Entanglement for gauge theories:
An operational point of view

I. Cirac, N. Schuch
PRL 91, 010404 (2003); PRL 92, 087904 (2004)
K. Van Acoleyen,N. Bultinck,J. Haegeman,M. Marien,V.Scholz
ArXiv 1511.04369

Quantum Many body physics: the 21st century point of view

Entanglement

Outline•••

Entanglement as a resource theory
Entanglement in the presence of superselection rules
Entanglement in gauge theories

Entanglement as a resource theory••

The beauty of quantum information theory stems from the subtle
interplay between–the Massive Hilbert space (“a convenient illusion”)–The limitations of the allowed quantum operations–

Resources that allow to overcome those limitations

The theory of entanglement is the resource theory when faced with thelimitation of local operations and classical communication (LOCC)ClassicalCommunication•

Basic premise: entanglement is a resource (just as e.g. energy inthermodynamics) which allows to perform certain information tasks much
more efficient than classically–

AMother of all tasks is quantum teleporation: entanglement + LOCC
allows to do any global operation locallyB•

Basic premise: entanglement is a resource (just as e.g. energy inthermodynamics) which allows to perform certain information tasks much
more efficient than classically–

Mother of all tasks is quantum teleporation: entanglement + LOCC
allows to do any global operation locallyC.C

AB–

•L.O.

A

Similarly: quantum cryptography using Bell states, dense coding,measurement based quantum computation

Basis unit of currency: Bell stateBEntanglement Entropy•

Theory of quantum information is to a large extent
concerned with constructing a resource theory ofentanglement: how (efficient) can I convert states
to Bell states? What about mixed states? What about
the entangling power of a Hamiltonian?

•Fundamental theorem in the theory of entanglement:•Given a pure bipartite quantum statewith marginal, then I can convert N
copies ofinto S.N copies of Bell states and vice versa using only local operations
and classical communication, with S the entanglement entropy of the state• Hence the amount of bipartite quantum correlations can be completely
quantified by the entanglement entropy– Open question: neccessary and sufficient conditions for interconvertibilityof pure multipartite entangled states●e.g. interconvertibility of W-states into GHZ-states is related toborder rank problem for multiplication of matrices)Entanglement for mixed states•

Consider a mixed bipartite state•

There are 2 natural entanglement measures with an operational meaning :•●The entanglement cost

●The enanglement of distillation

A state is seperable (unentangled) iff it can be written as a mixture ofproduct states:•In general, we have●●

•Otherwise, we could build an “entanglement perpetuum mobile”
There exist cases where
called bound entanglement; this phenomenon isFor pure states, we obviously have●

●But equality can also happen for mixed states iff there exists alocal basis in which it holds thatIn that case,Entanglement and superselection rules•What happens with the operational definitions of entanglement if there are
extra restrictions on the operations that one can implement?
–e.g. in atomic physics: all physical operations commute with particle
number operator–

In spirit of QIT: what is the resource which allows me to overcome thenew limitations?

Entanglement and superselection rules
FV, Cirac PRL '03•Let us give a consider the simplest example: a system of 2 qubits, but with aSSR of particle number–

All physical states are of the form–

Let us consider the state●This state is seperable (non-entangled) in the usual sense

but cannot be created using LOCC and local SSR, as its seperable
decomposition involves terms of the form

Entanglement and superselection rules•

General framework: given a SSR rule which decomposes the Hilbert space
into a direct sum of projectors

then under LOCC +local SSR operations, any bipartite stateis indistinguishable from•

Conversely, a state can be prepared using LOCC and local SSR iff

FV, Cirac PRL '03Entanglement and superselection rules• Central mantra in QIT: constraints lead to new resources (e.g. LOCC, SSR) andhence must have quantum information theoretic uses
–Indeed, let us consider the 2 bipartite states–If the operations at A and B have to commute with
it is impossible to detect the phasewith LOCC●●●

thenOff-diagonal elements are invisible for observables commuting with
QIT task: data hiding.
– The bit can only revealed when the two parties come together
Which resource do we need to overcome this limitation?
–– Indeed: a local entangled measurement can then reveal info

about the bit:

Entanglement and superselection rules•

In the case of 1 global superselection rule (e.g. U(1)):–

Quantification of resource is terms of superselection induced variance:–

Using standard QIT tools, we can then prove:

Schuch, FV, Cirac '04

Entanglement in symmetry protected phases of matter•

The same discussion can of course be done for other global symmetries (e.g.
SU(2), CPT, ...)•

When dividing the system into 2 regions, we assume that all local
observables and Hamiltonian terms commute with the global symmetry in
the respective regions

AEntanglement in symmetry protected phases of matter•Let r label the irreps ofand hence denote the charge in region A,
and r* the dual irrep denoting the charge in region B, then the group action
on the Hilbert space in A can be represented asA•

The physical Hilbert space is given by

and therfore the reduced density matrix on A is given byEntanglement in symmetry protected phases of matter•

The entanglement entropy of this state consists of 2 parts:•Any local observable/operation has to leave the irrep label invariant, andhence the entropy associated to r is purely “classical”(can only be used for
creating classical correlations); the useful entanglement (entanglement ofdistillation) is given by the second part:–Note however that this is not longer true when we have several copies
available: cfr. Superselection induced varianceGauging SPT phases: zero coupling gauge theories•Given an SPT state on a lattice, it is possible to lift the global symmetry to a
local one by introducing gauge degrees of freedom on the links(Levin & Gu '12, Haegeman et al. '15)–

Wavefunction analogue of minimal coupling procedure–

Lifts SPT phases to so-called quantum doubles●•Example: paramagnet

toric code (Z2 gauge theory)

This gauging procedure obviously leads to an extensive number ofsuperselection rules; how should we quantify “useful” entanglement inthose gauge theories?
–Spoiler: gauging leaves entanglement of distillation invariant (e.g. zerofor the case of toric code)

Entanglement in lattice gauge theories• Setting: system with extensive number of superselection rules–Operational definition of entanglement can be obtained along the lines
explained before

Entanglement in lattice gauge theories•

Question has been adressed (and solved) many times, although a precise
operational meaning of entanglement seems to have only been obtained
very recently:–

Buividovich and Polikarpov '08–

Donnely '12–

Casini, Huerta, Rosabal '14–

Radividic '14–

Gromov and Santos '14–

Aoki, Iritani, Nozaki, Numassawa, Shiba, Tasaki '15–

Ghosh, Soni, Trivedi '15–...

Entanglement in lattice gauge theories• We proceed as before (and as done in all references given before):–

Let us consider the gauge constraints crossing the border between
regions A and B–For region A, let's define the basiswith labeling theequivalence classes of irreps of G on the different links, enumeratinga basis in the associated representation space, and the multiplicity
space. The local Hilbert space therefore has the following direct sum
structure:
Note that the representation space is trivial in the Abelian case.

–Following the Peter-Weyl theorem, any physical state (exhibiting thegauge symmetry) is now defined in the physical space

wheredenotes the dimension of the direct product irrep

Entanglement in lattice gauge theories• All local observables have to commute with the projection

and hence the algebra of local observable has a nontrivial center spanned by(see e.g. papers of Casini et al.)• Due to the direct sum structure and the structure in representation space, theentropy of the reduced density matrix of any quantum state incan bedecomposed in three terms: (see also Donnely and other references)• As in the case of SSR, the first two terms are useless from the entanglement
point of view, as we cannot touch them. By depolarizing, we get a mixed stateand hence•Examples:–Z2 lattice gauge theory (toric code):–

Double semion string net:–

Pure Abelian lattice gauge theory at zero coupling:–

Pure non-Abelian gauge theory at zero coupling:withthe number of inequivalent ways the representations
can fuse to the scalar one–

Perturbed toric code (Z2 gauge) with a magnetic field:●Careful: entropy is not an observable (unstable underperturbations: Fannes inequality) and therefore we have to use
formalism of quasi-adiabatic evolution of Hastings

Notice the topological quantum entropy correction!

Entanglement of distillation violates subadditivity•

Let's consider a Z2 gauge theory and a quantum state which is asuperposition of the product state
and the stateobtained by applying two Wilson loops to it:•We haveand henceConclusion•

Theory of entanglement is a theory of resources•

In the presence of global superselection rules, the rules of the game change,
and we get new restrictions (but also new resources to overcome them)–•Distillable entanglement quantifies the “useful” entanglement (i.e.
number of Bell states which can be extracted), and is generically strictly
smaller than entanglement entropy in the presence of superselection rules

Lattice gauge theories exhibit an extensive number of superselection rules, andthe distillable entanglement can be obtained by looking at the average entropy
in the different superselection blocks–

For Abelian lattice gauge theories at zero coupling,
non-Abelian ones, we get non-zero distillable entanglement, while for–Distillable entanglement exhibits topological quantum entropy correction–Distillable entanglement violates (strong) subadditivity–To get universal quantity: look at regions A,B seperated by a region C