%
\documentclass{pramana}
%%
%%download pramana.cls and save it in the folder of your source file
%%
%%suggested packages to be included
\usepackage{graphicx,amsmath,bm}
\usepackage{dcolumn} % needed for some tables
\usepackage{color}
%%The following packages are included with the class file.
%%Please download if these packages are not included
%%in your local TeX distribution
%%txfonts,balance,textcase,float
%%
\begin{document}
\newcommand{\ket}[1]{\ensuremath{\left|#1\right\rangle}}
\newcommand{\bra}[1]{\ensuremath{\left\langle#1\right|}}
\newcommand\floor[1]{\lfloor#1\rfloor}
\newcommand\ceil[1]{\lceil#1\rceil}
%%paper title
%%For line breaks \\ can be used within title
\title{Supplementary Material}
\author{Arpita Maitra\textsuperscript{1,*}, Joseph Samuel\textsuperscript{2,3}\and Supurna Sinha\textsuperscript{2}}
\affilOne{\textsuperscript{1}C.R. Rao Advanced Institute of Mathematics, Statistics and Computer Science, University of Hyderabad Campus, Prof. C.R. Rao Road, Gachibowli, Hyderabad 500\,046, India\\}
\affilTwo{\textsuperscript{2}Raman Research Institute, Sadashivanagar, Bengaluru 560\,080, India\\}
\affilThree{\textsuperscript{3}International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Shivakote, Bengaluru 560\,089, India}
\onecolumn
\maketitle
\section*{}
The material below may not be of interest to the general reader, but would be invaluable to those attempting to reproduce (and perhaps improve on) our results.
This material includes the programs used, the raw data and a fine point, which would have cluttered the main text.
The programs are in pdf format. The Mathematica files have a .nb extension: \\
1preparation.nb\\
2aquantumadvantagedirect.nb\\
2bquantumsearch.nb \\
2cquantumkeep.nb \\
2dquantadvantageparameters.nb\\
3circuittwocnotgates.nb\\
4U3gates.nb\\
The files are to be run in order 1, 2a, 2b, 2c, 2d, 3c, 4. 1 deals with the preparation stage.
2a finds the best direct strategy. 2b numerically determines an entangling unitary matrix which improves on the direct strategy.
This program is to be run just once as it changes the random unitary. 2c and 2d remember the unitary being used and the parameters it yields.
3 numerically implements the mathematics described in the paper by Shende \textit{et al}, \textit{Phys. Rev. A} \textbf{69}, 062321 (2004), which explains how to convert a given unitary transformation
into a sequence of quantum logic gates with a minimum of CNOT gates. 4U3 converts the unitary transformation into a program which can directly be run on a quantum computer.
The two programs have syntax compatible with ibmqx4 and IBM Q16.
The remaining files (.csv files converted to pdf) show the raw data on which our graph (figure 3) is plotted.
The
numbers should be read as \\
$1.$ 00000 $\rightarrow$ 00,\\
$2.$ 00100 $ \rightarrow$ 10,\\
$3.$ 01000 $\rightarrow$ 01\\
$4.$ 01100 $\rightarrow$ 11\\
We briefly explain a fine point glossed over in the text.
In determining the optimal direct strategy, it is sufficient
to restrict attention to orthogonal matrices (rather than
unitaries). Equivalently, the optimal basis is a pair of
antipodal points in the $x$--$z$ plane of the Bloch sphere.
To see this, let $v=\{\cos{\phi/2},\sin{\phi/2}\exp{i \psi}\}$ represent an
arbitrary point on the Bloch sphere, which together with
its antipode gives us a general measurement basis. One
can show that the relative entropy attains a maximum as
a function of $\psi$ only at $\psi=0,\pi$.
In other words, any basis which is off the $x$--$z$ plane does not improve over some
basis in the $x$--$z$ plane. A simple calculation shows that
$p=\bra{v}\rho_A \ket{v}$
is independent of $\psi$ and $q=\bra{v}\rho_B \ket{v}$ has the form $A+B \cos{\psi}$, where $A,B$ are non-zero constants independent of $\psi$.
(We suppose $0 < p, q < 1$.)
The maximum
of $S(q(\psi))=p\log{p/q}+(1-p)\log{(1-p)/(1-q)}$ must have \begin{equation}\frac{\mathrm{d}S}{\mathrm{d}\psi}=\frac{\mathrm{d}S}{\mathrm{d}q} \frac{\mathrm{d}q}{\mathrm{d}\psi}=0.\notag\end{equation}$({\mathrm{d}S}/{\mathrm{d}q})$ cannot vanish at a maximum,
because if it did,
\begin{equation}\frac{\mathrm{d}^2S}{\mathrm{d}\psi^2}= \frac{\mathrm{d}^2S}{\mathrm{d} q^2} \left(\frac{\mathrm{d}q}{\mathrm{d}\psi}\right)^2+\frac{\mathrm{d}S}{\mathrm{d}q}\frac{\mathrm{d}^2q}{\mathrm{d}\psi^2}\notag\end{equation} would be positive and describe a minimum, which is
a contradiction. It follows that
\begin{equation}\frac{\mathrm{d}q}{\mathrm{d}\psi}= B\sin{\psi}= 0,\notag\end{equation} which implies the result.
\end{document}