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b/latex/chapters/chap2.tex index ce23229..8675acc 100644 --- a/latex/chapters/chap2.tex +++ b/latex/chapters/chap2.tex @@ -9,6 +9,14 @@ \chapter{Levy's family and observable diameters} +\begin{abstract} +The concentration of measure phenomenon come with important notation called the observable diameter, being the smallest number $D$ such that forall 1-lipschitz function $f$ from $X$ to $\R$, where $X$ is a metric measure space, except on a set with measure $\kappa$, the value of $f$ is concentrated in interval with length $D$. + +From Hayden's work, we know that a random pure state bipartite system, the entropy is nearly the maximal and the entropy function has Lipschitz constant upper bounded by $\sqrt{8}\log_2(d_A)$ for $d_A\geq 3$. We can use the entropy function as a proxy for the observable diameter. + +Altogether, we experimenting how entropy concentration reflects the geometry of high dimensional spheres, generic bipartite projective state spaces, and symmetric quantum state manifolds. +\end{abstract} + In this section, we will explore how the results from Hayden's concentration of measure theorem can be understood in terms of observable diameters from Gromov's perspective and what properties it reveals for entropy functions. We will try to use the results from previous sections to estimate the observable diameter for complex projective spaces. @@ -113,7 +121,7 @@ Few additional proposition in \cite{shioya2014metricmeasuregeometry} will help u and $$ - \diam(g_*\mu_X;1-\kappa)\leq \diam((f\circ g)_*\mu_Y;1-\kappa)\leq \obdiam(Y;1-\kappa) + \diam(g_*\mu_X;1-\kappa)\leq \diam((f\circ g)_*\mu_Y;1-\kappa)\leq \obdiam(Y;-\kappa) $$ \end{proof} diff --git a/latex/main.pdf b/latex/main.pdf index f0cc2f99280af89d56866f6e5355bc4b295abafc..465789df40db1000786eece95884e2e121679b16 100644 GIT binary patch delta 259954 zcmZs?Q*@?Hw6+`DwmNp+*tXF@$F|KUwr!(>j%}-B+v>QJjP}AqYkPL zs!rxT=Y7>fHSJ6|eZ4e11Scp(7G540?siS5r-#E46H#Fd9_1Hxb@B-VZt`p`jX9ex zr!MOu=+Wxwhe>n8OuCm~N>Al}K<-1R+3c{Z^D)OpIE6!IalfR3zQ+q}K6i6vftawY znIm(sNcD4^GEeSBDAzsEnYM|!5s6G|9sYdAm5kM26Rv$S7HC%sAzy~RsX@v$WfR6A zKf3!Mdn-KE5;IZ#fni`B%X95-J}q|pxjp#>UPDN->6_1&(!oCzo(xZ9i$%6L$BDq8hB1si%~xEy|jYPFN&|5&Ujd3xwON z{pa_w6$1rB*tO06=bvpDRj7#cX})jmCBN;MXF&Ju)q{)k-)U5EPuc=$XNlaA$*7;0 zYJ8}J{W^?!3D-}ZnQRAK3wr14hA{>ES^ean0)JB;@pu1HfPg1c3)1z-#-!ybQ#RKR zM@s{dNM9YgjzQ*Y=HmjP{Vl_H@Xirb;#aX3SM}`Qn z<;#g%!hT25xvOXJ?f*?~)j+5ejN_Kc7PFp4yCi2RU#JT;U7-+1fYUVM;IG|VV>lLn=SK!p$@2a7>VMV%AcJ3f~`7@OeR^C$9HlLFX;FuB!k znnrdyE}tgdgS8nd&o?w;cKLp$P=`(J1mSS7E_DL#lo(lZD7R{fvxB(p+b^j*^@$$> zM-AJ^ZUPJ|-Cj>v1ifJ}lRR8^X(XiZMlhppF{7G5QU+$uf$2L$qhxd{Qq5%$a98~@ z?MW`)(Ohr~SoK~`Z+MnLH5H>f)wKP)inis1T@Bu$!L#<3Px8s6Ex4|5|Ah$C>zjw4B z)WSl~wmJ^Xu#!;IcAmiT2sQM4GPZFon20EhY_-hqKh!MBx0x8aFw4%OnyeA%T%XjpKZ$ARSnH9OM5aio;<1V}-O{E1wb0!U9j3pg^zu--J zKc7$d3+n9Cp!*9#FRx?4DX*ZKskD(i8#S>vbTRo*52Kz!l01QHi=h{>1&j)lleV~fSb2@GQ(FBS5BR;zc9YsE85050! z)IyzjADII|eHEj-PwTLaH*!Y^NXXV)Y%#tK-OvvV;D9q0v*sdlS_+3q}WsqY52)c_uvLFGp1%;)DwTYGfTL9tW zFmY5FsR!QNY+$0j0+Jgb)%F0AnhU}Vj`tfnL@k*k& z8MA$3kRZN{lB8F7ykFH3y*vw^fL8afrts1>th8T0+Puhgjxl|NoeoZzk?g~hg5L_Z zEPo&Hlc3+_xSlOE$mV`UwWwuR5)9$)T9!``E|8FBiStWG-j}g!C|K46R<;Ya%Jp$!_fg( zdbX~cTsR-KhCz;-*HdatUep~@6qy`5tXpbga@54?D^TS; ziwh-cpsTQJy6M=3_SPw{iY%Ndn|MI)n^w5>Pz^G8T$Z~WD{Iu}sSuCpgq8ogfJcZYv~j5+kC3ZgoLeZ*sho zBG!y?=zEYh40d;NK6tQ}5Nkql`cLV3>ycoIcqP%IN!@VfAjDnBQnIk%c_%=bpfTS_ z)HNblN)#Fk32{J;Q2C07v)Beam!hpBu#+OAm?Mobt@|8a)P>OrPPT`!C4_+tJ{sW; z6Ba2I$ww4PE?9r8((FP2)yy$le+or9rL+`v<$BKCpE7ol^M7GzWi9#C~LLl(y)CP&aX zHFgwzTo{B5Ds#rtftZ0INp@t~4@wE85>PtN2ZIiby-u`8qOs*q&M$FGv@4Z1lwMjf zLGy$8T1&Sw^4-@=u4 zzy!S~B}`8|=8q;NpCc8KAy<+hwf0CS_hRh!y1q=ENt1vfqpFdo=Pf}h8Pmiwu zyYyjI(D{n^cX-Sy=(-br>$Hq6U8TVAr>7;uGh*#6|lhCjp7Br7y^ zPC+VdkK1kooAnUj0t=wc?DW$ssOV@M7bFP>Op#E=XdI#@ww&)$B95fkv5^x}pcNC2 zmAZJ7Yj62+;ailXzQRl4l<&HpuLBh)%!SIKPBpzerENAc2oMzL(uZkF9ggr@E?1 zU%MXT9w%H6j(>l|;@wkGW0CnKw$m&Ki}|EsvW#}4n&zV^@-o>9nB#|^C!@$Qh)|s4 z`7sVlD-udtTJtw7ht~wsxeLF^2TBpCL&8-d68}CN# zdfp0aNPcPcG2Bec!4Eya3~8U#OQ=?Izx7&_>tw$P>;qA2CKY-|-|4LMCVO;9KTLH5 z86$q%tmLMdz_5+xjS~nwyVhLpnn3EMO%y}1N5&cM>l=45&s#L~?55?7HsCt0cFoEw zl{H+qb!n)(FB^SSqH=-{7MF@&O|98`D_ z)K;;WXaUS}Pp#Fr4NWul!EsXRxBbto>1Bb|H0nW^x$U z?}a_qESFN#{C+%R-Q(c6Jv_e%!>Xg$u-mn1w1NiJI>3+~w>|IltzXkrnUF$JvqVx; z7xdp#RKfMpLQ8E0EL@iHs~qTT71{vkV^7t{&_G*?h8;4ej`)&&^Q=;F9QD!$o3Q(O&)zZ8j_{+(xV0{71wNV%8}VkTpMg!oWv!7?Hcou z2gxe~>@VpCjLjBe(#eD_Jch3&_<&{qV8b!4g5iC~l3M$n@n84UOn-bfr)u|mYZ}}w zf#z6HS8gevdgI2p;aDoc&Z4O<_46&_k&Nq<_Oe3n<*$Eyn}luMp4#|rRf2y+Dm3jj zJ*Ji)n>WnkOoV4VZk@2Y9*Tz!(ykq0(W=*24FN{!c6*h}K;zRCnEhogwH@x)UKV46 z5!aK}5TIIzJI4BfEkEiod#B`?L*2Rya7I{jHnG-o`FxQ@`+k%<%|AXg6#cFPkbz>k zIhzPj&l&SMZ(TS34pZfD3l&7*cM4ipdR;f0fA5?fqFFbqHOvurQBR$mWOKB& z&>j}8eLw?MBXh;P>5n)!IGpfhHI|%uPUCogonkhvaS(PbysGs?lFlv>P7$mC7pi_l z1j|rdf}Ytc^*hGKcH29S_yIdip)Ic^#(baEL}sm5Q18;bQ)h&0hpV+q`NuRA4TyW)G~m9lV4ZZ1~~~M->j?cAAMD1@*g&W-EC3%|A{vTz%Nn`!J_> zw;v3VnHLPcthur(QeI+(Op4Qio=&=hFP9dd2X;c4 z6)*e)>MIw*5q!U&QlopjL1A~%LcNVIl0(%4pBMiu40=EgTnWeS_s_mS4$URUtoZi| z)b*o~W@L8|Q~-_rn=*BZol>3KpA`N3d0`}LXEm7#pC8lxOaNw zv)KP{V5zA^?C4Fr`2Pmp0IIg}54G!XS=SHWkEW3>?|_fRs>Ez3n*wz%a69CUw~ zevy7uI7)n;xbQdRZI%~N7vv!r?a2zUn zj1Y&*vVtNZwu8_O4CUI-56l_{2v?Cx|HRz{%zNc)?`L7K@|D-2Xg77-1WS>C#UV!( z3QKXyf@wA1N3jOs>dN&wTxxcA;=ByDWSBbzs)$Wnqap)dDl>OoBa9mx*aw}mEy-X ze=mqA`G?{O|1pdB$4e*;NV|=CsNLp=)j>2DAq&bVRpfYlKtOD=GxQ%eP8W*`%fdv! zbvc+M!=UL5To*R966E5^NR)(t<%(%{^ceg27{jV;DBpLY4I8NqRH+EA$>bo53dCEK_7B2Vyr3UBa;mY{S0!atx$h&T} zqxW8D-C8K6!N`L_$<*4SK2q;^IadCphuwtriJYH1N&F#1X$$xhoWVqt8m-GtrcFu? 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A+5i9m diff --git a/latex/main.tex b/latex/main.tex index cabebc8..f628316 100644 --- a/latex/main.tex +++ b/latex/main.tex @@ -88,6 +88,17 @@ \newtheorem{prop}[theorem]{Proposition} \newtheorem{defn}[theorem]{Definition} +%%%%%%%%%%%%%%%%%%%%%% +% Add abstract for book +\newenvironment{abstract}{ + \begin{center} + \bfseries Abstract + \end{center} + \quotation +}{ + \endquotation +} + \title{Concentration of Measure And Quantum Entanglement} \author{Zheyuan Wu} \date{\today}

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