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\runninghead{\tiny Rosu, Planat, Saniga} {\tiny MUBs: From FPG to QPE}

\normalsize\textlineskip
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%\copyrightheading{}                     %{Vol. 0, No.0 (1992) 000--000}

\vspace*{0.88truein}

%\fpage{1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\centerline{Mod. Phys. Lett. A xx (2000) i-f
%[quant-ph/0003xxx v3]}
%\centerline{\footnotesize\it To understand hydrogen is to understand all of
%physics}
%\centerline{\footnotesize in ``The Yin and Yang of Hydrogen", D. Kleppner,
%Phys. Today, April 1999, pp. 11-12}
\bigskip
%\bigskip
\centerline{\large \bf MUBs: From Finite Projective Geometry to Quantum Phase Enciphering}

\vspace*{0.035truein}
%\centerline{\bf MANUSCRIPTS USING COMPUTER SOFTWARE\footnote{For
%the title, try not to use more than 3 lines. Typeset the title
%in 10 pt Times Roman, uppercase and boldface.}}
\vspace*{0.37truein}
%\centerline{\footnotesize NAME}
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%\centerline{}
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%Country\footnote{State completely without abbreviations, the
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%pt Times Italic.}}
\vspace*{10pt} \centerline{\footnotesize H. C.
ROSU\footnote{\tiny E-mail: hcr@ipicyt.edu.mx $\quad$ mubg04.tex \hfill Status: accepted at AIP Conf. QCMC 2004, Strathclyde, Glasgow}, M. PLANAT, M. SANIGA }
\vspace*{0.015truein}
%\centerline{\footnotesize [Received 17 August 1999] }
%\baselineskip=10pt
\centerline{\footnotesize  Applied Math-IPICyT, Institut FEMTO-ST, Slovak Astronomical Institute}
\centerline{\tiny [Mexico-Romania, France, Slovak Republic]}
\centerline{\small quant-ph/0409096}
\vspace*{0.225truein}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\publisher{(June 2002) - Mod Phys. Lett. A 17}
%{(May 14, 2000)}

%\centerline{\footnotesize Dated Sept. 15th, 2004 $\qquad$ file: MUBG04.TEX}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
\vspace*{0.21truein} \abstracts{ {\bf Abstract.} This short note highlights the most prominent mathematical problems and physical questions associated with the 
existence of the maximum sets of mutually unbiased bases (MUBs) in the Hilbert space of a given dimension. }{}{}
%%%%%%%%%%%%%%%%%%%%%%%%%%%

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\noindent

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%PACS number(s):  98.80.Hw, 11.30.Pb

\noindent
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\newpage

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%\section{The Main Text}



%%111111111111111111111111111111111111111111111111111111111111111111111111
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{\bf Introduction}

\medskip

\noindent
Technical problems in quantum information theory already connect
such distinct disciplines as number theory, abstract algebra and
projective geometry. For a partial list of open problems related
to the development of quantum computing technologies, see
{\em http://www.imaph.tu-bs.de/qi/problems}. In this stimulating
research area the issue of the so-called mutually unbiased bases
is an important one since it is related  to the complete state
determination of a quantum system.

\medskip

{\bf Definitions and Basic Facts}

\medskip

\noindent
Two different orthonormal bases $A$ and $B$ of a $d$-dimensional
Hilbert space are called {\em mutually unbiased} if and only if
\begin{equation}
|\langle a|b\rangle|=1/\sqrt{d}~,
%\label{gl1}
\end{equation}
for all $a \in A$ and all $b\in B$. An aggregate of MUBs is a set
of orthonormal bases which are pairwise mutually unbiased. It has
been found that the maximum number of such bases cannot be greater
than $d+1$ in $d$-dimensional Hilbert space \cite{1.}. It is also
known that this limit is reached if $d$ is a power of a prime.

Yet, a still unanswered question is if there are non prime power
values of $d$ for which this bound is attained. Based on numerical
calculations, it is generally agreed \cite{2.} that in the
latter cases the lower bound on the maximum number of such bases is
$$
 N_{\rm max}= 1+{\rm min} (p_i^{e_i}),
$$
where ${\rm min} (p_i^{e_i})$ is the lowest factor in the prime
number decomposition of $d=\prod _i p_{i}^{e_i}$.


Whether or not there exists a set of $d+1$ MUBs in a
$d$-dimensional Hilbert space if $d$ not a power of a prime could
be intimately linked with the question of the existence of
projective planes whose order is not a power of prime according to
a conjecture that we published recently \cite{3.}. Another interesting
recent result is that when the quantum measurements are performed in
MUBs, a simple linear and universal relation exists between the
post-measurement density matrix and the pre-measurement density matrix
\cite{4.}.

The main application of MUBs pertains to secure quantum key
exchange (quantum cryptography). This is because any attempt by an
eavesdropper (say Eve) to distinguish between two non-orthogonal
quantum states shared by two remote parties (say Alice and Bob)
will occur at the price of introducing a disturbance to the
signal, thus revealing the attack, and allowing to reject the
corrupted quantum data. Modern protocols, e.g., the original BB84
protocol, use only one qubit technologies implying dimension
$d=2$, usually the polarisation states of the photon. But the
security against eavesdropping increases when all the three bases
of qubits are used, or by using qudits, or entanglement-based
protocols.

Quantum state recovery and secure quantum key distribution can
also be achieved using positive operator valued measures (POVMs)
which are symmetric informationally complete (SIC-POVMs)
\cite{5.}. These are sets of $d^2$ normalized vectors $a$ and $b$ such
that
$$
 |\langle a|b\rangle|=  1/(d + 1)^{1/2}\,\, {\rm   when} \,\, a \neq b~.
$$
Unlike the MUBs, the SIC-POVMs could exist in all finite
dimensions. Recently, SIC-POVMs have been constructed  in
dimension $d=6$ \cite{6.}.

\medskip

{\bf MUB's and Finite Projective Planes}

\medskip

\noindent
As already mentioned, we have recently conjectured \cite{3.} that the
existence of the maximum set of MUBs in a given dimension $d$ and that
of a projective plane of the same dimension may well represent two
aspects of one and the same problem. Perhaps the most serious backing
of our surmise is found in a recent paper by Wootters
\cite{7.}. Associating a line in a finite geometry with a pure state
in the quantum problem, the author shows that a complete set of MUBs
is, in some respects, analogous to a finite {\it affine} plane, and
another kind of quantum measurement, the SIC-POVM, is also analogous
to the same configuration, but with the swapped roles of points and
lines. It represents no difficulty to show that this ``dual" view of
quantum measurement is deeply rooted in our conjecture. To this end,
it suffices to recall two facts \cite{8.}. First, any affine plane is
a particular subplane (subgeometry) of a projective plane, viz. a
plane which arises from the latter if one {\it line}, the so-called
``line at infinity," is deleted. Second, in a projective plane, there
is a perfect {\it duality} between points and lines; that means, to
every projective plane, $S_{2}$, there exists a dual projective plane,
$\Sigma_{2}$, whose points are the lines of $S_{2}$ and whose lines
are the points of $S_{2}$ \cite{9.}. So, {\it affinizing} $S_{2}$
means deleting a {\it point} of $\Sigma_{2}$ and thus, in light of our
conjecture, qualitatively recovering the results of Wootters, shedding
also important light on some other of the most recent findings
\cite{4.,10.}. The latter reference, in fact, gives several strong
arguments that there are no more than three MUBs in dimension six, the
latter being the smallest non-prime-power dimension.

\medskip

{\bf MUB's and Quantum Fourier Transforms (QFT's)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\medskip

\noindent
There is a useful relationship between MUBs and QFTs.  Consider a
basis B$_0$ =$(|0\rangle, |1\rangle,...,|d\,-1\rangle )$ with
indices $n$ in the ring $Z_{d}$ of integers modulo $d$. The dual
basis defined by the quantum Fourier transform is
$$
|\theta _k\rangle =\frac{1}{\sqrt{d}}\sum _{n=0}^{d\,-1} \omega
_{d} ^{nk}|n\rangle~, \quad \qquad \omega _{d}
=\exp\left(i\frac{2\pi}{d}\right).
$$
In the particular case $d=2$, $\omega _{\rm 2} =-1$ and the
$\theta$ basis acquires the form
$$
|\theta _0\rangle =\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)~;
\quad  |\theta _1\rangle = \frac{1}{\sqrt{2}}(|0\rangle -
|1\rangle).
$$
Note that the two orthogonal bases B$_0$ =$(|0\rangle, |1\rangle)$
and B$_1$ =$(|\theta _0\rangle, |\theta _1\rangle)$ are mutually
unbiased. The third base B$_2$ =$(|\psi _0\rangle, |\psi
_1\rangle)$, mutually unbiased to them, is obtained from $H$ (the
Hadamard matrix) amended by the action to the right of a $\pi$/2
rotation $S$
$$
H=\frac{1}{\sqrt{2}}\left( \begin{array}{cc}
1 & 1 \\
1 & -1\end{array} \right )~, \quad
S=\left( \begin{array}{cc}
1 & 0 \\
0 & i\end{array} \right ) \qquad  \rightarrow \qquad HS= \frac{1}{\sqrt{2}}\left( \begin{array}{cc}
1 & i \\
1 & -i\end{array} \right )
$$
as applied to B$_0$.

The fact that the B bases for $d=2$ are also the eigenvectors of
the Pauli matrices $\sigma _z$, $\sigma _x$, and $\sigma _y$,
respectively, has led to a method of MUBs' construction in
dimension $d$ in terms of the generalized Pauli operators
$$
X_{d}|n\rangle = |n+1\rangle~, \qquad Z_{d}|n\rangle = \omega
_{d}^{n} |n\rangle,
$$
known as shift and clock operators. For a prime dimension $d=p$,
it can be shown \cite{11.} that the eigenvectors of the set of unitary
operators $Z_{p}, X_{p}, X_{p}Z_{p}, ..., X_{p}Z_{p}^{p-1}$ generate
the corresponding $d+1$ MUBs.

Can MUBs be obtained for any $d$ and any (number) field by Fourier
transforms as in the case of $d=2$?  In principle, the answer is
yes. For this, one should employ such a quantum Fourier transform
whose exponent $\omega$ now acts on a finite (Galois) field,
$G=GF(p^m)$, having characteristic $p$ and $d=p^m$ elements.
Denote ``$\oplus$"  and ``$\bullet$" the two operations in the
field, corresponding to ``+" and ``$\cdot$" in the field of real
numbers. Then, the $G$-Fourier transform reads
$$
|\theta _k\rangle =\frac{1}{\sqrt{d}}\sum _{n=0}^{d\,-1} \omega
_{p}^{n\bullet k}|n\rangle~.
$$
Given any two polynomials $k$ and $n$ in $G$, there exists a
uniquely determined pair $a$ and $b$ in $G$ such that
$$
k=a \bullet n \oplus b,
$$
where deg $a$ $>$ deg $b$, so that the exponent in the $G$-quantum
Fourier transform can be written in  the form
$$
E=n\bullet (a\bullet n\oplus b).
$$
The last formula is valid for the case of a prime dimension $d=p$
for which $E$ is an integer. Otherwise, it has to be replaced by
the trace of $GF(p^m$), i.e., a map down to $GF(p)$ defined as
follows
$$
tr(E) = E +E^{p}+... +E^{p^{m-1}}~, \quad E\in GF(p^m),
$$
and so
$$
|\theta _b^a\rangle =\frac{1}{\sqrt{d}}\sum _{n=0}^{d \,-1} \omega
_{p} ^{tr [n \bullet (a \bullet n \oplus b)]}|n\rangle~.
$$
This general formula was first obtained by Wootters \& Fields and
further insights into it have recently been given \cite{12.}. In a
Galois field of odd characteristic, the latter formula provides us
with a set of $d$ bases of index $a$ for the base and index $b$ for
the vector in the base, mutually unbiased to each other as well as to
the computational base B$_0$. It is worth noting that this strategy of
constructing MUBs fails for characteristic two, since in this case
$$
|\sum _{n=0}^{d \,-1} \omega _{2} ^{tr [n \bullet (a \bullet n
\oplus b)]}|n \rangle|=0~,
$$
irrespectively of the values of $a$ and $b$. In this case, one has
to use Galois rings instead of Galois fields in order to find MUBs
\cite{12.,13.}.

Finally, we notice that the same formula provides an interesting
relationship between MUBs and quantum phase operators
\cite{14.}. Indeed, it is known that the Fourier basis $|\theta
_k\rangle$ can be derived as the set of eigenvectors of a quantum
phase operators $\Theta =\sum _{k=0}^{d-1} \theta _k|\theta _k\rangle
\langle \theta _k|$. Thus, viceversa, each base of index $a$ can be
associated to a quantum phase operator of the form
$$
\Theta ^a= \sum _{b=0}^{d-1}\theta _b^a|\theta _b^a\rangle \langle \theta _b ^a|~.
$$
The implementation of the MUB concept at the level of quantum
phase kets and operators could have important technological
consequences for defining generalized measurements of the quantum
phase, which is a key feature in quantum computing processes.

\medskip

{\bf Conclusion}

\medskip

\noindent
This short note highlights only the most prominent mathematical
problems and physical questions associated with the existence of
the maximum sets of MUBs in the Hilbert space of a given
dimension. Yet, it should give the reader a fairly good picture of
the state of the art of the topic and why the latter entails
steadily-increasing attention of both physicists and
mathematicians.

\begin{thebibliography}{99}
%\bibliography{sample}

\bibitem{1.} W.K. Wootters \& B.D. Fields, Ann. Phys. {\bf 191}, 363 (1989); I.D. Ivanovic, J. Phys. A {\bf 14}, 3241
(1981).
\bibitem{2.} G. Zauner, ``Quantendesigns", Dissertation in German, Wien
(1999).
\bibitem{3.} M. Saniga, M. Planat, and H. Rosu, J. Opt. B: Quant. Semiclass. Opt. {\bf 6}, L19 (2004), {\em Preprint}
math-ph/0403057.
\bibitem{4.} C. Dhara \& N.D. Hari Dass, A New Relation between Post and Pre-Optimal Measurement States, {\em Preprint}
quant-ph/0406169.
\bibitem{5.} J.M. Renes, R. Blume-Kohout, A.J. Scott, and C.M. Caves, J. Math. Phys. {\bf 45}, 2171
(2004).
 \bibitem{6.} M. Grassl, On SIC-POVMs and MUBs in Dimension 6, {\em Preprint}
 quant-ph/0406175.
\bibitem{7.} W.K. Wootters, Quantum Measurements and Finite
Geometry, {\em Preprint} quant-ph/0406032.
\bibitem{8.} A. Beutelspacher \& U. Rosenbaum, {\it Projective Geometry: From Foundations to
Applications}, Cambridge University Press, Cambridge, 1998.
\bibitem{9.} H. Levy, {\it Projective and Related Geometries},
Macmillan, New York, 1964, p. 108.
\bibitem{10.} I. Bengtsson, MUBs, Polytopes, and Finite
Geometries, {\em Preprint}
 quant-ph/0406174.
\bibitem{11.} S. Bandyopadhyay, P.O. Boykin, V. Roychowdhury, and F. Vatan, Algorithmica {\bf 34}, 512
(2002).
\bibitem{12.} A. Klappenecker \& M. R\"otteler, Lecture Notes in Computer Science {\bf 2948}, 137 (2004), {\em Preprint}
quant-ph/0309120.
\bibitem{13.} M. Planat, H. Rosu, S. Perrine, M. Saniga, Finite
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(2003).
\end{thebibliography}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%





\end{document}

quant-ph0409096.extracted_bib
\begin{thebibliography}{99}
%\bibliography{sample}

\bibitem{1.} W.K. Wootters \& B.D. Fields, Ann. Phys. {\bf 191}, 363 (1989); I.D. Ivanovic, J. Phys. A {\bf 14}, 3241
(1981).
\bibitem{2.} G. Zauner, ``Quantendesigns", Dissertation in German, Wien
(1999).
\bibitem{3.} M. Saniga, M. Planat, and H. Rosu, J. Opt. B: Quant. Semiclass. Opt. {\bf 6}, L19 (2004), {\em Preprint}
math-ph/0403057.
\bibitem{4.} C. Dhara \& N.D. Hari Dass, A New Relation between Post and Pre-Optimal Measurement States, {\em Preprint}
quant-ph/0406169.
\bibitem{5.} J.M. Renes, R. Blume-Kohout, A.J. Scott, and C.M. Caves, J. Math. Phys. {\bf 45}, 2171
(2004).
 \bibitem{6.} M. Grassl, On SIC-POVMs and MUBs in Dimension 6, {\em Preprint}
 quant-ph/0406175.
\bibitem{7.} W.K. Wootters, Quantum Measurements and Finite
Geometry, {\em Preprint} quant-ph/0406032.
\bibitem{8.} A. Beutelspacher \& U. Rosenbaum, {\it Projective Geometry: From Foundations to
Applications}, Cambridge University Press, Cambridge, 1998.
\bibitem{9.} H. Levy, {\it Projective and Related Geometries},
Macmillan, New York, 1964, p. 108.
\bibitem{10.} I. Bengtsson, MUBs, Polytopes, and Finite
Geometries, {\em Preprint}
 quant-ph/0406174.
\bibitem{11.} S. Bandyopadhyay, P.O. Boykin, V. Roychowdhury, and F. Vatan, Algorithmica {\bf 34}, 512
(2002).
\bibitem{12.} A. Klappenecker \& M. R\"otteler, Lecture Notes in Computer Science {\bf 2948}, 137 (2004), {\em Preprint}
quant-ph/0309120.
\bibitem{13.} M. Planat, H. Rosu, S. Perrine, M. Saniga, Finite
  Algebraic Geometrical Structures Underlying Mutually Unbiased
  Quantum Measurements, {\em Preprint} quant-ph/0409081, to be published.
\bibitem{14.} D.T. Pegg \& S.M. Barnett, Phys. Rev. A {\bf 39}, 1665 (1989); M. Planat \& H.C. Rosu, Phys. Lett. A {\bf 315}, 1
(2003).
\end{thebibliography}