Information dynamics in the long-range interacting
systems
Keiji Saito（Keio Univ.）

In collaboration with
Tomotaka Kuwahara (Riken）: Part 1
Takaaki Minato (Keio), Koudai Sugimoto (keio),
Tomotaka Kuwahara (Riken) : Part 2

Content•

Short-range versus Long-range interactions in many-body systems

Part 1•Long-range effect in information propagation
Lieb-Robinson physics

Part 2• Suppression of information propagation due to measurement
Measurement-induced phase transition•

Conclusion

Short-range versus Long-range interactions in many-body systems•

Short-range interacting many-body systems
e.g., nHeisenberg model
H=

X
i=1

•

[JSi · Si+1 + BSiz ]Long-range interacting many-body systems
Power-law interaction V (r) / r ↵ in our context<latexit sha1_base64="hqGO/I7bwRD+0zWC2z/2lXnioBw=">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</latexit>

e.g., Heisenberg model with the power-law interaction<latexit sha1_base64="UTuoGPJJi+VXKgtEkQXGBuDMLJo=">AAADDXichVFNS9xQFL1JrU79HGsXBVEeDn6AOryRwZZCQXQjXanjqDDRkMTnTJh8kbwZGGP2pZsuu+hKwYW4lm7ETRf1D7jwJ4jgZgrddOGdJOOgot6QvPPOvee+83JVx9A9TumlIL5qe93ekXjT2dXd09uX7H+75tkVV2N5zTZsd0NVPGboFstznRtsw3GZYqoGW1fLC438epW5nm5bq7zmsE1TKVr6jq4pHCk5+XXxs+RVTNn3JdUkbjBFYjAeEOJ/IZKNarIXk2T6Lr235UuK4ZSUIAipXCA3iyRt2+bkjm01nCTRWRExT3JNuLVL5GSKpmkY5DHIxCAFcSzZyV8gwTbYoEEFTGBgAUdsgAIePgXIAAUHuU3wkXMR6WGeQQCdqK1gFcMKBdkyfou4K8SshftGTy9Ua3iKga+LSgKj9IIe0To9p8f0iv5/spcf9mh4qeGqRlrmyH3f3uf+vagyceVQaqme9cxhBz6GXnX07oRM4xZapK/u/qjnPq2M+mP0gF6j/316SX/jDazqX+1wma38fMaPil7wj+GAMg/H8RiszaQzs+nscjY1Nx+PKgGDMAITOI8PMAeLsAR57H8jvBOGhGHxu3ginopnUakoxJoBuBfin1seQsKW</latexit>

H=

X
r,r

•

X
J
z
0 +
S
·
S
BS
r
r
r
|r r 0 |↵0rChange of thermodynamics due to long-range interaction
Nonadditivity : Entropy can be nonconcave
A. Campa, T. Dauxois, and S. Ruffo, Phys. Rep. (2009)

Phase transition :A. Campa, A. Giansanti, and D. Moroni, Phys. Rev. E. (2000)

Quasi-stationary state:

V. Latora, Rapisarda, C. Tsallis, Phys. Rev. E. (2001)
D. Baguchi and C. Tsallis, Phys. Rev. E. (2016)

Long-range interactions are ubiquitousLong-range interaction in nature V (r) / r↵ = 1 Gravitational eld↵ = 3 Magnetic dipole interaction<latexit sha1_base64="1JurDIQTo+C6m7LisKf+1slPwsc=">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</latexit>•↵<latexit sha1_base64="hh7SBdfjrdPvrUXFVZwbJfX1ibc=">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</latexit><latexit sha1_base64="nZxyh2NgAPeuIiMOADqmnUdozpI=">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</latexit>↵ = 6 Lenard-Jones potential<latexit sha1_base64="5RD6YU9XN6nTy+yUvVhf1JRSVvc=">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</latexit>

One can tailor interaction range in recent cold atomic gases, Rydberg atoms, and in ion trapped systems<latexit sha1_base64="TRGNP5JO27T8WFnds14MX0xD7tA=">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</latexit>

H=

XX
r

r

1
|r r 0 |↵
0

z z
r r0

Even global coupling !
J. W.Britton, et.al., Nature (2012)

They may provide Black-Hole-like physics !?
fi•The main theme in this talk
What is the effect of Long-range interaction in
information dynamics ?
Information ? and information dynamics ?
(1) In wider context, information dynamics
means operator spreading.
e

iHt

Ox e

iHt

= Ox + it [H, Ox ] + · · ·<latexit sha1_base64="vAXKFpEDmRC8f5s8aUaFELZM31A=">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</latexit>xt=0(2) More speci cally, in quantum information, it usuallymeans Shannon entropy. Information dynamics is the
dynamics of entropy, or dynamics of entanglement.
<latexit sha1_base64="dj3iRG0Itq/WbDrXCTwTER0w1KE=">AAACtXichVHNShxBEP4cjTEm0dVcArkMWQwKutSKRBEC/lw8quuq4CzDzNi7Ds4fPb0LuvgCPoAePCl4kDyAF2+i+AI5+Aji0UAuOaRmdsA/NDX0dNVX9VV/3WVHnhsrous2rb3jTefbrnfd7z987OnN9fUvx2FdOqLshF4oV20rFp4biLJylSdWIyks3/bEir05m+RXGkLGbhgsqa1IVHyrFrhV17EUQ2aOSoNqSP+hj+hNQ/r6ktwxp41h3ZAboTmtJznDCx6GZi5PBUpNf+4UMyePzObD3CkMrCOEgzp8CARQ7HuwEPO3hiIIEWMVNBmT7LlpXmAH3cytc5XgCovRTf7XOFrL0IDjpGecsh0+xeMlmaljgH7RCd3RFf2kG/r7Yq9m2iPRssW73eKKyOzd/Vz681+Wz7vCxj3rVc0KVUykWl3WHqVIcgunxW9s79+VJhcHmt/oiG5Z/yFd0znfIGj8do4XxOLBK3ps1sIvxgMqPh3Hc2d5tFD8XhhbGMtPzWSj6sIXfMUgz2McU5jDPMrcfw9nuMClNq5VtHWt2irV2jLOJzwyLfwHvY2ftA==</latexit>

S(t) =

TrA ⇢A (t) ln ⇢A (t)fi<latexit sha1_base64="b/ReXdUn1ovCL6thK59Jq56kAGM=">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</latexit>

t>0

Ox

A

B

Content•

Short-range versus Long-range interactions in many-body systems

Part 1•Long-range effect in information propagation
Lieb-Robinson physics

Part 2• Suppression of information propagation due to measurement
Measurement-induced phase transition•

Conclusion

Part I: Information propagation

Information propagation as operator spreading•

It has connection to Black-hole physics, especiallyin the context of scrambling, giving the time scalereleasing information from BHTime that information reaches•

treach ⇠ r<latexit sha1_base64="QfD2t5Gz+V1Ivz+JVClXNexPWBY=">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</latexit>

treach ⇠ rz
<latexit sha1_base64="qmJMEqZDL8hQYJWJ7QBlfc60hhQ=">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</latexit>: indicating linear light cone
: still light cone, butpolynomial light cone

treach ⇠ ln r or faster : fast propagation<latexit sha1_base64="l6t6TjDqaIBF4XoS2GW5L/gJcUw=">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</latexit>r<latexit sha1_base64="b+I65MOcTkFliL1xaq3uR8vLMwM=">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</latexit>

How to characterize ?
Important function•h<latexit sha1_base64="vjM330JeebAPtKdQ9dHdOSa20pI=">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</latexit>

k [Ox (t), Oy ] kp
<latexit sha1_base64="0epCv7E66KB8Xkndx/DBfmINtTM=">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</latexit>

Ox
x

kOkp := Tr(O† O)

t>0

…

i p1

p
2

Oy
y

t=0
<latexit sha1_base64="LrJa8/x8+LBi1DTOF+DnXJeombE=">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</latexit>

p = 1 Operator norm: gives the fastest propagation<latexit sha1_base64="T3WtaSldaSzLreKTPgtD+Snm610=">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</latexit><latexit sha1_base64="wW01ZlfnhkqYnxZUUhbKIRCDvys=">AAACmHichVHLSsNAFD2N72ejbkQ3waK4KpMiKoJQdKHubGtVUClJnNbQvEjSQg39AX/AhQtREBQ/wA9w4w+46CeISwU3LrxJA6Ki3jCZM2fuuXNmruoYuucz1koIHZ1d3T29ff0Dg0PDSXFkdNuza67Gi5pt2O6uqnjc0C1e9HXf4LuOyxVTNfiOWl0N93fq3PV029ryGw4/MJWKpZd1TfGJKoljwX5UJHD5YVNypGUp0yyJKZZmUUg/gRyDFOLYtMU77OMQNjTUYILDgk/YgAKPvj3IYHCIO0BAnEtIj/Y5mugnbY2yOGUoxFbpX6HVXsxatA5repFao1MMGi4pJUyzR3bDXtgDu2VP7P3XWkFUI/TSoFlta7lTSp6MF97+VZk0+zj6VP3p2UcZi5FXnbw7ERPeQmvr68enL4Wl/HQwwy7ZM/m/YC12Tzew6q/aVY7nz/7wo5IXejFqkPy9HT/BdiYtz6fncnOp7Ercql5MYgqz1I8FZLGOTRSpfgPnuMaNMCFkhTVho50qJGLNGL6EkP8A8u2XSg==</latexit>

p=2

Frobenius norm: gives typical propagation at=0This is the Out-of-time order correlator (OTOC)
J. Maldacena, SH. Shenker, and D. Stanford, JHEP (2016).

Lieb-Robinson bound for short-range interaction

E. Lieb and DW Robinson, CMP (1972)•

Information propagation in interacting systems
H=X

hi,i+1 + hie.g., H =

i

n
X
i=1

[JSi · Si+1 + BSiz ]Rigorous bound for k [Ox (t), Oy ] k1 : Lieb-Robinson bound<latexit sha1_base64="VmGoCSG/1xaYz8J80SzQbzZbhNM=">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</latexit>•<latexit sha1_base64="clALDF57w6a6lVmrdV3KaSnMoac=">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</latexit>

k [Ox (t), Oy ] k1 < const.kOx kkOy ke
Light cone t<latexit sha1_base64="5klOx5DxjrkEQBePBh3fM72YCoY=">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</latexit>

v/

X
j

khi,j kExponentially small
information propagationx•

vt |x y|yLight cone

LR velocity is roughly the order of one site
interacting operators<latexit sha1_base64="5klOx5DxjrkEQBePBh3fM72YCoY=">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</latexit>

v/

X
j

khi,j kLieb-Robinson bound for short-range interaction
LR bound is important !
1. Causality: fundamental
2. Many Applications: leading to crucial physics(2-1) Exponential decay in ground state in gapped system
Hasting, Koma, J.Stat.Phys.

(2-2) No-go theorem of long-range entanglement at nite temperatures
T. Kuwahara and KS, PRX (2022)(2-3) Estimation of computation time
…

fi

•

Question for information propagation
Long-range interacting system•<latexit sha1_base64="lu/6+QDNf8v3tLUMgSLCZIKF+yc=">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</latexit>

H=

X

g0
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hi,j

i,j2⇤

di,j
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e.g.,
•

<latexit sha1_base64="XltHo3cp7B8PLrOZgU1SK//tBsI=">AAAClnichVHLSsNAFD2N72db3RTcBIviqkylqLgQUUSXrbUqqJQkjjU2L/Ko1NAf8AdciAsFFfED/AA3/oCLfoK4VHDjwps0ICrqDZM5c+aeO2fmypamOi5jzZjQ1t7R2dXd09vXPzAYTySH1h3TsxVeUkzNtDdlyeGaavCSq7oa37RsLumyxjfk6mKwv1HjtqOaxppbt/iOLlUMdU9VJJeociLpb4dFfFnzeEM8EBvlRJplWBjiT5CNQBpR5M3EHbaxCxMKPOjgMOAS1iDBoW8LWTBYxO3AJ84mpIb7HA30ktajLE4ZErFV+ldotRWxBq2Dmk6oVugUjYZNShFj7JHdsBf2wG7ZE3v/tZYf1gi81GmWW1pulePHqeLbvyqdZhf7n6o/PbvYw0zoVSXvVsgEt1Ba+trRyUtxdnXMH2cX7Jn8n7Mmu6cbGLVX5bLAV0//8COTF3oxalD2ezt+gvXJTHYqkyvk0vMLUau6MYJRTFA/pjGPFeRRovqHOMMVroWUMCcsCcutVCEWaYbxJYT8B0Zplw4=</latexit>

H=

X J

Si · Sj↵di,ji,jQuestion

V (r) / r ↵<latexit sha1_base64="oMv3v6fiJbvVFMpoyJmKrBVt8cU=">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</latexit>

Long-range interaction instantly transfers the informationeven to distant locations.
But the amplitude of Long-range potential is small for large r.
What is the condition of the linear light cone ?

<latexit sha1_base64="boaBKsre7oLagn/1doJfEV85G7E=">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</latexit>

H=

Hand-waving argumentg0khi,j k  ↵
di,j

hi,j

i,j2⇤

To get the feeling, divide H into Hshort and Hlong<latexit sha1_base64="hBz0RgxRwRj0IuSwihl1+f98DQY=">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</latexit><latexit sha1_base64="fona54IsWszCYvYq2750Ar5is3Q=">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</latexit>•

X<latexit sha1_base64="kFdpS37iL+I9VNb8DzeStYhL5/I=">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</latexit>

Hshort =X<latexit sha1_base64="wJ99AG0EBqYS0wDxJoPlwv5AEq0=">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</latexit>hi,jHlong =

i,j:di,j `X<latexit sha1_base64="PRGMHjxAAlXLL1fHxB/eHdV1mco=">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</latexit>hi,ji,j:`<di,j<latexit sha1_base64="NA+1XuAswS65cXZfa23ex70vUuk=">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</latexit>i<latexit sha1_base64="1kkAtVYlMo07MsVqH+UhfxLnXnM=">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</latexit>

long

`

short
<latexit sha1_base64="cYrBLClGs8PXvB37DyRmbi6jp/A=">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</latexit>•

We want to know the operator spreading of e i(Hshort +Hlong )t(1) Easier problem to get the feeling: e iHshort t e iHlong t<latexit sha1_base64="NPNn99bEWnweM+UvCzVpxcqn544=">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</latexit>: gives the LR linear light cone evt r
e
0
0
d ↵+1
v
iHlong t : e `t r
v
`
⇠
`
e
iHshort t<latexit sha1_base64="XIOVVRUWkNeF6TNevgFbForximw=">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</latexit><latexit sha1_base64="av4Wz1JFIIzj/Gw/qcrcE3PY+5Q=">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</latexit><latexit sha1_base64="mt/HGRQ+bscZBD4szXZKDoyBzD4=">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</latexit><latexit sha1_base64="A8bX0Gk20tf+3ZTeq4MF0JITUho=">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</latexit><latexit sha1_base64="yAqsJ72cCwDk4tXWWxGUL+1xMTw=">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</latexit>

Amplitudeoperators
1
X of one-siteZ interacting
v0 ⇠khi,j k ⇠ g0
drrd 1 r ↵ ⇠ g0 `d ↵<latexit sha1_base64="oyLmaw2vNiJvazkHV9sJHPIrrnk=">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</latexit>`j:`<di,j

If ↵ > d + 1 , v 0 ` ⇠ `d ↵+1 ! 0 (` ! 1)
linear light cone is expected<latexit sha1_base64="KzJgpQPvWJDAPRY8rWTf7vMO7+I=">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</latexit><latexit sha1_base64="IOmKitN3eyg3gIohPAsN+KWxQxk=">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</latexit><latexit sha1_base64="NA+1XuAswS65cXZfa23ex70vUuk=">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</latexit>i<latexit sha1_base64="1kkAtVYlMo07MsVqH+UhfxLnXnM=">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</latexit>`

long<latexit sha1_base64="/sjbhtGodciJ0eLW4Sl/GVHMD7o=">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</latexit>j<latexit sha1_base64="boaBKsre7oLagn/1doJfEV85G7E=">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</latexit>

H=

Hand-waving argumentg0khi,j k  ↵
di,j

hi,j

i,j2⇤

To get the feeling, divide H into Hshort and Hlong<latexit sha1_base64="hBz0RgxRwRj0IuSwihl1+f98DQY=">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</latexit><latexit sha1_base64="fona54IsWszCYvYq2750Ar5is3Q=">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</latexit>

•

X

X

<latexit sha1_base64="kFdpS37iL+I9VNb8DzeStYhL5/I=">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</latexit>

Hshort =<latexit sha1_base64="F+/zY3rsSYNJGVb/CZKT/P/pPe4=">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</latexit>hi,jHlong =

i,j:di,j `X<latexit sha1_base64="PRGMHjxAAlXLL1fHxB/eHdV1mco=">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</latexit>

hi,j<latexit sha1_base64="NA+1XuAswS65cXZfa23ex70vUuk=">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</latexit>i,j:`<di,j <2`i<latexit sha1_base64="1kkAtVYlMo07MsVqH+UhfxLnXnM=">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</latexit>

`

long

short
<latexit sha1_base64="cYrBLClGs8PXvB37DyRmbi6jp/A=">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</latexit>•

We want to know the operator spreading of e i(Hshort +Hlong )t<latexit sha1_base64="0Iz5JJ1FIlM6tIYGwV22Q38xzeo=">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</latexit><latexit sha1_base64="cYrBLClGs8PXvB37DyRmbi6jp/A=">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</latexit>i(Hshort +Hlong )t? = e iHshort t ee(2) How is

Hlong (t) = eiHshort t Hlong e iHshort t<latexit sha1_base64="Ud13AL6RuDZp/3fdr7qQMrJvCEw=">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</latexit>

i

Rt
0

Amplitude of one-site interacting operators<latexit sha1_base64="LkWdhpvlWb0VnNvNsgJlIlZkjPo=">AAAC13ichVE9axRRFD0Zv2L8yEQLBZuHIWKhy12RKMEiaCPYJBs3iWTi8Gb2Zfdl33wwM7sQx8VOxM7KwkrBQvwBWiuClZ1FGgsbFcsINhbemVkQDYl3mHnnnXvPnfPe9WKj04xoc8Tas3ff/gOjB8cOHT5ydNyeOLaYRr3EV00/MlGy7MlUGR2qZqYzo5bjRMnAM2rJ614r8kt9laQ6Cm9mG7FaDWQ71GvalxlTrt3InbJJ7hnpdwfCSXuBm6+LGeEoY8QV0XJzfW59IDh1V3SqDaOiUgdCtF0qK2/nrfOONHFHDlx7kmpUhtgO6kMwOXvjhP3m3ecPc5H9Cg5aiOCjhwAKITLGBhIpPyuogxAzt4qcuYSRLvMKA4yxtsdViisks13+tnm3MmRD3hc901Lt818MvwkrBaboI72gLXpPL+kb/dqxV172KLxs8OpVWhW74w9PLvz8ryrgNUPnj2pXzxnWcLn0qtl7XDLFKfxK37/zeGthpjGVn6Fn9J39P6VNessnCPs//OfzqvFkFz8ee+Eb4wHV/x3HdrB4oVafrl2c50ldRRWjOIXTOMvzuIRZXMccmtz/NT7hC75at6x71n3rQVVqjQw1x/FXWI9+A7gss+4=</latexit>

t=0<latexit sha1_base64="nucC7YE1mjwl/lZz31xZkLeR34E=">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</latexit>!

Xj:`<di,jkhi,j k ⇠ g0 `d ↵dsHlong (s)<latexit sha1_base64="NA+1XuAswS65cXZfa23ex70vUuk=">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</latexit>i<latexit sha1_base64="1kkAtVYlMo07MsVqH+UhfxLnXnM=">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</latexit>`

long<latexit sha1_base64="/sjbhtGodciJ0eLW4Sl/GVHMD7o=">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</latexit>jg0 `d ↵ ⇥ td (Ballistic operator spreading)<latexit sha1_base64="oxIQxKhrJ2c0tgKi/dBomqhqdXY=">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</latexit>

t>0<latexit sha1_base64="S8n6XiWuVxSHJeOcs29d4127emY=">AAACpXichVHLSiNBFD224yu+orMZmE1hiLoKNyIqLlRmNrMZSNREwUjobivapNPddFcC2iTLWcwPuHDlgAvR7cx8gBt/wIWfMMwyggguvOkEREW9TXWdOnXPrVN1Dc+2AkV03aV1f+jp7esfiA0ODY+MxsfG84Fb9U2ZM13b9TcNPZC25cicspQtNz1f6hXDlhtG+Wtrf6Mm/cBynXW178ntir7rWCXL1BVTxbgIC1GR0LB1s1wXMSWWSAjRaDQKyhWxejGeoBRFIV6CdAcklu9KUWTc+F8UsAMXJqqoQMKBYmxDR8DfFtIgeMxtI2TOZ2RF+xJ1xFhb5SzJGTqzZf7v8mqrwzq8btUMIrXJp9g8fFYKJOmKTqlJl3RG/+j+1VphVKPlZZ9no62VXnH056e123dVFZ4V9h5Vb3pWKGEh8mqxdy9iWrcw2/rawWFzbXE1GU7SL/rP/o/pmi74Bk7txjzJytWjN/wY7IVfjBuUft6OlyA/k0rPpWazlFj5gnb04zMmMM39mMcKviGDHNf/gXP8xh9tSvuurWv5dqrW1dF8xJPQig+/+J9W</latexit><latexit sha1_base64="MoeMTgJEfNaP9uLf51dFmW70h7U=">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</latexit>

i

e

Rt
0

!

<latexit sha1_base64="mt/HGRQ+bscZBD4szXZKDoyBzD4=">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</latexit>

dsHlong (s): ev 0 `t rv 0 ` ⇠ `d ↵+1 td ⇠ `d ↵+1 `d = `2d ↵+1<latexit sha1_base64="KE1HMtx80Y3JMuQbQvHUqK6kgaQ=">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</latexit>

If ↵ > 2d + 1 , light cone is expected<latexit sha1_base64="3VRAKIE4tnJKPm3mx2rnlcVEBEg=">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</latexit>

(Rigorously proven) State-of-the-art diagram
State-of-the-art classi cation•

1. Operator norm k [OX (t), OY ] k1
<latexit sha1_base64="r1UZTLhqkJolnC+2HqWLn1MkYlc=">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</latexit>

Fast propagation
No light conedLight-cone

2d + 1<latexit sha1_base64="qSuUiavbcwdprn0q3S0uNXvGTEE=">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</latexit>

2d<latexit sha1_base64="T1fjqNiHNCGe1ujkEgJbN2k9L/w=">AAACh3ichVHLSsNAFD2Nz9ZHq24EN8WiCEKdlFKLKx8bl221KmiRJI46NE1CMi3U4g+4cKvgSsGF+AF+gBt/wIWfIC4V3LjwJg2IivWGyZw5c8+dM3N1xxSeZOwponR19/T29UdjA4NDw/HEyOiGZ9ddg5cN27TdLV3zuCksXpZCmnzLcblW002+qVdX/P3NBnc9YVvrsunwSk07sMS+MDTpU5m9WXU3kWJpFkTyN1BDkEIYBTtxhx3swYaBOmrgsCAJm9Dg0bcNFQwOcRW0iHMJiWCf4xgx0tYpi1OGRmyV/ge02g5Zi9Z+TS9QG3SKScMlZRJT7JHdsFf2wG7ZM/v4s1YrqOF7adKst7Xc2Y2fjK+9/6uq0Sxx+KXq6FliH/nAqyDvTsD4tzDa+sbR2evaQmmqNc2u2Av5v2RP7J5uYDXejOsiL1108KOTF3oxapD6sx2/wUYmrebS2WI2tbgctqofE5jEDPVjHotYRQFlqn+IU5zhXIkqc0pOybdTlUioGcO3UJY+AUuwkJA=</latexit><latexit sha1_base64="LMsqHIJ0B4eaqFizz50WCZhBlkk=">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</latexit><latexit sha1_base64="snIq9q+EGYQ2yTi//6269xbki8Y=">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</latexit>

ct

⇠e
r↵ d

⇠ t0
<latexit sha1_base64="x8BTFBXlNksclGBMMifljb3V4Jg=">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</latexit>

Polinomial
Light cone<latexit sha1_base64="Nu+fdVMOLWZTsIw7xOfQM7t5ujE=">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</latexit>⇠↵<latexit sha1_base64="+eddQvtEhZgSQ+22IuWdLwTlGcA=">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</latexit>

LinearLight cone

⇠ evt r<latexit sha1_base64="WbyqRya5F7bMT/f9IOQknOYRSQM=">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</latexit>trmin(1,↵ 2d)

M. Turan et al., PRL (2021)

T. Kuwahara and KS, PRX (2020)

2. OTOC k [OX (t), OY ] k2
<latexit sha1_base64="flAzUrf2v+PduTG9OW9ptuKYZXA=">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</latexit>(SYK
model)↵<latexit sha1_base64="+eddQvtEhZgSQ+22IuWdLwTlGcA=">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</latexit>

T. Kuwahara and KS, PRL (2021)d<latexit sha1_base64="qSuUiavbcwdprn0q3S0uNXvGTEE=">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</latexit>

Details in light-cone regime are not very clear…
fi

0<latexit sha1_base64="rGYSXFsFQerr0Gq3RKXFIr2Gt7A=">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</latexit>

Light-cone (No fast scrambling regime)

Content•

Short-range versus Long-range interactions in many-body systems

Part 1•Long-range effect in information propagation
Lieb-Robinson physics

Part 2• Suppression of information propagation due to measurement
Measurement-induced phase transition•

Conclusion

Part II : Suppression of information propagation•

So far, we have seen operator spreading
Here, let’s see on the entanglement spreading•

Quanti ed by the Von-Neumann entropyAB<latexit sha1_base64="0RBHhpJaP2Dt+XZ3RdhDhCkZ1yw=">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</latexit><latexit sha1_base64="AB46Sg+GkNBQf6Eq92jz6Idmh9w=">AAAC/XichVG9LwRBFH/W9/ehkWgmLoTmMieCSCQ+GuUdDomVze4adxOzH5ndu4R10fsHFCoShRCtP0Cj01Ao6EVJolF4u7ciiPMmM+/N773f+8gzXME9n9L7GqW2rr6hsam5pbWtvaMz0dW97DlFabKc6QhHrhq6xwS3Wc7nvmCrrmS6ZQi2YmzNhf6VEpMed+wlf9tl65aet/kmN3UfIS0xH6hRksAQRVYmqiw42gyZIoEqLbIky9osGSK7akmXboGrUrfzghFVxLoCk10yXNYSSZqikZDfRjo2khBLxkncgAob4IAJRbCAgQ0+2gJ08PCsQRoouIitQ4CYRItHfgZlaEFuEaMYRuiIbuGbx99ajNr4D3N6EdvEKgKvRCaBAXpHT+kLvaZn9Im+/5kriHKEvWyjNipc5mqd+72Lb/+yLNQ+FL5YVRifsRK9m3irz+djzEQ0F8c53QgJJzYrtUo7By+LkwsDwSA9ps846xG9p1c4rV16NU+ybOGwSicG9hJWx4MLTf9c329jeSSVHkuNZkeT07PxapugD/phCPc3DtMwDxnIYY0LuIUHeFT2lBPlXLmohCo1MacHvoly+QGJZ7aa</latexit>

S=⇢A = TrB (|'ih'|)

TrA ⇢A ln ⇢A

S=0<latexit sha1_base64="A+RzsIDF8fpvSW/1JkJwLqqPWho=">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</latexit>

Product state:

Above singlet state: S = log 2 > 0fi<latexit sha1_base64="YSf/G7Ho53SsQWybEANp0wh47Mg=">AAAC0nichVE7SwNBEJ6crxhfURtBhMOgWIWJBBVBEW0s8zQBI+HuXOORzd1xdwkkRwqxE3sLKwWLYKc/wcbOyiI/QSwVbCycu5yIinGW3Z39Zr6Z/RjZ4KplI7YDQk9vX/9AcDA0NDwyOhYen9ix9KqpsKyic93My5LFuKqxrK3anOUNk0kVmbOcXN5y47kaMy1V1zJ23WB7FamkqQeqItkEFcMzTsEr4shcUspNMb0mFrheEhfFdRGbxXAEo+iZ+NuJ+U4EfEvo4QcowD7ooEAVKsBAA5t8DhJYtHYhBggGYXvgEGaSp3pxBk0IEbdKWYwyJELLdJboteujGr3dmpbHVqgLp20SU4Q5fMQWvuA9XuMTvv9Zy/FquH+p0y13uMwojp1Mpd/+ZVXotuHwi9WF8ZlrUvSAdnd9NuWseLpU0ml4iKtY6fSqNc5e0qupOWceL/GZtF5gG+9IrVZ7Va6SLHXe5Scy/cXtTosGGvs5vt/OzmI0thSNJ+ORjU1/tEGYhllYoPktwwZsQwKy1OMYWnADt0JGaAhHwnEnVQj4nEn4ZsLpB0gapJo=</latexit>

1|'i = p | "#i2<latexit sha1_base64="mh8K3IwFCLpmKJgmpJMUzRb7B10=">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</latexit>| #"iEntanglement usually grows up to the system size in pure dynamics
Quantum entanglement in many-body systems
L
X
z z
z
x
e.g., hi,i+1 =
+
h
+
g
H=
hi,i+1
i i+1
i
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Outcomeor

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Local measurement: Entangled states → Product states

Phase transition of entanglement•

Pure dynamics•

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time
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Kills entanglement

Enhances entanglement•

Phase transition as a result of the competition
Measurement-induced phase transition (MIP)
O(L)

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Measurement frequency<latexit sha1_base64="Quax4McR0QxsAwbPoSfcGQJVxuo=">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</latexit>CMeasurement-induced Phase Transition (MIP)
Li, et al., PRB (2019) Skinner, PRX (2019)•

Circuit model: Unitary gate + projective measurementwith probability p

time

Volume law phase
Area law phasesize L• Measurement-induced phase tr.

<latexit sha1_base64="MLyNzC35KHSRXy307/B6Xgw0+yg=">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</latexit>p < pc/ O(L)<latexit sha1_base64="pPi24Gk4jTnWOnWxgjsz3Qe+dQw=">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</latexit><latexit sha1_base64="4G7t6q/2bGjFHQ1RBTDQh7ECL50=">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</latexit>

pc<latexit sha1_base64="tWIe47HNYudLDuqlgNtblNMHPgs=">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</latexit>p > pc/ O(1)<latexit sha1_base64="zqg0s1+FQHXTgIeYb3OfYg6rvYE=">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</latexit>

Time-evolution of S:
realization of steady stateSize-dependence of S
at steady state20

Measurement-induce phase transition (MIP)•

Finite-size scaling works well →Quantum dynamical phase tran.
Numerical simulation
Li, et al., PRB (2019) Skinner, PRX (2019)•

First experiment recently exists !: Noel et al., arXiv:2106.05881
Ion-trapped system 4-8 qubits

21

Question

Minato, Sugimoto, Kuwahara, KS, PRL 2022

Effect of long-range interaction with r<latexit sha1_base64="JJFMjU3+TIiU5sb9SyG8bceCw4o=">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</latexit>•<latexit sha1_base64="nkS0R7kH4vnOrVe9zndvIeUiD3w=">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</latexit><latexit sha1_base64="8fq3Q2XgkFjkjnp8/asX3yNTXKQ=">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</latexit>?

1d example

H=

•

↵

L/2
L X
X
1 h
↵
r
j=1 r=1c†j+r cjc†j cj+r + V nj+r nj

Background of this motivation
As we have seen in Part I,
long-range interaction enhances operator spreadingThis indicates thatthe entanglement growth should be affected by long-range int.
What is the effect in the MIP ?
22

i

,

Main statement
Main statement•

Suf cient condition to observe the MIP

d
↵> +1
2
↵>d+1
<latexit sha1_base64="nkr9YKD+MBhKCsXPEpgcpc651S0=">AAACrHichVE9LwRBGH6sr/N9KBDNhPhIJJfZiyAKERqlr0Nyd7nsrjk25nY3u3uXsLk/IFErVCQKUSo0VBqlRkGjJUoSjcJ7e5cIgnczO8888z7vPDOv7kjT8zm/q1Kqa2rr6iMNjU3NLa1t0faOFc/Ou4ZIGLa03TVd84Q0LZHwTV+KNccVWk6XYlXfmi3trxaE65m2texvOyKd0zYsM2samk9UJjoUpMIigS7zoshSmnQ2NTbFgnWWsknI4kU2wtRiJtrPYzwM9hOoFdA/PfFw03X/2DNvR8+RwjpsGMgjBwELPmEJDR59SajgcIhLIyDOJWSG+wJFNJI2T1mCMjRit+i/QatkhbVoXarphWqDTpE0XFIyDPBbfsJf+DU/5U/8/ddaQVij5GWbZr2sFU6mbbd76e1fVY5mH5ufqj89+8hiIvRqkncnZEq3MMr6ws7+y9Lk4kAwyI/4M/k/5Hf8im5gFV6N4wWxePCHH5280ItRg9Tv7fgJVuIxdSw2ukCdmkE5IuhFH4apH+OYxhzmkaD6ezjDBS6VmLKsJJV0OVWpqmg68SWU7Af1NqLX</latexit>

Bilinear Fermion systems<latexit sha1_base64="QnRz/rzS2e370EzLAA0wTmzSmyk=">AAACo3ichVHLSsNAFD3G97vqRhAkWCqCUCYiKi6k6EZwo9W2oi0liaMGp0lI0mIN3bnyB0RcKbhQFy4U/AA3/oALP0FcVnAj6G1aEBX1hsmcOXPPnTNzNVsYrsfYY51U39DY1NzS2tbe0dnVHerpTbpW3tF5QreE5axqqsuFYfKEZ3iCr9oOV3Oa4CltZ66ynypwxzUsc8Ur2jyTU7dMY9PQVY+obGjQTwdFfE3keUlOq8LeVuUZeUOWR2WllA2FWZQFIf8ESg2EY6320drV7vuiFbpFGhuwoCOPHDhMeIQFVLj0rUMBg01cBj5xDiEj2OcooY20ecrilKESu0P/LVqt11iT1pWabqDW6RRBwyGljAh7YOeszO7ZJXtib7/W8oMaFS9FmrWqltvZ7oP+5dd/VTmaPWx/qv707GETU4FXg7zbAVO5hV7VF/YOy8vT8Yg/zE7ZM/k/YY/sjm5gFl70syUeP/7Dj0Ze6MWoQcr3dvwEybGoMhEdX6JOzaIaLRjAEEaoH5OIYR6LSFD9fVzgGjdSRFqQ4tJKNVWqq2n68CWkzAdvq57U</latexit>

General non-integrable systems

Background of this statement
H = HA + HB + HAB
Formal expression of entanglement growth rate•

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ṠA = ikHAB k (⇢)(⇢) := TrAB (hAB [⇢AB , ln ⇢A ⌦ 1B ])
HAB
hAB =
kHAB kkHAB k < 1 →strong measurement beats entanglement growth<latexit sha1_base64="srOsnx1r61STC2da3O1c9XYY0zI=">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</latexit><latexit sha1_base64="eQd1s065f1AIlCiHMmtHBl0GD7E=">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</latexit><latexit sha1_base64="zxXGAOxEclc1jXtcHssH5pyQ/GU=">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</latexit>→MIP should exist
kHAB k ! 1 →measurement cannot entanglement growth→No MIPfi<latexit sha1_base64="GGk8s33IRiiWPa3S0GoutUDWoiU=">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</latexit>

23

Numerical check•

1D Setup<latexit sha1_base64="nkS0R7kH4vnOrVe9zndvIeUiD3w=">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</latexit>

H=

L/2
L X
X
1 h
j=1 r=1•c†j+r cjr↵c†j cj+r + V nj+r nj

i,Quantum jump process with the quantum measurement
Measures the onsite particle<latexit sha1_base64="GS6KMPEKKUiFKPdpJyQkYIpBrEg=">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</latexit>d| (t)i =L hX

inj | (t)ipiH| (t)i +dwj (t)h (t)|nj | (t)ij=1<latexit sha1_base64="gm7dE9zumwFNpjHIq8cZ+6Vmjwk=">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</latexit><latexit sha1_base64="gizcufkjG1Uk8KO9ZGbdkARy9XY=">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</latexit>

dwj (t) = 0 or 1

dwj (t) = hnj idt

24

Demonstration for free fermion system V=0•

Stationary state entanglement entropy from the Neel state

the MIP exists

No MIP exists

25

α dependence of the critical point•

Calculate: mutual information•

Cross point
indicates the critical point•

Consistent result in the theory (↵ > 3/2 )<latexit sha1_base64="hOQo1xzgXOukJMG5/lBegxfHnTY=">AAAC0HichVHNSwJBFH9uX2YfWl0CL4tSBIGNFhUdQurS0Y80QUV2t7EWx91ld5VskfDqOejQqSAiutTf0KVbpw7+CdHRoEuH3q5GlGRvmJk3v/d+782PJ2pMNkxCWi5uYHBoeMQ96hkbn5j0+qam04Za0SWaklSm6hlRMCiTFZoyZZPRjKZToSwyuieWtu34XpXqhqwqu2ZNo/mycKDIRVkSTIQKPr+Vc4pYIqvQek5g2qHAb/LLSxG+XvAFSYg4xvc64a4TjAZyi6etaC2m+p4gB/ugggQVKAMFBUz0GQhg4MpCGAhoiOXBQkxHT3biFOrgQW4FsyhmCIiW8DzAV7aLKvi2axoOW8IuDLeOTB7myDO5IW3ySG7JC/n4s5bl1LD/UsNb7HCpVvA2Z5Pv/7LKeJtw+M3qw/jK1TFaxN1fn4k5644uGXVqDmIrljq9qsdn7eRGYs6aJ5fkFbVekBZ5QLVK9U26itPEeZ+fiPgXuzsuHGj49/h6nXQkFF4NrcRxslvQMTf4IQALOL81iMIOxCCFPRpwDXdwzyW4I+6Ea3RSOVeXMwM/jGt+AgzBp2I=</latexit>

26

Summary Part I and II•

Information propagation in the long-range systems is considered.

•

In the rst part, the operator spreading is considered
Operator norm version is classi ed

In the second part, growth and suppression of entanglement
is considered.

fi

General criterion to observe the measurement-induced phasetransition is provided.

fi•Free Fermion systems show BKT transition•

Calculate: mutual information

Finite-size scaling of BKT transition•

For noninteracting Fermion case: BKT transition.

Noninteracting case•Nonintegrable systems shows the 2nd order PT.