Fine-structure classification of multiqubit entanglement by algebraic geometry

Masoud Gharahi,Giorgio Ottaviani,Stefano Mancini

doi:10.1103/physrevresearch.2.043003

Masoud Gharahi, Giorgio Ottaviani + Show 1 more

Open Access

https://doi.org/10.1103/physrevresearch.2.043003

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Abstract

The authors use algebraic-geometry tools, secant varieties, and l-multilinear ranks, to present an entanglement classification of generic n-qubit pure states under stochastic local operation and classical communication that is based on a finite number of families and subfamilies.

Highlights

Classification, intended as the process in which ideas and objects are recognized, differentiated, and understood, plays a central role in natural sciences [1]
We present a fine-structure entanglement classification under stochastic local operation and classical communication (SLOCC) for multiqubit pure states
As entangled states are a basis for quantum-enhanced applications, it becomes of key importance to know which of these states are equivalent in the sense that they are capable of performing the same tasks almost well

Summary

INTRODUCTION

Classification, intended as the process in which ideas and objects are recognized, differentiated, and understood, plays a central role in natural sciences [1]. [17,18,19], where the authors investigated the geometry of entanglement and considered small systems (up to C3 ⊗ C2 ⊗ C2) to lighten it It has been recently realized the existence, for four qubit systems, of families, each including an infinite number of SLOCC classes with common properties [20,21,22]. We introduce an entanglement classification of “generic” n-qubit pure states under SLOCC that is based on a finite number of families and subfamilies (i.e., a finestructure classification). We do this by employing tools of algebraic geometry that are SLOCC invariants.

THE MAIN RESULT

CONCLUSION