Organising the space of entanglement structures of a multipartite quantum system is a much more challenging task than its bipartite version: while the local unitary (LU) orbit of a bipartite pure state can be conveniently characterized by its entanglement spectrum, invariants of multipartite entanglement structures are comparatively difficult to define and work with. The root cause of this difference is that the bipartite problem can be reduced to the analysis of matrix invariants, while its multipartite version is governed by a much richer space of tensor invariants. The present work explores the latter through the lens of so-called trace-invariants, which are in one-to-one correspondence with combinatorial objects known as colored graphs. We first explain why trace-invariant evaluations can serve as labels of LU-orbits of multipartite pure states, how this strategy extends to random states, and how the effect of local operations (LO) can be analyzed through such data. We then focus on entanglement classification within an (infinite-dimensional) subspace of reference states, whose basic building blocks are GHZ states of various dimensions. We show that relatively simple subclasses of trace-invariants are sufficient to separate the LU-orbits of reference states, and enable a complete (resp. an incomplete) characterization of their relations in the LO (resp. LOCC) resource theory of entanglement. Finally, we investigate how a (still infinite) subclass of reference states of local dimension N can be efficiently distinguished at leading and subleading orders in an asymptotic large-N expansion (among themselves, or from Haar-random states). This analysis relies crucially on combinatorial quantities associated to colored graphs, some of which have already played instrumental roles in the recent literature on random tensors. Results of broader relevance are reported along the way.
The entanglement structure of a bipartite pure state is characterized by its entanglement spectrum, or equivalently, by the family of Rényi-
k entanglement entropies with integer
k. Taking the difference of two such entropies defines a quantity known as an entanglement spread, that is monotonous under local operations, and that vanishes if and only if the entanglement spectrum of the state is flat. We introduce a multipartite generalization of this notion which, for a fixed number
D≥3 of parties, results in a collection of maps indexed by pairs of
D-edge-colored graphs (obeying some condition). Each member of this collection takes the form of a difference of two multipartite Rényi entanglement entropies, each associated to a local unitary polynomial invariant (known in this context as a trace-invariant), and is shown to be monotonous under local operations. We then argue that a previously introduced family of so-called hypergraph-tensor (HT) states can be understood as a multipartite counterpart to the family of flat bipartite states. Indeed, we first prove that, when
D=3, a tripartite pure state is HT if and only if it minimizes a particular (infinite) family of entanglement spreads. Second, we embed the set of HT states into the much larger family of spectral hypergraph-tensor (SHT) states, defined by a new Ansatz we introduce. We then prove that HT states can be uniquely characterized as the minimizers of some fixed entanglement spread among SHT states, for arbitrary
D≥3. We also investigate SHT states in their own right: in particular, we characterize their orbits under local unitary transformations. Finally, along the way, the asymptotic large-
N expectation values of a number of entanglement spreads are computed in the Haar-random state of local dimension
N.
Research on quantum states often focuses on the correlation between nonlocal effects and local unitary invariants, among which local unitary equivalence plays a significant role in quantum state classification and resource theories. This paper focuses on the local unitary equivalence of multipartite quantum states in quantum information theory, aiming to determine a complete set of invariants that identify their local unitary orbits; these invariants are crucial for deriving polynomial invariants and describing the physical properties preserved under local unitary transformations. The study deeply explores the characterization of local unitary equivalence and the method of detecting entanglement using local unitary Bargmann invariants. Taking two-qubit systems as an example, it verifies the measurability of invariants that determine equivalence and establishes a connection between Makhlin fundamental invariants (a complete set of 18 local unitary invariants for two-qubit states) and local unitary Bargmann invariants. These Bargmann invariants, related to the traces of products of density operators and marginal states, can be measured through cycle tests (an extended form of swap tests).
Local unitary equivalence is an important ingredient for quantifying and classifying entanglement. Verifying whether or not two quantum states are local unitary equivalent is a crucial problem, where only the case of multipartite pure states is solved. For mixed states, however, the verification of local unitary equivalence is still a challenging problem. In this paper, based on the coefficient matrices of generalized Bloch representations of quantum states, we find a variety of local unitary invariants for arbitrary-dimensional bipartite quantum states. These invariants are operational and can be used as necessary conditions for verifying the local unitary equivalence of two quantum states. Furthermore, we extend the construction to the arbitrary-dimensional multipartite case. We finally apply these invariants to estimate concurrence, a vital entanglement measure, showing the practicability of local unitary invariants in characterizing entanglement.
Quantum Entanglement is one of the key manifestations of quantum mechanics that separate the quantum realm from the classical one. Characterization of entanglement as a physical resource for quantum technology became of uppermost importance. While the entanglement of bipartite systems is already well understood, the ultimate goal to cope with the properties of entanglement of multipartite systems is still far from being realized. This paper covers characterization of multipartite entanglement using algebraic-geometric tools. First, we establish an algorithm to classify multipartite entanglement by [Formula: see text]-secant varieties of the Segre variety and [Formula: see text]-multilinear ranks that are invariant under Stochastic Local Operations with Classical Communication (SLOCC). We present a fine-structure classification of multiqubit and tripartite entanglement based on this algorithm. Another fundamental problem in quantum information theory is entanglement transformation that is quite challenging regarding to multipartite systems. It is captivating that the proposed entanglement classification by algebraic geometry can be considered as a reference to study SLOCC and asymptotic SLOCC interconversions among different resources based on tensor rank and border rank, respectively. In this regard, we also introduce a new class of tensors that we call persistent tensors and construct a lower bound for their tensor rank. We further cover SLOCC convertibility of multipartite systems considering several families of persistent tensors.
Entanglement cohomology assigns a graded cohomology ring to a multipartite pure state, providing homological invariants that are stable under local unitaries and characterize inequivalent patterns of entanglement. In this work we derive exact expressions for the dimensions of these cohomology groups in two canonical entanglement classes, generalized GHZ and W states on an arbitrary number of parties and local Hilbert space dimensions, thus proving conjectures of arXiv:1901.02011. Using the additional structure of the Hodge star and wedge product operations, we propose two new classes of local unitary invariants: the spectrum of the natural Laplacian acting on entanglement
k-forms, and the intersection numbers obtained from wedge products of representatives for cohomology classes. We present numerical experiments which investigate these invariants in particular states, suggesting that they may provide useful quantities for describing multipartite entanglement.
Existing methods for deciding whether a bipartite quantum state is separable or entangled typically fall into one of two categories: they are either complete but require access to an explicit density matrix followed by numerical optimization, or they can be evaluated directly by measuring the quantum system but are incomplete, in the sense that they cannot detect all forms of entanglement. In this work, we overcome both limitations in a unified framework. First, we bypass numerical optimization by deriving separability criteria in the form of universal bounds on tensor powers of separable states. We prove that these bounds are complete: every entangled state violates them for sufficiently large tensor powers. Second, we explicitly construct a corresponding complete family of nonlinear entanglement witnesses, which can detect all forms of entanglement without requiring an explicit density matrix. The witnesses we construct are moreover basis-independent, in the sense that they are invariant under conjugation by local unitaries. Altogether, our results expand the toolbox for entanglement detection in arbitrary local dimensions in a manifestly invariant way.