1. The document discusses classical communication over a quantum channel. It defines key concepts like quantum entropy, mutual information, Holevo information, and typical subspaces. 2. It describes a protocol where Alice selects a classical message, encodes it into a quantum codeword, and sends it through uses of the quantum channel. Bob then measures the output state to recover the message. 3. The direct coding theorem shows that a rate up to the Holevo information of the channel is achievable. The converse theorem proves that the classical capacity of a quantum channel is equal to the regularization of the Holevo information.

1. Classical communication over
quantum channel
Ma Yihsin 2020/06/11

2. Quantum Entropy 、
Quautum
Eutwpyi let 有 EDIHA ) ,
TheentropyofthestatefnisdefinedasHA 4 -_-
TrhlogfA }
If 有 ⼆
交 Pxlx ) IXYXI , where IMAI isorthonormal
_
mut
cwfo 。
HIA ) f
=
H ( X )
IlXiBiGnditioualQuantumEntropyiktfABEDlHA@HB7.define
HIAIB )
f
i =
HIAB ) e
-
H 1 B ) f
l
=
-
Trl ftp.logfAB} -

3. Quantum Mutual Iuformation .
A
measureofcorrelationktfn.BEDIHA HB )
betweentwosubsystem 、
Define Il Ai B )
e-HIAYHIBYHIABYHIAKHIBY.tn?DIlAiB)f

4. The Holevo Information.nu State
Alicepreparessomeclassicalensemble E : =
{ Pxlx ) , eg }
andhandsthestateto Bobwithouttellinghīm
thedassicalindexx.ForBob.thestatelookslikefB-F.de仍
Theaaessibleinformationquantifies Bobbinformationgainafter
performingsomeoptimalmeasurement { y} .
Iacc ( E ) : = max Ilx ; Y ) ldassical .
{ y}
Thequantīty Iaccishardtocompute ,
butfortuhateythereisanotherauantītycalledthe Hokwinformationthatprovidesanupperbound ,
Theholevobound Byconcavityof
Iacc ( E ) E X ( E ) : ⼆
HlfB ) -
[ x
Pxlx) Hleg ) > 0
entropy ,

5. The Hokvoinformation Xk)
ofadasi.ca/-quantumstateP3l86xB=iPxlx)lxxxlQ1ffisequaltothemutual information
X ( E ) = I ( X i B ) 。
P 384
The Holevoentropyofaquantumchannel NECPTPIHA ,
HBIXW) : =
hi Ilxi B )
。
BwherefxB
isoftheform
I Pxlx )
IXYI Nm Ble! !
Iacc (N ) : = max
APPHHolevoboundipxm.amIMY ) E X ( N )

6. Thewholestory ,
Alice
E A
'
r B
' Bob The protocolfor
M
必 → 9*
M
'
classicalcommunicationover-o-0.O-anum.namei
iABix-ex.tlProduct Measurementl Asimplercase )
Alīceselectsthemsg
Mandencodesthemsgasaquantumcodeword.milI-tuy-xnlm-fxncmj-fx.cm) exnlm)
PbpertmsīndiidudNM 何 0⼼ "
0tthechamdon-PrynM.xM.tn以 1 以,
不 ⼼ ) =
丌 i, Trlnyrlexicm ) ) }
Theoptimalratetheycancow.mu
hicatetheaccessibleinformatiouofthecham.IN,
Iaccw )
亨 ǐyexm Il Xi Y )

7. Themaximalerrorprobpi _
riu I - Pr { M
'
= m 1 M = m
} 、
Therateofthecommunication R -
Èloglul 。
Arate Risachievableforaquantumchannelrif
比 E ( 0,1 ) ,
Ǔ 8>0 , nsufficiehtlylarge
⼆ anln.RS ,
E ) code
Theclassicalcapacityofaquantumchannel r :
(W) : =
sup { R 1 Risachievable }
Theoremltlolevo-Schumacher-Westmore.land 1 HSW )
Theclassicalcapacityofaquantumchannelisequaltothe
regulari3ationoftheHolevoinformationofthechanne.li
(W) ⼆
Xregw ) where
Xregw) -
fmiixlfk)

8. Quantum Typicamy <
EHA
Considertheensemble IRIX) , 比 幻 xc-x.tt 有 ⼆
Exllx><划
ffn = [ Rnlxn ) 以 ><划 .
where Bnlxh) =
丌 Rlxi )
Xh E Xh
Typicalsubspacei The ftypicalsubspace
TfnisasubspaceofthefullHilbertspaceHAn-HA.O.IQH An
Tfn : =
Spanllxiixne T T }
Thetypicalprojector 丌 f _ I lxh><
il.EETj

9. (ouditioual Typicality
Gnsider
adassical.quantuneusemble.IR/lX),lxxxlQef3xcx.exnlxB:=IxPx ⼼情 、
zEinneiEIxnexPxliPxlxn1hxkxnD@lepiox.rxoy)
Nowconsiderthespectraldeompositionofffe.FRlx MIXJHYI
惗 =
唟yn
Pynlynlxylyh >< ynl
Cohditionallgtypicalsubspacei The
Snonditionaytypicalsubspace Tfnlxn
Tfny : =
spanlljiyil Fllyhlxh) -
HIBIX ) 1 : S }
Theconditioualtypicalprojector 丌 前i
-
汞TYITM
" 你 1

10. Typical Subspace Gndìtiona1 Typical Subspace
( Unit Probabilīty ) ( Unit Probabilīty )
Trhfntn.nl?1-E Exntrln 8
⾨ } : 1 -
E
Bhlxh
BnlExponentially Smdl Dimension ) ( Exponentially Small Dimension )
下 [ 丌 f ] 三 2
h HIA )
下 [ 丌
S
] 三 zn
HIBIXHS
BhilEquipartition ) l Equipartition )
Tlfnfgn Tfn 三
znHHTfn.in/nf,ixn=z-hHlBlXY8Blxn

11. The Packinglemma Chapter 1 6 .
let E :-( Rcx) ,
Glx betheensemble ,
where GEDIH ) ,
Supposeacodesubspaceprojector
Tandcodewordsubspaceprojectors{ Tx } exist , theyprojectontosubspaceof H
andsatisfgthefdlowingi
Theodespaacontainseachmsg GX
(I) Tr { 丌 G } : 1 -
E
withhighprob
(2) Tr { Tlxbx} : 1 -
E Eachcodewordsubspace - _ -
(3) Tr { T } Ed Thedimofeachcodewordsubspaceisrestricted .
(4) TIGT t.TW
hen 6
isprojectedontothesubspacewith T.it
isapproximatelymaximaymixed.where.EE10,1 ) ,
D > 0 ,
d E 1 0 ,
D )

12. Suppose M : =
L . _ _ _
,
⼼ } .
Wegeneratedassicalcodebook C =
{ Cm }
mauwhereeachCm EX issampledfrom Pxlx)
⼆ acorresponding POVM { m}
muthatreliablydistinguishesthestateslahmc-u.ieEdhǖihmam} } : 1 -
2 ( Etzi ) -4 Mlf ,
Corollay
FuvthermoreǒmEM Trhm 6cm
} 2 1 - 4 ( E +2 冷 ) -16
ldlf

13. The Dīrect Coding Theorem : C ( N) ⼆
Xreg (N) P549-17553
Alīcefirstrandomlysekctswldassicdcodewords {
xhnhīndependentlyaccordingtothedistribution
吣妙 =
{ i
⼼ 11
㾵,
1不 ⼼ 了 ,
如 E 㙾
,
如
GTYAlicesendtheensembelpxnii.fi…
7 to Bb ,
Theclassīcalcodewordsareassociatedtothequantumstatescxnlm) -
f
如 ( M)
=
fXilmo.n@fnlmIfNHnox.orle"
吵

14. (I) 下 { Tlfn 6 發 } 2 1 -
E
fxi 不 Pxcx) 必必 恬
(2) Tr { 丌前xn
6 砻 } : 1 -
E
13) Tr { 丌 前xn } E znl
HIBIX ) + c S ) =
d
14) Tlfn Exn { 6 笳 } 丌 品 :
2
" " "
-_-
…
逢 ⼀
古 欣
l-EWiththeaboveconditionshold.itfdlowsthe Packinglemmai
⼆ POVM { m } suchthat Ilxi 13知 的
corrollay.PEE 4 ktzf ) + 1 6
-1M| IMBIHIBIX) tlctcgs )
1 -
E

15. Choosing Ml = zh
( ⼯ ( X ; B) - ( Ctc
'
+ 1 ) 8 )
withtherate R = Ilxi B ) -
lctc
'
+ 1 ) 8
Pet ⼆ 4 ( Etzf ) t 1 6 古 zhf
The mutual īnformation ICX; B )
9
withrespecttotheclassical -
quantumstate
如 :
⼆点中⼼
lx ><xl Nlex )
isachievablerateforthetransmissionofdassical.info
rmationoverr

16. Wecanmaximizge Ilxi
BkbysekctingarandomcodeaccordingtotheensembleE :-( Rlx) ,
比 }
toachievetheHolew XW) :=
mqx
Ilxi
Bkofachannel N.
請 參考 P545.
Furthermore ,
sīnce XW)
isanupperboundoftheaccessibleīnformationIacc (N ) : = max
1吣, ex.tn
Ilxi Y ) E xm
Sofar,
wehaveshownthat Iacccr )
isanachievablerateforusīngthestrategy
thatwecodingforasingler.Butthisisnothessarilyopt.imalifweconsidertheentanglementbetweenthechanneluse.

17. Entanglementbetween
Alice Bob,
Aliethemareallowed.ph,
oo-oo-F.no
tfti! ! ! ! i
o_o 0 _
Iacclr) EÌ
IacclN@MThismotivatesustohavethedefinition.of
theregularyationoftheaccessibleinformationIreg (N ) ofthechannel N ,
Ireg W ) ⼀
想。
六 Iaccwg
Ireg W) isachievablebymakingtheblocksforwhichtheyare
codingarbitrarilylarge .
→
Xregtr ): =
蕊⼈
些少
isachievbleo

18. The Converse Theorem : C ( N) ㄑㄧ
Xreg (N)
Randomness distvibution ,
Īmi ⼀点uǜimyml lmiml
MaximallgcorrelatedstatelAliceencodesthemsgmwithffoxnandtransmitsit throughtheuses
WMBI ⼆点uǜimyml Nalfioxn)
ofthechannel ,
1 Bobmeasuresthestatewithmeasuremeuthm}
Wmi
前⾼uǜimyml mimi.Trhm.NU剥 了
The state Wnuishouldbe
E-doseintracedistancetotheoriginalstateIMM.foranln.GE) code.ie ÌNĪ
mi-WMMILEE.howtochooseeioxn.IMm
'
} .
Bydef.thedassicalcapacityofachannelcanr.eu exceed
theapacityofrandomhessdistvibutiow ,

19. Theorem ,
AFW inequalīty 11.10.3 可以多 介紹
intuition.lt9AB , GABEDIHAQHB ) ,
binaryentropy
SUPPOSethatillfAB-G.ph EE fr EEGIJ
fmction
IHIAIB4-HIAIBJ.IE 2
ElogdimlHA) + 1 1 + E ) 成長)
Pfoftheconversepart
Considertherate Ritloglul =
CW) -8 foracorresponding 8>0
Since ÌHĪ
mi-WMMILEE.WecanapplytheAFWinequality.tlIIMMhiIIMM7wl-IIHIY-HIMMhh-lglw.tl MIM
'
⼈ 11
⼆
IHMIM7w-HMMhnliiloghu.lt ⼼ ) hz 漄 ) -
fwu,
E )

20. R =
CW) -8 ⼆
六 logldll
=
ÈIMM
'
) 重
E Il Mj M
'
) wtflldll ,
E )
EĪIIM; Bywtfwud ←
低
Dataprocessinginequahg.siXW 炒 +
Èflldll ,
E ) ,
cr) -
S) (1 -2Ekilogldllll-29) EÈXW炒 +
Ě ⼼ ) hz 漄 )
Considerasequenceoflh.ly/En)dassicalcommunicatipwtocolswithrate C Sn ⼆
Ègul Suchthatfmfriǐifn⼆ 0
CW ) E
卡 xcfn ) 0