tensor product. The resulting piecewise analytic expression recovers the standard MABK Tsirelson bound 2(n−1)/22^{(n-1)/2}2(n−1)/2 in the maximally entangled limit and approaches the classical local-hidden-variable bound in the product-state limits. It further shows that every entangled generalized GHZ state exhibits a strict quantum-classical separation under the EMABK inequality, thereby eliminating the nonviolation region of the standard MABK inequality in the partially entangled regime.
In chapter 2 various parameterizations for the orbits under local unitary transformations of three-qubit pure states are analyzed. It is shown that the entanglement monotones of any multipartite pure state uniquely determine the orbit of that state. It follows that there must be an entanglement monotone for three-qubit pure states which depends on the Kempe invariant defined in [1]. A form for such an entanglement monotone is proposed. A theorem is proved that significantly reduces the number of entanglement monotones that must be looked at to find the maximal probability of transforming one multipartite state to another. In chapter 3 Grover's unstructured quantum search algorithm is generalized to use an arbitrary starting superposition and an arbitrary unitary matrix. A formula for the probability of the generalized Grover's algorithm succeeding after n iterations is derived. This formula is used to determine the optimal strategy for using the unstructured quantum search algorithm. The speedup obtained illustrates that a hybrid use of quantum computing and classical computing techniques can yield a performance that is better than either alone. The analysis is extended to the case of a society of k quantum searches acting in parallel. In chapter 4 the positive map Г : p → (Trρ) - ρ is introduced as a separability criterion. Any separable state is mapped by the tensor product of Г and the identity in to a non-negative operator, which provides a necessary condition for separability. If Г acts on a two-dimensional subsystem, then it is equivalent to partial transposition and therefore also sufficient for 2 x 2 and 2 x 3 systems. Finally, a connection between this map for two qubits and complex conjugation in the magic basis [2] is displayed.
In quantum physics, multiparticle systems are described by quantum states acting on tensor products of Hilbert spaces. This product structure leads to the distinction between product states and entangled states; moreover, one can quantify entanglement by considering the distance of a quantum state to the set of product states. The underlying optimization problem occurs frequently in physics and beyond, for instance in the computation of the injective tensor norm in multilinear algebra. Here, we introduce a method to determine the maximal overlap of a pure multiparticle quantum state with product states based on considering several copies of the pure state. This leads to three types of hierarchical approximations to the problem, all of which we prove to converge to the actual value. Besides allowing for the computation of the geometric measure of entanglement, our results can be used to tackle optimizations over stochastic local transformations, to find entanglement witnesses for weakly entangled bipartite states, and to design strong separability tests for mixed multiparticle states. Finally, our approach sheds light on the complexity of separability tests.
A multi-qubit pure quantum state is called separable when it can be factored as the tensor product of 1-qubit pure quantum states. Factorizing a general multi-qubit pure quantum state into the tensor product of its factors (pure states containing a smaller number of qubits) can be a challenging task, especially for highly entangled states. A new criterion based on the proportionality of the rows of certain associated matrices for the existence of certain factorization and a factorization algorithm that follows from this criterion for systematically extracting all the factors is developed in this paper. 3-qubit pure states play a crucial role in quantum computing and quantum information processing. For various applications, the well-known 3-qubit GHZ state which contains two nonzero terms, and the 3-qubit W state which contains three nonzero terms, have been studied extensively. Using the new factorization algorithm developed here we perform a complete analysis vis-à-vis entanglement of 3-qubit states that contain exactly two nonzero terms and exactly three nonzero terms.