From 8adda985f8832d77954ba37f288ce819a6f41654 Mon Sep 17 00:00:00 2001 From: Zentrik Date: Tue, 14 May 2024 20:39:37 +0100 Subject: [PATCH] Fixup some more stuff --- CodingAndCryptography/01_noiseless_coding.tex | 2 +- .../03_information_theory.tex | 23 ++++++++++++------ CodingAndCryptography/cc.pdf | Bin 350831 -> 353372 bytes QuantumInfoAndComputing/qic.pdf | Bin 277570 -> 62511 bytes QuantumInfoAndComputing/qic.synctex(busy) | Bin 0 -> 131072 bytes 5 files changed, 16 insertions(+), 9 deletions(-) create mode 100644 QuantumInfoAndComputing/qic.synctex(busy) diff --git a/CodingAndCryptography/01_noiseless_coding.tex b/CodingAndCryptography/01_noiseless_coding.tex index e44ba09..c8b9c95 100644 --- a/CodingAndCryptography/01_noiseless_coding.tex +++ b/CodingAndCryptography/01_noiseless_coding.tex @@ -184,7 +184,7 @@ \subsection{Gibbs' inequality} As the $p_i$ form a probability distribution, $\sum_{i \in I} p_i = 1$ and $\sum_{i \in I} q_i \leq 1$, so the right hand side is at most 0. Therefore, \begin{align*} - -\sum_{i=1}^n p_i \ln p_i = -\sum_{i \in I} p_i \ln p_i \leq -\sum_{i \in I} p_i \ln q_i \leq -\sum_{i=1}^n p_i \ln q_i + -\sum_{i=1}^n p_i \ln p_i = -\sum_{i \in I} p_i \ln p_i \leq -\sum_{i \in I} p_i \ln q_i = -\sum_{i=1}^n p_i \ln q_i \end{align*} If equality holds, we must have $\sum_{i \in I} q_i = 1$ and $\frac{q_i}{p_i} = 1$ for all $i \in I$, giving that $p_i = q_i$ for all $i$. \end{proof} diff --git a/CodingAndCryptography/03_information_theory.tex b/CodingAndCryptography/03_information_theory.tex index c348c8c..79adf9d 100644 --- a/CodingAndCryptography/03_information_theory.tex +++ b/CodingAndCryptography/03_information_theory.tex @@ -164,20 +164,27 @@ \subsection{Shannon's first coding theorem} \end{theorem} \begin{proof} + % Let $\varepsilon > \frac{1}{n}$, for each $\epsilon_n$ $\exists n_0(\epsilon_n)$ s.t. $\exists$ typical sets $T_n \; \forall n \geq n_0(\epsilon_0)$. + % Let $T_k \subseteq \mathcal A^n$ be $\epsilon_n$ typical sets where $n_0(\epsilon_n) \leq k \leq n_0(\epsilon_{n+1})$. + % Define $n_k = \min \qty{n : T_k}$ + % Then, $\forall \; n \geq n_0(\varepsilon)$, $\forall \; (x_1, \dots, x_n) \in T_n$ we have $p(x_1, \dots, x_n) \geq 2^{-n(H + \varepsilon)}$. + % Therefore, $1 \geq \prob{T_n} \geq \abs{T_n} 2^{-n(H + \varepsilon)}$, giving $\frac{1}{n} \log \abs{T_n} \leq H + \varepsilon$. + % Taking $A_n = T_n$ in the defn of reliable encoding shows that the source is reliably encodable at rate $H + \varepsilon$. + Let $\varepsilon > 0$, and let $T_n \subseteq \mathcal A^n$ be typical sets. Then, $\forall \; n \geq n_0(\varepsilon)$, $\forall \; (x_1, \dots, x_n) \in T_n$ we have $p(x_1, \dots, x_n) \geq 2^{-n(H + \varepsilon)}$. Therefore, $1 \geq \prob{T_n} \geq \abs{T_n} 2^{-n(H + \varepsilon)}$, giving $\frac{1}{n} \log \abs{T_n} \leq H + \varepsilon$. Taking $A_n = T_n$ in the defn of reliable encoding shows that the source is reliably encodable at rate $H + \varepsilon$. - Conversely, if $H = 0$ the proof concludes, so we may assume $H > 0$. + If $H = 0$ the proof concludes, so we may assume $H > 0$. Let $0 < \varepsilon < \frac{H}{2}$, and suppose that the source is reliably encodable at rate $H - 2\varepsilon$ with sets $A_n \subseteq \mathcal A^n$. Let $T_n \subseteq \mathcal A^n$ be typical sets. Then, $\forall \; (x_1, \dots, x_n) \in T_n$, $p(x_1, \dots, x_n) \leq 2^{-n(H - \varepsilon)}$, so $\prob{A_n \cap T_n} \leq 2^{-n(H - \varepsilon)} \abs{A_n}$, giving \begin{align*} \frac{1}{n} \log \prob{A_n \cap T_n} \leq -(H - \varepsilon) + \frac{1}{n} \log \abs{A_n} \to -(H - \varepsilon) + (H - 2 \varepsilon) = -\varepsilon \end{align*} - Then, $\log \prob{A_n \cap T_n} \to -\infty$, so $\prob{A_n \cap T_n} \to 0$. - But $\prob{T_n} \leq \prob{A_n \cap T_n} + \prob{\mathcal A^n \setminus A_n} \to 0 + 0$, contradicting typicality. + Then, $\log \prob{A_n \cap T_n} \to -\infty$ as $n \to \infty$, so $\prob{A_n \cap T_n} \to 0$. + But $\prob{T_n} \to 1$ and $\prob{A_n} \to 1$ as $n \to \infty$ \Lightning\footnote{$\prob{T_n} \leq \prob{A_n \cap T_n} + \prob{\mathcal A^n \setminus A_n} \to 0 + 0$ as $n \to \infty$, contradicting typicality.}. So we cannot reliably encode at rate $H - 2\varepsilon$, so the information rate is at least $H$. \end{proof} @@ -228,11 +235,11 @@ \subsection{Capacity} \begin{proof} Let $\delta$ be s.t. $2p < \delta < \frac{1}{2}$. - We claim that we can reliably transmit at rate $R = 1 - H(\delta) > 0$. - Let $C_n$ be a code of length $n$, and suppose it has minimum distance $\floor*{n\delta}$ of maximal size. + We claim that we can reliably transmit at rate $R = 1 - H(\delta)\footnote{$-\delta \log \delta$} > 0$. + Let $C_n$ be a code of length $n$, and suppose it has minimum distance $\floor*{n\delta}$ and it's of maximal size. Then, by the GSV bound, \begin{align*} - \abs{C_n} = A(n, \floor*{n\delta}) \geq 2^{-n(1-H(\delta))} = 2^{nR} + \abs{C_n} = A(n, \floor*{n\delta}) \geq \frac{2^n}{V(n, \floor*{n\delta} - 1)} \geq \frac{2^n}{2^{-n\delta \log \delta\footnote{$-\log \delta > 1$.}}} = 2^{n(1 - H(\delta))} = 2^{nR} \end{align*} Replacing $C_n$ with a subcode if necessary, we can assume $\abs{C_n} = \floor*{2^{nR}}$, with minimum distance at least $\floor*{n\delta}$. Using minimum distance decoding, @@ -285,7 +292,7 @@ \subsection{Conditional entropy} \end{definition} \begin{lemma} - $H(X,Y) = H(X \mid Y) - H(Y)$. + $H(X,Y) = H(X \mid Y) + H(Y)$. \end{lemma} \begin{proof} @@ -312,7 +319,7 @@ \subsection{Conditional entropy} \end{corollary} \begin{proof} - Combine this result with the fact that $H(X,Y) \leq H(X) + H(Y)$ where equality holds iff $H(X), H(Y)$ are independent. + Combine the previous result with the fact that $H(X,Y) \leq H(X) + H(Y)$ where equality holds iff $H(X), H(Y)$ are independent. \end{proof} Now, replace r.v.s $X$ and $Y$ with random vectors $X^{(r)} = (X_1, \dots, X_r)$ and $Y^{(s)} = (Y_1, \dots, Y_s)$. diff --git a/CodingAndCryptography/cc.pdf b/CodingAndCryptography/cc.pdf index 2076912dfaef9b8c8fb2de731746ff5184d0f3f7..607886a61d096ce9e173bcfe97706e46d9f11f4c 100644 GIT binary patch delta 112605 zcmY(J<8mg7qC{icwrzW2+qP|ev2EM7Z95a&wlR0t{dXRst7~ds!jAG786!%GNwNaUETpd4Driqj^wO{l8BH*0$WN=PUmqc5w^cK~NP;@tR+ zGgg!HR1f?hSck)80w&5ySv;~}`1$C@^{r#}J6#~ZKSSVCH?WwiCI-OP+mVenXALrU z85(Eq?esL8nvzx0;;n!@uYPK9^l)bXZzsB6Hoy1WV&x`89>&g+zi<$2?Y(!HQLvkB zQ6e9;mLvF%iCJ9l`#ydxy;H^VD*%)DbLs4CdXY_u%0ebyK{}qx3%Pejh3bHL2nyX! z|I#j3f^#Q`iNO{2U>|Ttfk&%}p>+~MLXnF+N1nKKNeP0BL6wYbN>Q6=xp$wO0NUfqa;Ix`qQvUIdz z2}4=JOD^!h_fQzdZr_2+O~@&ghL^t?xzW`};1s5y7jvcd2*qooIlsE2Zw6@8;MgKh z;>fOAv~R&2Q)zuEH@i4{aDOc$QSgE&-)C+?z@@;+F0NkY z=)(i1Z(l3Efqb7a|H92`cf3ya9U^899MU}}i{-rmqH$#KL~vZq(uDsrlWhPr*0#?r zyjureuL9r&0xOnnQf=cj1y+sSr9_1T;6HJQ`Bu8yQ%`wia%2B$ zv%k#Q@o_netwmqZhI_LAJzEgu-dz2r6$7- zTp-EiuKpwr+`_?uzJwPvEp20tx$+c{jn~OauCsvB^Z?tEDpp%rw-js*vtS4(Mt?3M zEdyXF`8#CM5M_nJ+YU~EW)K5Ue2jbfN4?jYe)S`oD1i>eHCHf99KvvIA^cfEdK5Mc zOswb2Ll8r4m0ohLZ{cgw%Z1^%Sd7F6viPXZFHpguU1oD30;nOId)QOqZ2X#1hBs2u z7DkURQ7Y?-mt5&gWU8qtw$cLLdE!lbOaL~7UNc-tt0eo2fB0A=etBF=SgaNi^2Zjx zQjuorL)tGF+mIIwIPr{J%Tn>XMaWTMvi%Ba)~9dB2dfq@IZHEpA!iAURR5CM)Y8=Y z!j!Yd2Siik)Iby5X{pP>RB@oSREeUWCiFM96_{C8L)*0Ej4|LT1^*vaSI7=Q-Hv>?S8xm1B^!*ewE1PXa^0n6|QcV|#2-C*D zh!RG_XOoLB@+3@Q*Gk&|ftK{=cLD&?cJKYA-T-Km+M)&lBb}qG-sCR%IP-GqyyTs! zUgN<4&JFY?ota72_AdKHv?^5BUR1ih5ntqF?VsI=>eth1iBzch962cJ93jbHPBF<( zyuNo*Wuy=|?(0XHsdx?{axjGr(t;*EeXEf(hMOTi2944lG@7P0;2}q#P(Z{ozh)xC zoAk1JTNU;dIJUQBm$jQ0wO)2z7hIm#TxSTQtpALOb8s&b0E4+(LpdZp!G z3Kf;bzD^4`HB_IPYID_YZ2}i9Bie~p1rRS)%ElvBoqH!r(}xN~PCa>~6o36Yr_52a zKf|pB2pqDl^J+e&{pZ2NBeo zdY=Z+Iov8Br^c0cP8pxsywIa^*G|Gm#@K}G zG_1#R_#^Zi-*|WCC0tG6Vs$Jo0>H8o&itWU_V~}cv&5Vp-w|TI;OResIl@2DKr{R7 zt<$|b0W9Bs!5xg8&NWk4!_yPPHlj^$RL)+!95{bN1aum-07}I#%`jYXy0;sub&o*1 zFJJ){p(**`McKSsQ?j)R0)p_+wnoR_m-d3TAkR6JsuGH z^_th|{@5W@+5~Bz8j5guBZP^%k|K>*m@OlAbxv3aNTOcX5m%wVG8YT|i_(2`B15SqCfFe%HukedYRp|t86o$ua~#shF-kYsh(=NqdRy|GpYX!;Vo z4kD?vd7}3kewIpOpSrbd%@b-5nTzf?tQ;C>Dkl~W_0LhB;*7r^5gGz-EVMuQ9$XAP zi46V&9hb8a($Px1!vFw7!%m5{UA`la@d@0PMl8N9C&mEUZSjyhEL72a_)W%=mG~K+ z?!C3%uu_M8N80wRnsoa{;%9a#MEmEyZu<1^8Yh!pI89(1X%33?r&!N5bbj6{3nVNQ zF;2ey&LWo(BH}1`+R{Ac1RN@tE_+!-<4zAR!YuuIHGBpB8DQ(gLdV121$d)8owxY> z@a^O1V_~?<^`;;&(%JeZ9=DV~CpXiFozr0YAPuTV^B5xV!3RZ)gCfgp=(RLe=w+(@ z!a`=7gZ9Ht@0@K@83l}gKO(a^@6@`H_oSc3{=(4RwK#*9;>_E}^HgBgstPN35( zQPxlNec-8A^YcxGg`!qNE_+4bp5DKvb?AB{n}T;bIrAZ{#EOK`Hjr zhd+XO5uyfCnX=2p>HTTlm;05V90nv@brSF(pQTF=ABuL=<({%-(4j-=aU4hRU+&duZ4}d3 zmB%X712SmHz<-h0nuQU$fw%|OqA>*ILAyUuR?)OGA4w#lt#yhg7$HYfp2?8m>{KID z$@49nXVrPefyLSzUVJ@rm#_Y5-zG{TmGc9n3G&tnbpjBumvmy3kPPi-ecA|E_<#(3-hd0pQ8R$m+yoc25i}_q}59y-l zp;Y@(cOG4Wm&K})AhQqIZhiuMbhVG@54*vx3RpY$A1sGf@!_)qd0JqQ6#P9Lau0gz z|JkvQW^3EaTYKqRef*sk)42G0+&$7=>zP%40;6M>(42Buvc5Jpr#2|&QK0myiV+j^ z0|+SmWR+iCCu^+SKO?5|R!k;!+*&i^)o{8F1+ZrZkp8IZo*rvWSA_!JhT35~F{ z+tOGr&!@^98i>1#YxQM$?BMt^9IvLN2uRF7&w3ba`TyMg)sZiz>fp#`(O){VJ1c6@_sMa zF-8Ue;DTYqw)FFCjsZgLICu;1$Lk~W@jS}@cSP`GxH-&xHPh6Bk_CO|q|0PKa}Yj0 zjH>mSoF2aSqeK8!H_*$~bh0FcwL#UyYEHAs$oW9pa}paRqpgi{^+grxQW;QF?rXDpirANDgQK zhpFQ(Ck`|MEcPG_Ag84MvPBTT0nPBLn-$ev$+8`ziO(0zhBUDd+h3ROL?M{x06 ziNL_KN;V!YI3iG1YgVZikIYSIfp8>OmYyA8e8<%n%ce%9dhL!1lw~Am48&){&+Zy? zG>B$+0WYqMyPAL2v#1jxM-FU3F!YG<<}jNyryUYd_x)2 zN=D}_u;L}+mY#HpbgcFV{^>?-K;8Os=tc2F__n2N%S8)6gyOE(iU!an(B>tP&ASM9 zw+X!0Mwu^BpxXq%ohBVLFmgb_As4L<>NgqObRIR$ED|#Jq;d8PVJmdV(wSLmo5v^T zR1oYT0s!G}DD@iq3L!arstiy7;5LlFO~zwbvR=3Y#wA_JBj1y8seelKN6~^!QXXlE zg{fmD^a;g+=qlbJY-k>B$(^-`y$@oHKj((txnEoiO|`ofU2NPfHYgBy!EJ{KS@4_h@s)EAg&SfjgXXA78a zu_c+-w&hJiM)>73lwhY-L;UEYPw9rQ^}a3^j8kskbeL3*3Wii{cTsjm#RJ#JEx zKDy(_7|I2WL;kL8x>6nx#LpASS@UDvk|{CxYVZ3cA2foJZYG}w(%R&2V0tm1RQioQ z(FCqjAVcKp^|S+(j^cMsyC4@obvKry{5r*#guh`NyHQggEOr?tFs|e{ybvdhiIN%) zL}HA~C@h0!76`(F?{s zf8g$_sB40e!a-?}|;@FD+g>t2cDAKw%D;N2Whu>g9vmz491BOefM<0d>huh$61 zJFu1jns`7d!H;AJGFzIZLR$!8Q&&a-FF^xx$`enw$p2H~D_t^=j^>_T(BM@caf3LN zy(RQ%M&k(jU=Im69Od2T?~7Hy_|Cdy))!BvO*bZ<+VQ{7Tqb=MB*>=@k(&>UJ~TV` zBrFxy3p2;L9kG@)yR~2sPnD!PX~=$D%}K5Q{HM~F_4J4?2t#!Vjx8rcxJWI2uFSQ; 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