FormalismotQ.in ormalionTheory 2 3Watrous

TO eoeycgjsyskmxcueassocialeahilbertspace.seThis course dim Decoo adjoint

EXDirac notation 14 Ese 401 107 C TE i 50147 prindheadt CLASS

Ket bra bra Ket11 11 11

seedd

05 Ed Iqikianti linear

11411 1 1442411 orthog projection onto Q 147 CLEAR

texts KI 145 14 41 k 44441 14 41 ran Qk2

Register I finite set no D FOND 1 1 2 Stafford

e.g qubit 2 20,1 D Q2 w standard basis 107 f ID f

A G stale or X is an element of

DER L SE Posed H Eff position semidefinite densityoperator

S pure 5 14 41 for some 147 Ese 444 1maximally mixed State T Ed A dim 2 TesS classical on 2 52 g Ep k I f Poo f qubit

probdist

Doe is convex extreme po Indeednts pure states

SPECTRALTHM VSFOND kid prob dish pit pikiK4

What does DC look like for a qubit

O LT

Qubits Pauli matrices E Co9 ji E fiare basis of 2 2 Herm matrices Ctedvectorspace

5 I I Hermitian tr i r E Bloch vector

eigenvalues pil ps for PERpCtp def 44 2

y E 4C Urk1001 ay

S State PCI p 20 Urkel Bloch ball

s

a

LEX.CL classical

For a G system composed of subsystems Xiii Xn ItheHilbertspaceiso2n.fRegister 2 2 x x En D GEE 4 0 EEN

w product basis IX KD KD

Eg n qubits I Lois see ton

product State 5 8 Sn where Sie Daeiotherwisecorrelated e.g 40,030,01 t 14134,11Ces entanglement

ooze C DCE E

A measurement CPour on X with outcomes in RCfiniteset is

µ 1 a POSER S th plut IseWestportZuce If measure in states obtain outcome

g weeWIIwith probability pCw tr pass probdist N

UD Post rheas stale NOTspecifiedCyet

µ projective µCw projectionsdw your QM class

basismeasurement few l4uK4w1 for OND 140 Swee of Dle g Standard basis of no µ I a Poste

Qubits can measure in standard basis 103 1175 but also in

Hadamard basis HS I B where I If lo us

e.g S C i s Pstd toLToxol CannotDOTH

factsread HEtexts I E BE g

Discriminating g states

IsoerceT So OR Sc 50 each 011

WANT i Measuement µ Lois Roscoe s th

Buccess Etr 4067So t I tr 40478 maximal

e g for a qubit80 10 01 51 11 4 i psuccess 100 wa Std basis means

O O a optimalgo_Eff us SF us Psuccess in general

In general Set 1 77 Then

Psuccess Etr Qso c It CI Q Si L Etr EQCso Sc

E Etty Kgo Siktracedistance

HereHAH f tail trace norm of Hermitian A with eigenvaluesLais

for A fo S Iai l4iX4ili

HEE AI Fai tto 4iD gift KilECQl Fai 0

equality Q oki ceilCprojection

thromlineoptimalprob.otsuccessisztytsIt is achieved by a projective measurement

PICTURESQ.ch Sir Soot III gJ g identitychannel OILS Agut for unitary a

basechange

OILS trEg3o for state a OIEST 5 00 for stale 0discards input andprepares stale 6 prepareadditionalsystem in state of

measureandprepare channel

EE ELSIEtrepang ooo S o Tete ooo

whee pie Pos 2 mees lowSweeStates on Yg

depolarizing channel replacesbyOIEg Cfp ftp.Edtr g for PE Io D al probability p

General math definition In a later lecture we will define the set

CCoergof Channels from system X to system Y

For today the homework onlyimportant thatall examples above are indeed q Channelsevery gchannel OIECCI Y is a linearmap Ii Lose LEYlinearops