I ddetes

In this chapter we will consider one dimensional

spin chairs i.e Ted w

ee a o o

Ed Ed

with steaksd I

147 Ice in lie inin iN 0

t.ca mchm

Consider 147 2 Ci in lie in

We can Hunk of ee n in as a tensor

with N indices each index can take drones

Graphical reotaha I in in

1 l I r I l

Cri in MaBox _tensor

legs indices

free 147 and Cie in are the same offect

once we fix a Sens can can also write

I In n I

7We can also consider

Ciaa in Ca Crias a matrix

U row rolex rig and coherence molex

iz in G e a multi index

Now perform an AD of cilia ein

Cia iz in Nina Aqa VaiLiz inri

qfzva.cior C UAVoN

with A a diagonal matrix Aage daai da

and U V isometrics

Find Elie a pi 7

Tp in Va Liz in Be

Graphically

iiiiiiiiiz.ae

c

ef un I2 legs

matrix

comedylegsdenotes

contraction The legs arei

Eg iff TBF

Ibeaten

The tours read prodeTI

graphically

TpKian in Va Liz in B o

ntif I 1in

atgo

and fruple line

identity matrix

consistent

Ff Da AB

Du AIB

I A B n

Express state hpu.RU A V

143 I Ui e da Va liz in thin yinn'pig gin

I

It atpUiuahis Vaii inliu ina

Ira

f

tea

eeplea I UiaUjp4ijigs oh

2 Ui Ttip rap

tea and Is ONIis the Blewitt decomponton of ly

on the partition 1123 N

z 7with

Aqa eight the fluid coefficients

Now call U UH 1 1 V v

Consider M iz ti in I ha giVd 4 iz n.in

new routed radices

perform WD of M r v

is in is in

a tugg

d di

Al is diagonal 20

UH UH isometrics

f

L2

or E MY a If snap

IT1

Together

iii

ii inEiffel ME

what is the form ofthe decompossha

n the cat 1213 N

4 HomebyD right bans q

is oars

i

ri in

D T T dz

is alto au isometry as a map for 22 rial

D the is again a Schmidt decompositra

wow m the cat 12 3 N

Roreoverg if

we consider

iii Eia

iii in

I ME

across the cut 112 N then this still

gives a Schmidt Like decomposition

14k 21dL 15

with e Ile's orgs

and F 5pm Aaa dad

Orthonormal and the Schmidt

coeffrarat assorted m 1576

We can cede that this scheme aeedget

I de int UN

Batte tnfhFt7Ewith 5

iii t.ci z ccand thus also

heat FEtufting tug

L

L

i.e this representation gives a quasi Schmidt

decompositionin everycent 1 46th N k t n l

I le II

e.pk leak orgs

Erf Ii MY dad

wide coefsfor cut k

Alternatively we can consider

a as a setof row vectors Ua

and ftp as matrices UlLLada

andan

as a setof ed vectors UK

Then

i'in in e in

ne Engine iitifthu u i u

vector matrix matrix vector

ftp.ui.uiii.ui.imxPAe npjTT

Ingeneralwe have a

Cl l xD vector

Uk D xD crassa

U D xD mama

wentin Dmx 1 vector

in

i thela Ilona

hle call Di theuension

Observation we have re pleated Ci in as

2Mr

Did this reduce Eeekn

NE this caouette this decomposition

IS exact cannot reduce of parolees

Infect At each cat the bored de2eeee.tk

will generically be wire dew keptdeafight

e.g cure dk DN k

since Doe is the

Sumatran range of Kee flecuilt decompose ha

D the bond delueenan will kexpoueuballgt.ge

When we decompose an arbitrary state

147 I ai cn lie in