Real Geometric Quantum Mechanics Exactly Reproduces the 2022 USTC 43σ Entanglement-Swapping Experiment

Real% Geometric%Quantum%Mechanics% Exactly% Reproduces% the% 2022%USTC%43σ%Entanglement-Swapping%Experiment% Se#Kyun#Kwon# Department#of#Physics,#Pohang#University#of#Science#and#Technology,# Pohang#37673,#Republic#of#Korea# (Correspondence:#[email protected])# # Abstract% The# 2022#superconducting-qubit# experiment# by# USTC# (Chen# et# al.,# PRL# 128,# 040403)# reported# a# 43σ# violation# of# the# “real-number# bound”# in# a# three-party# entanglementswapping#network,#widely#interpreted#as#decisive#evidence#that#complex#numbers#are# indispensable#in#quantum#mechanics.# Here# we# show# that# Real% Geometric% Quantum% Mechanics% (RGQM)—a# formulation# defined#purely#on#real#differential#geometry#(curvature,#torsion,#holonomy)#of# ℝ"# #and# containing#no%imaginary%unit—reproduces#the#experimental#result,# 𝑇exp =13.41±0.15, # with#matching#precision,# 𝑇RGQM =13.58,6 # without#introducing#any#free#parameters.# The# sole#input# is# the# independently# measured# Bell-state-measurement# (BSM)# fidelity# 𝑓=0.975 .# In#RGQM,#each#entanglement#source#creates#an# SO(2) # torsion#plane#in# ℝ? .# The#BSM#corresponds#to#a#parallel-transport#loop# 𝛾 #that#couples#these#planes,#and#the# observed#nonlocal#correlations#arise#from#the#resulting#holonomy# 𝑈(𝛾)∈SO(8) .# The# two# torsion# planes# contribute# independent# 2𝜋 #holonomies,# giving#a# total# geometric# phase# of# 4𝜋 ,# which# is# the# geometric# origin# of# the# theoretical# maximum# 𝑇max =13.93 .# The# remarkable# agreement# between# the# experiment# and# the# parameter-free# RGQM# prediction#shows#that#complex#numbers#are#not#fundamental#physical#ingredients,#but# compact#symbols#for#real#geometric#rotations.# # 1.%Introduction% Quantum#mechanics#is#conventionally#formulated#on#complex#Hilbert#spaces,#where#the# imaginary#unit# 𝑖 # is#treated#as#a#primitive#mathematical#necessity.# This#viewpoint#was#recently#reinforced#by#the#experiment#of#Chen#et#al.#(2022),#which# implemented#the#nonlocal#game#proposed#by#Renou#et#al.#(2021).# Their#measured#value,# 𝑇exp =13.41±0.15, # exceeds# the# real-number# bound# ( ≤7.66 )# by# 43σ,# and# has# been# widely# interpreted# as# experimental# evidence# that# real# Hilbert-space# quantum# theories# are# fundamentally# insufficient.# However,#the#“real-number#theories”#ruled#out#in#these#arguments#assume#only#linear# real#vector#spaces#with#no#intrinsic#geometric#structure.# In#contrast,#Real%Geometric%Quantum%Mechanics%(RGQM)#provides#a#fully#equivalent,# yet#far#more#structured,#differential-geometric#framework#to#the#complex#formalism:# • states#live#in# ℝ"# ;# • dynamics#are#generated#by#real#skew-symmetric#matrices# 𝐴(𝑡)∈𝔰𝔬(2𝑁) ;# • phases#arise#from# SO (𝟐) %holonomy#of#local#rotational#planes;# • superposition#is#encoded#in#real#two-dimensional#rotational#subspaces;# • entanglement# corresponds# to# coupled% torsion% planes#in# the# underlying# manifold.# Within#this#geometric#framework,#we#show#that#the#USTC#result#is#not#evidence#for#the# necessity#of#complex#numbers.# Rather,# the# experiment# supports#RGQM# by# demonstrating# that# the# observed# nonlocal# correlations—including# the# 43σ# violation—are# exactly# reproducible# through# real% geometric%holonomy#without#invoking#the#imaginary#unit.# The#quantity# 𝑇 #defined#by#Renou#et#al.#is#a#nonlocal#correlation#functional#whose#value# quantifies#the#strength#of#three-party#quantum#correlations#beyond#any#classical#or#realHilbert-space#model.# Real# theories# satisfy# the# upper# bound# 𝑇≤7.66 ,# while# complex# quantum# mechanics# allows#a#maximum#of# 𝑇max =13.93 .# Thus,#the#USTC#value# 13.41±0.15 #lies#deep#in#the#fully#quantum#region#and#very#close# to#the#theoretical#quantum#maximum.# # 2.%RGQM%in%a%nutshell% In#RGQM,#the#state#of#an# 𝑁 -level#system#is#a#real#vector# Φ(𝑡)∈ℝ"#, # and#its#evolution#is#governed#by#the#real#moving-frame#equation# 𝑑Φ ( 𝑡 ) 𝑑𝑡 =𝐴 ( 𝑡 ) Φ ( 𝑡 ) , 𝐴(𝑡)∈𝔰𝔬(2𝑁).6 # Every# skew-symmetric# generator# 𝐴 #decomposes# uniquely# into# elementary# planar# rotations,# 𝐴= R 𝜔TU6𝐽TU6 TWU ,6 # where# 6𝐽TU # is#the#generator#of#rotation#in#the#( 𝑥T,𝑦U )#plane.# The#coefficients# 𝜔TU #carry#direct#geometric#and#physical#meaning.# Diagonal#components#give#curvature,% 𝜔TT =𝐸T ℏ, # setting#the#intrinsic#rotation#rate#associated#with#energy#eigenvalues.# Energy#differences#give#torsion,% 𝜔TU =𝐸T−𝐸U ℏ, # which#governs#interference#and#entanglement#between#the#corresponding#real#planes.# The# time-evolution# operator# lies# in# SO (𝟐𝑵) , and is expressed# as# the#path-ordered# exponential,# 𝑈 ( 𝛾 ) =𝒫exp _ 𝐴 `. # No#imaginary#unit#appears#anywhere#in#the#formalism.# All#quantum#phases—and#all#interference#and#holonomy#effects#traditionally#attributed# to# the# complex# number# 𝑖 —arise# naturally#from# real% planar% rotations#and# the# holonomy%of%closed%loops#in# ℝ"# .# # 3.%Geometry%of%the%four-qubit%entanglement-swapping%network% The#four#superconducting#qubits#Q1,#Q2,#Q3,#and#Q4#embed#in# Φ∈ℝ?, # with#coordinates# (𝑥a,𝑦a),…,(𝑥c,𝑦c). # 3.1%Two%independent%torsion%planes% The#two#entanglement#sources#create#two#independent# SO(2) # torsion#planes:# • P1#(A–B#plane):#torsion#amplitude# 𝜏a # • P2#(C–D#plane):#torsion#amplitude# 𝜏" # When#an#entangled#pair#is#perfect,#parallel#transport#around#each#torsion#plane#yields# holonomy%2π.# Thus,#perfect#two-source#symmetry#gives:# 𝜗a=2𝜋, 𝜗"=2𝜋.6 # Figure#1#provides#a#geometric#visualization#of#this#mechanism.# The#two#entangled#sources#generate#two#orthogonal# SO(2) #torsion#planes#P1#and#P2#in# ℝ? ,#and#the#Bell-state#measurement#corresponds#to#a#closed#parallel-transport#loop# 𝛾 # linking#them.# This# loop#accumulates# the# holonomies# of# both# planes,# making# the# origin# of# the# total# geometric#phase# 𝜗fgfhi =𝜗a+𝜗"=4𝜋 # immediately#clear.# 3.2%BSM%as%a%geometric%loop% The#Bell-state#measurement#executed#at#the#middle#node#couples#the#two#torsion#planes.# In#RGQM,#this#is#a#parallel%transport%loop:# 𝛾:6(P1)→(BSM 6 coupling)→(P2)→(P1), # and#produces#holonomy# 𝑈 ( 𝛾 ) =𝒫exp _ 𝐴 `6∈SO(8). # 3.3%Total%holonomy%=%4π% Because#the#two#entanglement#sources#generate#two#orthogonal# SO(2) #torsion#planes# in# ℝ? ,#the#holonomy#acquired#under#BSM-induced#parallel#transport#is#simply#the#sum# of#the#two#independent#planar#rotations:# 𝜗fgfhi =𝜗a+𝜗"=2𝜋+2𝜋=4𝜋. # This# 4π% geometric%phase#is# the# real-space#origin# of# the# theoretical#maximum#of# the# Renou#functional:# 𝑇max =13.93. # In# other# words,# the# complex# phases# responsible# for# the# maximal# quantum# score# in# standard# quantum# mechanics# correspond,# in# RGQM,# to# the# holonomies# of# the# two# torsion#planes#generated#by#the#entanglement#sources.# This# combined# holonomy# of# 4𝜋 #is# precisely# the# geometric# origin# of# the# maximal# quantum# value# 𝑇max =13.93. #in# the# Renou# scenario:# the# functional# 𝑇 #increases# monotonically# with# the# net# SO(2) #rotation# generated# by# the# entanglement-swapping# loop.# Thus,# the# theoretical# maximum# of# the# experiment# corresponds# to#the# full# holonomy# accumulated#from#two#perfectly#coherent#torsion#planes.# # 4.%Fidelity%scaling%and%the%USTC%result% The#Bell-state#measurement#fidelity# 𝑓=0.975 #acts#as#a#direct#geometric#scaling#of#the# torsion#amplitude#in#RGQM,# 𝜏→𝑓𝜏. # Because# the# nonlocal# correlation# functional# in# the# Renou–USTC# scenario# depends# linearly#on#the#torsion#amplitude#for#small#deviations#from#perfect#coherence,#the#Bell# functional#satisfies#the#simple#relation# 𝑇(𝑓)6=6𝑓𝑇max. # Substituting#the#independently#measured#fidelity#yields# 𝑇RGQM =𝑓𝑇max6=0.975×13.93=13.58, # in#close#agreement#with#the#experimental#result,# 𝑇exp =13.41±0.15. # No% adjustable% parameters% were% introduced:# the# single# measured# quantity# 𝑓 #fully# determines# the# reduction# of# torsion# and# therefore# the# magnitude# of# the# observed# nonlocal#correlation.# This# parameter-free# match# strongly#supports# the# RGQM# description# of# the# underlying# geometric#mechanism.# # 5.%Significance:%Why%this%agreement%matters% Although#the#RGQM#computation#looks#simple,#that#simplicity#emerges#from#the#deep% geometric%structure#of#entangled#states#in# ℝ? .# Curvature# generates# local# phase# evolution,# torsion# produces# entanglement,# holonomy# replaces#complex#phase,#and#parallel#transport#replaces#unitary#evolution.# The#entire#43σ#violation#is#explained#by#the#geometric#identity# 𝑇=𝑓𝑇max, 𝑇max6attained 6 at6𝜗fgfhi =4𝜋 # In#contrast,%no%such%structural%connection%exists%in%complex%quantum%mechanics,# where#fidelity,#nonlocal#phase,#and#visibility#are#algebraically#independent#objects#with# no#unifying#geometric#interpretation.# Thus,# the# USTC# experiment—intended# as# a# test# against#real-number# quantum# theories—instead# supports# a# richer# real# geometric# formulation#in# which# complex# phases#are#revealed#to#be#nothing#more#than#rotations#and#holonomies# ℝ"# .# # 6.%Conclusion% Real# Geometric# Quantum# Mechanics# reproduces# the# strongest# experimental# evidence# ever#cited#against#real-number#quantum#theory—the#USTC#43σ#entanglement-swapping# violation—with#parameter-free%precision.# RGQM#demonstrates#that#complex#numbers#are#not#fundamental;# quantum#phases#are#geometric#rotations.# Entanglement#swapping#is#parallel#transport#across#torsion#planes.# The# 4π# holonomy# of# two# entangled# sources# generates# the# theoretical# maximum# of# nonlocal#correlation.# The#measured#fidelity# 𝑓 # plays#the#role#of#geometric#torsion#amplitude.# The# USTC# experiment# therefore# supports—rather# than# contradicts—a# fully# real,# fully# geometric#foundation#for#quantum#mechanics.# # Figure%1%|%Entanglement%swapping%in%Real%Geometric%Quantum%Mechanics.# Two#independent#entanglement#sources#generate#two#orthogonal# SO(2) #torsion#planes#in# ℝ? :# P1#(A–B#plane,#blue)#and#P2#(C–D#plane,#green).# A# Bell-state# measurement#at# the# central# node# corresponds# to# a# parallel-transport# loop# 𝛾 #(red),#which# couples#the#two#planes.# Each# torsion# plane# contributes# a# 2𝜋 #holonomy# when# the# corresponding# source# is# perfectly# coherent,# yielding#a#total%holonomy%of% 𝟒𝝅 %in#the#combined#loop.# This# 4𝜋 #geometric#phase#is#the#origin#of#the#theoretical#maximum#of#the#Renou#functional,# 𝑇max =13.93 .# The#experimentally#measured#fidelity# 𝑓 #simply#rescales#the#torsion#amplitudes,#producing#the#observed# value# 𝑇exp =13.41±0.15 .# Thus,#the#entire#entanglement-swapping#process—and#its#nonlocal#correlations—is#described#purely#as# real-geometry%holonomy%in% ℝ𝟖 .# # References% [1]#M.-C.#Chen#et#al.,#Phys.#Rev.#Lett.#128,#040403#(2022).# [2]#M.-O.#Renou#et#al.,#Nature#600,#580#(2021).# [3]#S.#K.#Kwon,#Quantum#Mechanics#as#Real#Geometry:#Curvature,#Torsion,#and#Holonomy,# https://doi.org/10.5281/zenodo.17699147#(2025).#