Entanglement in QFT

Stefan Hollands, Ko Sanders

Aitor Lewkowycz, Robert Cobb Myers, Michael Smolkin

Edward B. Manoukian

Diego Pontello

In this thesis, we study aspects of entanglement theory of quantum field theories from an algebraic point of view. The main motivation is to gain insights about the general structure of the entanglement in QFT, towards a definition of an entropic version of QFT. In the opposite direction, we are also interested in exploring any consequence of the entanglement in algebraic QFT. This may help us to reveal unknown features of QFT, with the final aim of finding a dynamical principle which allows us to construct non-trivial and rigorous models of QFT. The algebraic approach is the natural framework to define and study entanglement in QFT, and hence, to pose the above inquiries. After a self-contained review of algebraic QFT and quantum information theory in operator algebras, we focus on our results. We compute, in a mathematically rigorous way, exact solutions of entanglement measures and modular Hamiltonians for specific QFT models, using algebraic tools from modular theory of von Neumann algebras. These calculations show explicitly non-local features of modular Hamiltonians and help us to solve ambiguities that arise in other non-rigorous computations. We also study aspects of entanglement entropy in theories having superselection sectors coming from global symmetries. We follow the algebraic perspective of Doplicher, Haag, and Roberts. In this way, we find an entropic order parameter that "measures" the size of the symmetry group, which is made out of a difference of two mutual informations. Moreover, we identify the main operators that take account of such a difference, and we obtain a new quantum information quantity, the entropic certainty relation, involving algebras containing such operators. This certainty relation keeps an intrinsic connection with subfactor theory of von Neumann algebras.

Jielun Chen, Edwin Miles Stoudenmire, Steven R. White

The quantum Fourier transform (QFT) is a key component of many important quantum algorithms, most famously being the essential ingredient in Shor's algorithm for factoring products of primes.Given its remarkable capability, one would think it can introduce large entanglement to qubit systems and would be difficult to simulate classically.While early results showed the QFT indeed has maximal operator entanglement, we show that this is entirely due to the bit reversal in the QFT.The core part of the QFT has Schmidt coefficients decaying exponentially quickly, and thus it can generate only a constant amount of entanglement regardless of the number of qubits.In addition, we show the entangling power of the QFT is the same as the time evolution of a Hamiltonian with exponentially decaying interactions, and thus a variant of the area law for dynamics can be used to understand the low entanglement intuitively.Using the low entanglement property of the QFT, we show that classical simulations of the QFT on a matrix product state with low bond dimension take time linear in the number of qubits, providing a potential speedup over the classical fast Fourier transform on many classes of functions.We demonstrate this speedup in test calculations on some simple functions.For data vectors of length 10 6 -10 8 , the speedup can be a few orders of magnitude.

Paul Cummins