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arXiv:2302.06574 · Physical Review A

Hierarchies among genuine multipartite entangling capabilities of quantum gates

Mrinmoy Samanta, Sudipta Mondal, Samir Kumar Hazra, Aditi Sen (De)

Published Dec 12, 2025• 0 citations
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Abstract

We classify quantum gates according to their capability to generate genuine multipartite entanglement (GME), using a hierarchy based on multipartite separable states. In particular, when a fixed unitary operator acts on the set of k-separable states, the maximal genuine multipartite entanglement content produced via that particular unitary operator is determined after maximizing over the set of k-separable input states. We identify unitary operators that are beneficial for generating high GME when the input states are entangled in some bipartition, although the picture can also be reversed, where such initial entanglement offers no advantage. We investigate the maximum entangling power of a broad range of unitary operators, encompassing special classes of quantum gates, as well as diagonal, permutation, and Haar-uniformly generated unitaries by computing generalized geometric measure (GGM) as a GME quantifier. Additionally, we observe a notable distinction in entangling power based on the nature of the input states: when maximization is restricted to separable states with real coefficients, the entangling power is lower than when the optimization is carried out over arbitrary separable states with complex coefficients, thereby highlighting the role of complex amplitudes in entanglement creation. Furthermore, we determine which unitary operators, along with their corresponding optimal inputs, yield output states with the highest achievable GGM.

DOI
10.1103/vhvd-jbnk

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We classify quantum gates according to their capability to generate genuine multipartite entanglement (GME), using a hierarchy based on multipartite separable states. In particular, when a fixed unitary operator acts on the set of k-separable states, the maximal genuine multipartite entanglement content produced via that particular unitary operator is determined after maximizing over the set of k-separable input states. We identify unitary operators that are beneficial for generating high GME when the input states are entangled in some bipartition, although the picture can also be reversed, where such initial entanglement offers no advantage. We investigate the maximum entangling power of a broad range of unitary operators, encompassing special classes of quantum gates, as well as diagonal, permutation, and Haar-uniformly generated unitaries by computing generalized geometric measure (GGM) as a GME quantifier. Additionally, we observe a notable distinction in entangling power based on the nature of the input states: when maximization is restricted to separable states with real coefficients, the entangling power is lower than when the optimization is carried out over arbitrary separable states with complex coefficients, thereby highlighting the role of complex amplitudes in entanglement creation. Furthermore, we determine which unitary operators, along with their corresponding optimal inputs, yield output states with the highest achievable GGM.
We propose a deterministic scheme of generating genuine multiparty entangled states in quantum networks of arbitrary size having various geometric structures---we refer to it as entanglement circulation. The procedure involves optimization over a set of two-qubit arbitrary unitary operators and the entanglement of the initial resource state. We report that the set of unitary operators that maximize the genuine multipartite entanglement quantified via generalized geometric measure (GGM) is not unique. We prove that the GGM of the resulting state of arbitrary qubits coincides with the minimum GGM of the initial resource states. By fixing the output state as the six-qubit one, we find the optimal way to create such states according to the available resource. Moreover, we show that the method proposed here can be implemented by using logic gates, or by using the time dynamics of realizable spin Hamiltonians. In case of an ordered system, GGM varies periodically with time while the evolution via disordered models lead to a low but constant multipartite entanglement in outputs at a critical time, which decreases exponentially with the increase of the strength of the disorder.
Multipartite entanglement is a fundamental aspect of quantum mechanics, crucial to advancements in quantum information processing and quantum computation. Within this field, Genuinely Multipartite Entanglement (GME), being entangled in all bipartitions, and Absolutely Maximally Entanglement (AME), maximally entangled in all bipartitions, represent two significant types of entanglement with diverse applications. In this work, we introduce a new measure called the GME-AME multipartite entanglement measure, with a non-zero value representing the GME states and the maximum value is reached only by the AME states. The measure is applied to study the multipartite entanglement of four partite systems using the operator to state mapping, and the four partite permutation qutrit states are classified according to the measure. With various examples, we show that our measure is robust in classifying the four partite entangled states.
Genuine multipartite entanglement (GME) is a valuable resource for quantum technologies, but quantifying it in many-body systems can be demanding since measures defined through optimizations over bipartitions are often assumed to require large joint measurements and many measurement settings. We introduce the concepts of GME evaluation length and GME evaluation number, which capture, respectively, the largest subsystem that must be jointly measured and the number of distinct subsystems that need to be measured to evaluate multipartite entanglement measures defined via bipartition optimizations. Taking the genuine multipartite concurrence (GMC) as a canonical example, we show that, for NNN-qubit Haar-random pure states, the GMC can be evaluated using only NNN single-qubit purity measurements. Thus, for typical many-body states, evaluating the GMC requires considering only the NNN single-site bipartitions, rather than the exponentially many distinct bipartitions. We further corroborate this behavior numerically for physically accessible states generated by local random quantum circuits. Our results show that the practical cost of quantifying GME can be dramatically lower than suggested by its formal definition, establishing measurement complexity as a complementary operational lens for multipartite entanglement.
Genuine multipartite entanglement (GME) is considered a powerful form of entanglement since it corresponds to those states that are not biseparable, i.e. a mixture of partially separable states across different bipartitions of the parties. In this work we study this phenomenon in the multiple-copy regime, where many perfect copies of a given state can be produced and controlled. In this scenario the above definition leads to subtle intricacies as biseparable states can be GME-activatable, i.e. several copies of a biseparable state can display GME. We show that the set of GME-activatable states admits a simple characterization: a state is GME-activatable if and only if it is not partially separable across one bipartition of the parties. This leads to the second question of whether there is a general upper bound in the number of copies that needs to be considered in order to observe the activation of GME, which we answer in the negative. In particular, by providing an explicit construction, we prove that for any number of parties and any number k∈N there exist GME-activatable multipartite states of fixed (i.e. independent of k ) local dimensions such that k copies of them remain biseparable.
Genuine multipartite entanglement (GME) is considered a powerful form of entanglement since it corresponds to those states that are not biseparable, i.e. a mixture of partially separable states across different bipartitions of the parties. In this work we study this phenomenon in the multiple-copy regime, where many perfect copies of a given state can be produced and controlled. In this scenario the above definition leads to subtle intricacies as biseparable states can be GME-activatable, i.e. several copies of a biseparable state can display GME. We show that the set of GMEactivatable states admits a simple characterization: a state is GME-activatable if and only if it is not partially separable across one bipartition of the parties. This leads to the second question of whether there is a general upper bound in the number of copies that needs to be considered in order to observe the activation of GME, which we answer in the negative. In particular, by providing an explicit construction, we prove that for any number of parties and any number k 2 N there exist GME-activatable multipartite states of fixed (i.e. independent of k) local dimensions such that k copies of them remain biseparable.
We compare the multipartite entangling and disentangling powers of unitary operators by assessing their ability to generate or eliminate genuine multipartite entanglement. Our findings reveal that while diagonal unitary operators can exhibit equal entangling and disentangling powers, certain non-diagonal unitaries demonstrate an imbalance when acting on fully separable states, thereby extending the known disparity from bipartite systems to those with any number of parties. Counterintuitively, we construct classes of unitaries and their adjoints that display unequal entanglement generation capacities, behaving differently when applied to systems with an even number of qubits compared to those with an odd number. Further, we illustrate that this asymmetry can be simulated using physically realizable Hamiltonians: systems with an even number of qubits employ nearest-neighbor Dzyaloshinskii-Moriya (DM) interactions, while those with an odd number utilize a combination of Heisenberg and DM interactions. Additionally, we present a circuit composed of random noncommuting unitaries, constructed from alternating layers of two-qubit Haar-random gates, to illustrate the discrepancy in the entangling and disentangling capabilities of unitaries.