Entanglement can show fundamentally different behavior in fermionic systems. However, while bipartite measures of fermionic entanglement have been established, genuine multiparty entanglement (GME) in fermionic systems is much less understood. We introduce an efficiently computable measure (via semi-definite programming) of fermionic GME, fermion genuine multiparty negativity (fGMN), and show that it is an entanglement monotone and a natural multipartite extension of the fermionic negativity. Using this measure, we find many states which have fGMN but no non-fermionic GME, including large classes of fermionic stabilizer states. The fGMN also displays some major phenomenological differences to the bipartite fermionic negativity, such as finite sudden death points over separation and temperature.
Entanglement can show fundamentally different behavior in fermionic systems. However, while bipartite measures of fermionic entanglement have been established, genuine multiparty entanglement (GME) in fermionic systems is much less understood. We introduce an efficiently computable measure (via semi-definite programming) of fermionic GME, fermion genuine multiparty negativity (fGMN), and show that it is an entanglement monotone and a natural multipartite extension of the fermionic negativity. Using this measure, we find many states which have fGMN but no non-fermionic GME, including large classes of fermionic stabilizer states. The fGMN also displays some major phenomenological differences to the bipartite fermionic negativity, such as finite sudden death points over separation and temperature.
Genuine Multipartite Entanglement Measure Based on α-concurrence
Quantifying genuine entanglement is a crucial task in quantum information theory. Based on the geometric mean of bipartite α-concurrences among all bipartitions, we present a class of well-defined genuine multipartite entanglement (GME) measures GαC with one parameter α for arbitrary multipartite states. We show that the GαC is of continuity for any multipartite pure states. By utilizing the related symmetry, analytical results of GαC are derived for any n-qubit GHZ states and W states, which show that the GHZ states are more genuinely entangled than the W states. With explicit examples, we demonstrate that the GαC can distinguish different GME states that other GME measures fail to. These results justify the potential applications of GαC in characterizing genuine multipartite entanglements.
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.
We introduce the genuine multipartite Rains entanglement (GMRE) as a measure of genuine multipartite entanglement that can be computed using semi-definite programming. Similar to the Rains relative entropy (its bipartite counterpart), the GMRE is monotone under selective quantum operations that completely preserve the positivity of the partial transpose, implying that it is a multipartite entanglement monotone. As a consequence, we show that the GMRE bounds both the one-shot standard and probabilistic approximate GHZ-distillable entanglement from above. We also develop a generalization of this quantity that incorporates other entropies, including quantum Renyi relative entropies.
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 N-qubit Haar-random pure states, the GMC can be evaluated using only N single-qubit purity measurements. Thus, for typical many-body states, evaluating the GMC requires considering only the N 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.
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.
A complete characterization and quantification of entanglement, particularly the multipartite entanglement, remains an unfinished long-term goal in quantum information theory. As long as the multipartite system is concerned, the relation between the entanglement contained in different partitions or different subsystems need to take into account. The complete multipartite entanglement measure and the complete monogamy relation is a framework that just deals with such a issue. In this paper, we put forward conditions to justify whether the multipartite entanglement monotone (MEM) and genuine multipartite entanglement monotone (GMEM) are complete, completely monogamous, and tightly complete monogamous according to the feature of the reduced function. Especially, we proposed a class of complete MEMs and a class of complete GMEMs via the maximal reduced function for the first time. By comparison, it is shown that, for the tripartite case, this class of GMEMs is better than the one defined from the minimal bipartite entanglement in literature under the framework of complete MEM and complete monogamy relation. In addition, the relation between monogamy, complete monogamy, and the tightly complete monogamy are revealed in light of different kinds of MEMs and GMEMs.