The rank of a tensor is analysed in the context of quantum entanglement. A pure quantum state v of a composite system consisting of d subsystems with n levels each is viewed as a vector in the d-fold tensor product of n-dimensional Hilbert space and can be identified with a tensor with d indices, each running from 1 to n. We discuss the notions of the generic rank and the maximal rank of a tensor and review results known for the low dimensions. Another variant of this notion, called the border rank of a tensor, is shown to be relevant for the characterization of orbits of quantum states generated by the group of special linear transformations. A quantum state v is called entangled, if it cannot be written in the product form, v≠v1⊗v2⊗⋯⊗vd, what implies correlations between physical subsystems. A relation between various ranks and norms of a tensor and the entanglement of the corresponding quantum state is revealed.
The rank of a tensor is analysed in the context of quantum entanglement. A pure quantum state v of a composite system consisting of d subsystems with n levels each is viewed as a vector in the d-fold tensor product of n-dimensional Hilbert space and can be identified with a tensor with d indices, each running from 1 to n. We discuss the notions of the generic rank and the maximal rank of a tensor and review results known for the low dimensions. Another variant of this notion, called the border rank of a tensor, is shown to be relevant for the characterization of orbits of quantum states generated by the group of special linear transformations. A quantum state v is called entangled, if it cannot be written in the product form, v≠v1⊗v2⊗⋯⊗vd, what implies correlations between physical subsystems. A relation between various ranks and norms of a tensor and the entanglement of the corresponding quantum state is revealed.
The rank of a tensor is analyzed in context of quantum entanglement. A pure quantum state v of a composite system consisting of d subsystems with n levels each is viewed as a vector in the d-fold tensor product of n-dimensional Hilbert space and can be identified with a tensor with d indices, each running from 1 to n. We discuss the notions of the generic rank and the maximal rank of a tensor and review results known for the low dimensions. Another variant of this notion, called the border rank of a tensor, is shown to be relevant for characterization of orbits of quantum states generated by the group of special linear transformations. A quantum state v is called {\sl entangled}, if it {\sl cannot} be written in the product form, v=v1⊗v2⊗⋯⊗vd, what implies correlations between physical subsystems. A relation between various ranks and norms of a tensor and the entanglement of the corresponding quantum state is revealed..
The usefulness of entanglement as a resource in quantum technologies increases for larger systems, that is, if more particles or higher-dimensional quantum systems are considered. Yet, the interplay between dimensionality and multiparticle entanglement is not well understood. Only for two-particle systems an unambiguous and coherent notion of entanglement dimensionality, based on the Schmidt decomposition, is known. We introduce a concept to characterize the entanglement dimensionality of multiparticle states based on decompositions of pure states into superpositions of states without genuine multiparticle entanglement. We provide constructive methods to characterize the resulting partition rank for pure and mixed states. This allows the identification of novel maximally correlated states as well as a discrete classification of quantum states under stochastic local operations and classical communication. From a mathematical perspective, our approach can be formulated in terms of the slice rank and partition rank of tensors and our results allow to characterize these by connecting them to a generalized injective tensor norm.
Quantum computing is fundamentally based on the mathematical structure of complex Hilbert spaces, linear operators, tensor products, and unitary transformations. Among these concepts, tensor products provide the formal mechanism for constructing the state space of composite quantum systems, whereas linear and unitary operators describe quantum transformations, measurements, and computational gates. This article examines the mathematical relationship between linear operators and tensor products and analyzes their role in the formation, manipulation, and characterization of quantum entanglement. Particular attention is given to two-qubit systems, separable and entangled states, Schmidt decomposition, controlled operations, and the role of non-factorizable operators in quantum circuits. The analysis demonstrates that tensor products perform a dual function: they provide the mathematical representation of composite systems and enable the construction of local operations acting on individual subsystems. At the same time, genuinely entangling transformations, such as the controlled-NOT operation, cannot in general be represented as a single tensor product of independent local operators. Recent research further demonstrates that tensor-product and operator-Schmidt decompositions can be used to characterize operator entanglement, reduce circuit complexity, and construct efficient representations of quantum processes. The results establish that linear algebra and tensor-product structures are not merely formal tools but constitute the mathematical foundation connecting quantum entanglement with quantum computation.
We introduce a class of states of a composite quantum system, the so-called cross states, that turn out to play a major role in the theory of entanglement for a genuinely infinite-dimensional bipartite system. In the case where at least one of the Hilbert spaces of the bipartition is finite-dimensional, all states are cross states, whereas, in the genuinely infinite-dimensional setting where the dimension of both Hilbert spaces is not finite, the cross states form a trace-norm dense, convex, proper subset of the set of all states. In the latter case, the cross states can be regarded as those physical states that possess a finite amount of entanglement; accordingly, all separable states are of this kind. We prove that, for any Hilbert space dimension, the separable states can be characterized as those cross states that minimize a suitable norm, i.e., the projective norm associated with the projective tensor product of two trace classes; all other cross states are density operators belonging to the projective tensor product space. This is a generalization of the classical cross norm criterion of separability. Finally, we define an extended real-valued entanglement function and study its main properties. Coherently with the interpretation of cross states as finitely entangled states, this function is finite, and coincides with the projective norm, precisely on the cross states of the system.
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.
Abstract The determination of quantum state separability is the basic problem of quantum entanglement theory. By judging the rank of the matrix vector group formed by the density matrix block, the necessary and sufficient conditions for the separation of quantum pure States are given, and the necessary conditions for the separation of quantum mixed States are given, too. In the composite system 2 × 2, the rank of the vector group corresponding to the divisible pure state is 1, and the rank corresponding to the entangled state is 4.