Masoud Gharahi - Research

My interests and experience in scientific research span physics, mathematics, and computer science, reflecting the inherently interdisciplinary nature of quantum information science. This field lies at the intersection of these disciplines, making it a rich area for exploration and discovery.

Driven by a strong enthusiasm for mathematics, I am particularly drawn to the theoretical and mathematical foundations of quantum information. My current focus is on entanglement theory, a central topic in understanding the structure and capabilities of quantum systems.

Entanglement through the lens of Algebraic Geometry

Entanglement Classification:

Multipartite entangled states feature complex quantum correlations that are considered as resources for quantum-enhanced applications. From the quantum information processing viewpoint, it is of key importance to know which of these states are equivalent in the sense that they are capable of performing the same tasks almost equally well. Finding such equivalence classes, that will provide an entanglement classification based on a ﬁnite number of entanglement families, is a long-standing open problem in quantum information theory.

Multipartite entanglement is a nonlocal quantum correlation between two or more qudits (d-level quantum systems), shared by a network of spatially separated parties. The most general local operations that can be implemented by the parties and which do not deteriorate entanglement can be described within the framework of Stochastic Local Operations and Classical Communication (SLOCC). Thus, it seems natural to seek a ﬁnite entanglement classiﬁcation under SLOCC. Two multiqudit states are SLOCC equivalent if one can be obtained with nonzero probability from the other one using local invertible operations. On the ground of group theory, SLOCC equivalence classes are orbits under the action of SL(d,C) on the set of n-qudit states.

Entanglement Interconversions:

A central problem in quantum information theory concerns the interconversion between different resources by SLOCC and asymptotic SLOCC. Using the Schmidt rank, one can characterize the SLOCC convertibility of bipartite systems. In fact, a bipartite quantum state is SLOCC convertible to another bipartite quantum state iff the Schmidt rank of the initial state is no smaller than that of the latter one. A generalization of Schmidt rank in multipartite systems is the tensor rank. Using Dirac bra-ket notation, the tensor rank can be considered as the length of the shortest bra-ket representation of a quantum state. Another tool relevant to SLOCC entanglement transformation is tensor border rank (border rank, for short). The border rank of a tensor T is deﬁned as the smallest r such that T is a limit of tensors of rank r. An important property of the tensor rank is that it is an SLOCC monotone, i.e., if a source quantum state |ψ〉 can be transformed into a target quantum state |ϕ〉 via SLOCC, then the tensor rank of the source is no smaller than that of the target. Indeed, the border rank has also the same property as the tensor rank that is an SLOCC monotone, i.e., a target quantum state |ϕ〉 can be approximated from a source quantum state |ψ〉 via SLOCC (it is actually an asymptotic SLOCC transformation), then the border rank of the source is no smaller than that of the target.

Entangled Subspaces:

Although entanglement is traditionally understood as a property of individual quantum states it can also be extended to entire subspaces of the Hilbert space. This broader viewpoint facilitates a more holistic study of entanglement, focusing not just on states, but on entire subspaces where all states are entangled. A central notion in this context is the Completely Entangled Subspace (CES), defined as a subspace where no vector is fully separable. One of the most well-established and structured methods for constructing CESs relies on Unextendible Product Bases (UPB). A UPB is a finite set of product states that spans a proper subspace of the Hilbert space, with the unique property that no additional product state exists orthogonal to all members of the set. The absence of extra orthogonal product states implies that the orthocomplement of the subspace spanned by the UPB contains only entangled states, thus forming a CES. Although fully separable states are, by definition, excluded from a CES, states with partial separability may still reside within it. This consideration motivates the notion of subspaces that exclude all forms of partial separability, thereby containing only Genuinely Multipartite Entangled (GME) states (also known as indecomposable states). Such a space is referred to a Genuinely Entangled Subspace (GES). The concepts of CES and GES not only provide deep insights into the structural complexity of multipartite entanglement, but have also been proven to be valuable tools in various quantum information processing tasks, such as quantum error correction and quantum cryptography. In light of their theoretical and practical significance, the development of systematic methods for constructing CESs and GESs, particularly those with explicit bases that elucidate the types of entanglement they support, remains a challenging and actively pursued objective within the field of quantum information science.