Entanglement distillation of boundary states of large N SU(N)1, Chern-Simons theory and Riemann surfaces

TL;DR

This paper introduces a tree tensor network method for entanglement distillation in large N SU(N)1 Chern-Simons theory and Riemann surfaces, providing insights into entanglement entropy in complex quantum systems.

Contribution

It proposes a novel tensor network approach for entanglement distillation in specific topological quantum field theories, extending previous proposals.

Findings

Successfully applied to bipartite systems with entanglement entropy log N

Demonstrates the effectiveness of the tensor network in complex topological settings

Provides a new computational tool for studying entanglement in large N theories

Abstract

A tree tensor network is proposed for the entanglement distillation of large N SU(N)1 Chern-Simons theory and Riemann surfaces, adopting a proposal of Bao, et al. This is illustrated for the entanglement entropy S(A) of a bipartite many-body system A, where here S(A) = log N.

Equations

$$S(A)$$

$$=\log N$$

$$S_{\alpha }(\rho )=\frac{1}{1-\alpha }\log \mathrm{Tr}{\rho }^{\alpha }(\alpha \ge 0)$$

$$S_{\text{max}}(\rho )$$

$$S_{\text{min}}(\rho )$$

$$S_{\text{min}}$$

$$=\log N-\mathcal{O}[(\log N{)}^{\frac{1}{2}}]$$

$$S_{\text{max}}$$

$$=\log N+\mathcal{O}[(\log N{)}^{\frac{1}{2}}],$$

$$\mathrm{rank}({\psi }^{ϵ})$$

$${\lambda }_{\text{max}}({\psi }^{ϵ})$$

$$|\Psi ⟩=\sum _{n=0}^{\exp \mathcal{O}(\sqrt{S})}\sum _{m=0}^{\exp [S-\mathcal{O}(\sqrt{S})]}\sqrt{{\lambda }_n}{|n,m⟩}_A{|\bar{n},\bar{m}⟩}_{A^c}$$

$$n:\exp \mathcal{O}\sqrt{S}=\exp [(\log N{)}^{\frac{1}{2}}]$$

$$m:\exp [S-\mathcal{O}(\sqrt{S})]=N\exp -[\mathcal{O}(\log N{)}^{\frac{1}{2}}]$$

$$|⟨\Psi ⟩\psi {|}^2$$

$$=1-ϵ-\exp [-\mathcal{O}(\log N{)}^{\frac{1}{2}}]$$

$$\dim {\mathcal{H}}_f$$

$$\dim {\mathcal{H}}_{\gamma }$$

$$V{|n⟩}_f{|m⟩}_{\gamma }={|n,m⟩}_A$$

$$W{|\overline{n}⟩}_{\overline{f}}{|\overline{m}⟩}_{\overline{\gamma }}={|\overline{n},\overline{m}⟩}_{A^c}$$

$$\exp [m\log N-\mathcal{O}(\sqrt{m})]$$

$$\exp [\mathcal{O}(\sqrt{m})]$$

$$|\Psi ⟩=(V\otimes W)(|\varphi ⟩\otimes |\sigma ⟩)$$

$$|\varphi ⟩=\sum _{m=0}^{N\exp [-\mathcal{O}(\log N{)}^{\frac{1}{2}}]}{|m\overline{m}⟩}_{\gamma \overline{\gamma }}$$

$$|\sigma ⟩=\sum _{n=0}^{\exp [\mathcal{O}(\log N{)}^{\frac{1}{2}}]}\sqrt{{\lambda }_n}{|n\overline{n}⟩}_{f\overline{f}}.$$

$${\Psi }^{A{A}^c}=V_{f\gamma }^A{W}_{\overline{f}\overline{\gamma }}^{A^c}{\varphi }^{\gamma \overline{\gamma }}{\sigma }^{f\overline{f}}$$

$$|\gamma |=|\overline{\gamma }|=N\exp -\mathcal{O}[(\log N{)}^{\frac{1}{2}}]$$

$$|f|=|\overline{f}|=\exp \mathcal{O}[(\log N{)}^{\frac{1}{2}}].$$

$$V_{f\gamma }^A$$

$$W_{\overline{f}\overline{\gamma }}^{A^c}$$

$$S(\Psi )=\log \dim A=\log N+\dots .$$

$$X:{\mathcal{H}}_{T^2}^{\otimes 2}\to {\mathcal{H}}_{{\Sigma }_2}.$$

$$Y=XU{X}^{†}+(I-X{X}^{†})$$

$$ab{a}^{-1}{b}^{-1}cd{c}^{-1}{d}^{-1}.$$

Full text

I Introduction

There is a mapping $Z$, the functional integral, for Chern-Simons theory which relates the 3-dimensional manifold $M$ to a probability amplitude $Z(M)$. If $M$ has a boundary, the path integral selects a state $\ket{M}$ in a Hilbert space $\mathcal{H}_{\partial M}$ associated to the boundary field configurations. For $\mathrm{SU}(N)_{1}$ Chern-Simons theory one can construct generators of the Clifford group and stabilizer states, and compute the entanglement entropy $S(A)$ of a bipartite many-torus subsystem $A\subseteq\partial M$ with only a single replica [1]. [2] show that

[TABLE]

for $\mathrm{SU}(N)_{1}$ [1].

Entanglement distillation, as presented by \mtextciteBaoSet,*Bao:2018pvs,*Bao2019, describes a state $\ket{\psi}$ in the boundary Hilbert space, which can be approximated by a state $\ket{\Psi}$ to arbitrary precision, by means of a tree tensor network relating the bulk and boundary theories. We explore this strategy for $\mathrm{SU}(N)_{1}$ Chern-Simons theory in this paper.

II Entanglement distillation

Entanglement distillation, as discussed by \mtextciteBaoSet, presents a general procedure for constructing tensor networks for geometric states, which we adapt for a CFT/Chern-Simons theory correspondence. We review, and paraphrase, the essential features needed for our discussion.

Consider a state $\ket{\psi}$ in large $N$ $\mathrm{SU}(N)_{1}$ CFT, for a many-torus subsystem $A$, with the entanglement entropy of $\ket{\psi}$ between $A$ and its complement $A^{c}$. For $\mathrm{SU}(N)_{1}$ in the large $N$ limit, this is described by (I.1) and (I.2). In addition to the von Neumann entropy, consider the Rényi entropies,

[TABLE]

where $\rho$ is the density matrix of the subsystem $A$. $S_{\alpha}(\rho)$ is a monotonically decreasing function of $\alpha$, where

[TABLE]

Smooth min. and max. entropies satisfy [1, 3]

[TABLE]

and

[TABLE]

which is flat to leading order in large $N$. There exists a normalized state $\psi^{\epsilon}$ within $\epsilon$ trace distance of $\psi$ satisfying

[TABLE]

The constraints on smooth min. and max. entropies for the multitorus Hilbert space permit entanglement distillation, where a bipartite state is approximated by a nearby state in which the entanglement between two regions is made manifest. The properties of smooth min. and max. entropies provide an approximate state

[TABLE]

where $(n,m)$ is an approximate division of eigenvalues into blocks

[TABLE]

of width

[TABLE]

for $\mathrm{SU}(N)_{1}$. Then

[TABLE]

where $\Psi$ approximates $\psi$ to the accuracy of (II.13).

Introduce auxiliary Hilbert spaces to distill EPR states $\mathcal{H}_{\gamma}$ and $\mathcal{H}_{f}$ with dimensions

[TABLE]

which define maps $\mathcal{H}_{f}\otimes\mathcal{H}_{\gamma}\to\mathcal{H}_{A}$ and $\overline{\mathcal{H}}_{f}\otimes\overline{\mathcal{H}}_{\gamma}\to\mathcal{H}_{A^{c}}$, where the bond dimensions are $\dim\mathcal{H}_{\gamma}\gg\dim\mathcal{H}_{f}$ at large $N$. These maps are given by

[TABLE]

and

[TABLE]

where $V$ and $W$ are isometries which embed Hilbert spaces of size

[TABLE]

and

[TABLE]

and their complex conjugates into the physical space $\mathcal{H}_{A}\otimes\mathcal{H}_{A^{c}}$.

For a bipartite state of $\mathrm{SU}(N)_{1}$

[TABLE]

with

[TABLE]

and

[TABLE]

Then $\ket{\Psi}$ is a tree tensor network of the form

[TABLE]

with

[TABLE]

and

[TABLE]

Identify

[TABLE]

where $N^{A}_{f\gamma}$ is the fusion matrix of $\mathrm{SU}(N)_{1}$ satisfying $A=f+\gamma\mod N$, where $f,\gamma$ and $A$ label the number of boxes of a single column Young tableau representation of $\mathrm{SU}(N)_{1}$, and where $\dim A=\dim\mathcal{H}_{\gamma}+\dim\mathcal{H}_{f}$. For the distilled state, to leading order in large $N$,

[TABLE]

Therefore (II.20)–(II.28) represents a bipartite state on $\mathcal{H}_{A}\otimes\mathcal{H}_{A^{c}}$ to $\exp-\mathcal{O}[(\log N)^{\frac{1}{2}}]$ accuracy.

As an application we can discuss this in the context of a genus two Hilbert space $\mathcal{H}_{\Sigma_{2}}$ and a two-torus Hilbert space $\mathcal{H}_{T^{2}}^{\otimes 2}$. There is an isometry [2]

[TABLE]

If $U$ is a unitary operator on $\mathcal{H}_{T^{2}}^{\otimes 2}$, then

[TABLE]

is unitary on $\mathcal{H}_{\Sigma_{2}}$. A representation of the genus two Riemann surface $\Sigma_{2}$ without boundaries is given by an octagon with identified sides labeled by

[TABLE]

Consider a bipartite division of this surface, where the isometry $X$ induces a bipartite division of $\mathcal{H}_{T^{2}}^{\otimes 2}$. [See Fig. 3 of ref. [2].] For the distilled state on $\mathcal{H}_{\Sigma_{2}}$, $U$ is approximated on $\mathcal{H}_{T^{2}}^{\otimes 2}$ to the same accuracy as $X^{\dagger}UX$ on $\mathcal{H}_{\Sigma_{2}}$. There is an analogous discussion for $\mathcal{H}_{T^{2}}^{\otimes q}\to\mathcal{H}_{\Sigma_{q}}$, reflecting that (I.1) and (I.2) holds for a many-torus system.

III Concluding remark

We proposed a tree tensor network for the entanglement distillation for large $N$ $\mathrm{SU}(N)_{1}$ Chern-Simons theory for a bipartite many-torus system. The next step in this program is the extension of the analysis to multipartite systems [4, 5, 6], in particular for stabilizer states. Note that equation (I.1) is reminiscent of the RT [7] and HRT [8] relations for holography.

Other related applications of entanglement in Chern-Simons theory are discussed in [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20].

IV Acknowledgements

We thank Isaac Cohen, Alastair Grant-Stuart, and Andrew Rolph for their assistance in the preparation of the paper, which provides a contribution to the author’s 85th birthday. {Bao:2018pvs,Bao2019}

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