Minimal Entanglement and Emergent Symmetries in Low-energy QCD Qiaofeng Liuab Ian Lowbcand Thomas Mehena

2025-05-02 0 0 605.73KB 42 页 10玖币

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Minimal Entanglement and Emergent Symmetries in Low-energy

QCD

Qiaofeng Liua,b, Ian Low b,c and Thomas Mehen a

aDepartment of Physics, Duke University, Durham, NC 27708, USA

bDepartment of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA

cHigh Energy Physics Division, Argonne National Laboratory, Argonne, IL 60439, USA

Abstract

We study low-energy scattering of spin-1

2baryons from the perspective of quantum information

science, focusing on the correlation between entanglement minimization and the appearance of

accidental symmetries. The baryon transforms as an octet under the SU(3) ﬂavor symmetry and

its interactions below the pion threshold are described by contact operators in an eﬀective ﬁeld

theory (EFT) of QCD. Despite there being 64 channels in the 2-to-2 scattering, only six independent

operators in the EFT are predicted by SU(3). We show that successive entanglement minimization

in SU(3)-symmetric channels are correlated with increasingly large emergent symmetries in the

EFT. In particular, we identify scattering channels whose entanglement suppression are indicative

of emergent SU(6), SO(8), SU(8), and SU(16) symmetries. We also observe the appearance of non-

relativistic conformal invariance in channels with unnaturally large scattering lengths. Improved

precision from lattice simulations could help determine the degree of entanglement suppression,

and consequently the amount of accidental symmetry, in low-energy QCD.

arXiv:2210.12085v2 [quant-ph] 16 Nov 2023

CONTENTS

I. Introduction 2

II. Entanglement and Entanglement Power 5

III. Natural and Unnatural Scattering Lengths 8

IV. EFT for Nucleons and Baryons 10

V. S-matrices in Baryon-baryon scattering 15

A. Flavors in np scattering 15

B. Baryon-baryon scattering 17

C. Minimally entangling S-matrix 21

VI. Entanglement Minimum and Symmetry 24

A. Minimal entanglement in 1-dim sectors – SU(6) 26

B. Minimal entanglement in 3-dim sectors – SO(8) 27

C. Minimal entanglement in 6-dim sector – SU(8) and SU(16) 29

VII. Results from Lattice QCD 30

VIII. Conclusion 31

Acknowledgments 32

A. Pionless EFT for nucleons and baryons 32

B. Baryon-baryon scattering channels 35

References 41

I. INTRODUCTION

Symmetry is among the most fundamental concepts underlying all branches of physics. It

is the most powerful guiding principle in formulating laws of nature. Traditionally, there are

two views on how symmetry enters into a physical system: those emergent from long-range

ﬂuctuations at long distances and those taken as fundamental at very high energy or short

distances. The ﬁrst viewpoint is exempliﬁed in the appearance of scale invariance during

a second-order phase transition while the second is embodied in the fact that all known

fundamental interactions in nature are dictated by symmetry principles. However, for such

a pillar of modern physics there have been very few studies on where the symmetry comes

from. Can symmetry be derived from even more fundamental principles?

In pondering the origin of symmetry, a promising line of thought stems from applying

tools in quantum information, in particular the concept of entanglement, to study systems

with accidental, emergent symmetries in the infrared [1, 2]. The system of interest is low-

energy scattering of nucleons (protons and neutrons) which exhibit accidental approximate

symmetries not transparent in the fundamental QCD Lagrangian. They include Wigner’s

supermultiplet SU(4)sm [3, 4], where the two spin states of protons and neutrons com-

bine into a fundamental representation of SU(4)sm, as well as the Schr¨odinger invariance

which is the non-relativistic conformal group and the largest symmetry group preserving the

Schr¨odinger equation [5]. In addition, simulations from lattice QCD suggest that, for spin-1

baryons transforming as the octet of SU(3) ﬂavor symmetry,1there could be an emergent

SU(16)sm symmetry analogous to Wigner’s SU(4)sm, where the two spin states of the eight

baryons furnish a 16-dimensional fundamental representation of SU(16)sm [6].2Reference [1]

made the fascinating observation that the regions of parameter space where the accidental

symmetries emerge coincide with regions where the spin entanglement in the 2-to-2 scat-

tering of a proton and a neutron is suppressed, while Ref. [2] studied the observation in

a quantum information-theoretic setting and showed that the S-matrix in the spin space,

when viewed as a quantum logic gate, corresponds to an Identity gate in the case of SU(4)sm

and SU(16)sm and a SWAP gate in the case of Schr¨odinger symmetry.3

These initial ﬁndings hint at a rich interplay between entanglement and symmetry and

suggest a potentially fruitful probe for the emergence of accidental symmetries using quan-

1The SU(3) ﬂavor symmetry acts on the u-, d- and s-quark, which together transform as the fundamental

representation of SU(3). It is an exact symmetry of the QCD Lagrangian when the quark masses are

neglected, which is a good approximation for (u, d, s) quarks. Spin- 1

2octet baryons are three-quark bound

states of (u, d, s).

2Although Wigner’s SU(4)sm can be seen as a consequence of large Ncexpansion [7], no similar explanation

exists for SU(16)sm .

3An Identity gate preserves the spin of qubits (nucleons) while a SWAP gate interchanges the spin of the

two qubits. Moreover, these are the only two minimal entanglers for a two-qubit system [2].

tum information science. In this work we extend the analyses of Refs. [1, 2] on neutrons (n)

and protons (p) to the eight spin-1

2baryons transforming as an octet under the SU(3) ﬂavor

symmetry: {n, p, Σ+,Σ0,Σ−,Ξ−,Ξ0,Λ}. As we will see, because of the rich theoretical struc-

ture, the octet baryon oﬀers a fertile playground to further explore the correlation between

entanglement minimization and emergent symmetries. Moreover, unlike the scattering of

a neutron and a proton, the scattering of two baryons in general can change ﬂavors and

the outgoing particles do not have to be the same as the incoming particles. For example,

nand Λ have a non-zero probability of scattering into pand Σ−. This feature together

with the Pauli exclusion principle, which forces the total wave function of the two incom-

ing/outgoing fermions to be totally antisymmetric, create a subtle interplay between ﬂavor

and spin quantum numbers that is not present in the np scattering.

We will focus on the very-low-energy scattering of spin-1

2baryons, below the energy

threshold for pion production. In this case the process is described by an EFT of QCD using

only contact interactions and the leading order Lagrangian contains only six independent

operators [8]. The number six is predicted by SU(3) group theory because the product of

two octets contains six irreducible representations (irrep): 8×8=27⊕10⊕10⊕8S⊕8A⊕1.

Moreover, since the electric charge and the strangeness quantum number are both conserved

in strong interactions, we classify the initial states according to the total electric charge Q

and the strangeness S, which must remain the same throughout the scattering process.

Our analysis uncovers an intriguing pattern that, successive entanglement minimization in

SU(3)-symmetric and Q/S-preserving channels is achieved, an increasingly larger symmetry

group appears in the low-energy EFT. We will identify the scattering channels whose spin-

entanglement need to be minimized in order to obtain SU(6) spin-ﬂavor symmetries, SO(8)

and SU(8) ﬂavor symmetries, as well as SU(16)em spin-ﬂavor symmetry. In addition, in

the case of unnatural scattering length, entanglement suppression leads to non-relativistic

conformal symmetry in some scattering channels, similar to the np scattering.4Our ﬁndings

call for improved precision in lattice QCD simulations of baryon-baryon interactions in the

case of natural scattering length, in order to determine the amount of emergent symmetry

in low-energy QCD.

This paper is organized as follows. In Sec. II we review measures of entanglement and the

4There are other intriguing works on entanglement suppression in scattering of diverse objects, ranging

from black holes [9, 10] to pions [11].

deﬁnition of entanglement power of operators. We will be applying this measure to ﬁnd the

minimally entangling baryon S-matrices. In Sec. III we discuss the momentum expansion of

the S-matrix for the cases of natural and unnatural scattering length. Succinctly, when the

scattering length is natural, i.e., of order the range of the forces, the S-matrix is expanded

in a power series in the momentum. When the scattering length is unnaturally large the

momentum expansion is modiﬁed so that powers of the momentum times scattering length

are summed to all orders. Eﬀective ﬁeld theories of baryons for dealing with these two

scenarios are described in Sec. IV. In Sec. V, the structure of the baryon-baryon S-matrix is

presented. Then minimally entangling S-matrices and their constraints on phase shifts are

considered. In Sec. VI the symmetries of the Lagrangian at each entanglement minimum

are considered, followed by a comparison with the numerical simulation in lattice QCD in

Sec. VII. Our conclusions are given in Sec. VIII. This paper has two appendices. Appendix

A gives a review of the pionless EFT for nucleon scattering and Appendix B gives the

composition of baryon states for each irrep of SU(3).

II. ENTANGLEMENT AND ENTANGLEMENT POWER

In this section, we brieﬂy review and summarize some of the key concepts in quantum

information which are needed in our analysis. We will start with entanglement, which is a

property associated with quantum states, and proceed to introduce the entanglement power

of an operator.

A quantum state of a system is entangled if it cannot be written as a tensor-product

state of its sub-systems. In this case a measurement on a sub-system can modify the state

of the rest of the system. Speciﬁcally let us consider a bipartite system H12, such as the

two-particle system in scattering, whose Hilbert space can be written as a tensor-product

space: H12 =H1⊗H2. A state vector |ψ⟩ ∈ H12 is entangled if it is not separable, i.e., there

do not exist |ψ1⟩ ∈ H1and |ψ2⟩∈H2such that |ψ⟩=|ψ1⟩⊗|ψ2⟩.

An entanglement measure is a way to quantify the degree of entanglement of any given

state. There are multiple entanglement measures. For a bipartite system, the commonly

employed von Neumann entropy is deﬁned as

E(ρ) = −Tr(ρ1ln ρ1) = −Tr(ρ2ln ρ2),(1)

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MinimalEntanglementandEmergentSymmetriesinLow-energyQCDQiaofengLiua,b,IanLowb,candThomasMehenaaDepartmentofPhysics,DukeUniversity,Durham,NC27708,USAbDepartmentofPhysicsandAstronomy,NorthwesternUniversity,Evanston,IL60208,USAcHighEnergyPhysicsDivision,ArgonneNationalLaboratory,Argonne,IL60439,USAAbst...

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Minimal Entanglement and Emergent Symmetries in Low-energy QCD Qiaofeng Liuab Ian Lowbcand Thomas Mehena

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