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#LyX file created by tex2lyx 2.3
\lyxformat 544
\begin_document
\begin_header
\save_transient_properties true
\origin C:/Users/Yu-Tsung/eclipse-workspace/Erratum-Geometry-of-discrete-quantum-computing/
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\begin_body
\begin_layout Title
Erratum: Geometry of discrete quantum computing
\end_layout
\begin_layout Author
Andrew J. Hanson
\begin_inset Formula $^{1}$
\end_inset
, Gerardo Ortiz
\begin_inset Formula $^{2}$
\end_inset
, Amr Sabry
\begin_inset Formula $^{1}$
\end_inset
, Yu-Tsung Tai
\begin_inset Formula $^{3,1}$
\end_inset
\end_layout
\begin_layout Address
\begin_inset Formula $^{1}$
\end_inset
School of Informatics and Computing, Indiana University, Bloomington, IN 47405, U.S.A
\end_layout
\begin_layout Address
\begin_inset Formula $^{2}$
\end_inset
Department of Physics, Indiana University, Bloomington, IN 47405, U.S.A
\end_layout
\begin_layout Address
\begin_inset Formula $^{3}$
\end_inset
Department of Mathematics, Indiana University, Bloomington, IN 47405, U.S.A
\end_layout
\begin_layout Email
ortizg@indiana.edu
\end_layout
\begin_layout submitto
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
jpa
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\noindent
Section
\begin_inset space ~
\end_inset
5.4 of our paper
\begin_inset space ~
\end_inset
\begin_inset CommandInset citation
LatexCommand cite
after ""
key "geometry"
literal "true"
\end_inset
requires a clarification and a correction.
\end_layout
\begin_layout Paragraph*
Clarification: Unentangled vs.
\begin_inset space \space{}
\end_inset
product states.
\end_layout
\begin_layout Standard
In conventional quantum mechanics, using the field of complex numbers, a state is unentangled when it can be expressed as a product state or, equivalently, when equation
\begin_inset space ~
\end_inset
(27) reports its purity to be 1
\begin_inset space ~
\end_inset
\begin_inset CommandInset citation
LatexCommand cite
after ""
key "PhysRevLett.92.107902, PhysRevA.68.032308"
literal "true"
\end_inset
. When using finite Galois fields
\begin_inset Formula $\ff{p^2}$
\end_inset
, for particular choices of
\begin_inset Formula $p$
\end_inset
, it is possible for equation
\begin_inset space ~
\end_inset
(27) to produce a purity of 1 for some entangled states. For example, the normalized entangled state
\begin_inset Formula $\ket{\Psi} = \ket{011}+\left(2+\rmi\right)\ket{100}+\ket{101}+\ket{110}$
\end_inset
has
\begin_inset Formula $P_{\fh}=1$
\end_inset
for
\begin_inset Formula $p=7$
\end_inset
. In addition, the process of determining whether a
\emph on
given
\emph default
state
\begin_inset Formula $\ket{\Phi}$
\end_inset
is a product state may depend on
\begin_inset Formula $p$
\end_inset
.
\end_layout
\begin_layout Standard
Thus, in finite fields, the simplest way to calculate the number of unentangled states is to disregard equation
\begin_inset space ~
\end_inset
(27) and count the number of product states. This is exactly how the counting in section
\begin_inset space ~
\end_inset
5.4 was done, but the paper did not point out that counting relying only on equation
\begin_inset space ~
\end_inset
(27) might not lead to the same result.
\end_layout
\begin_layout Paragraph*
Correction: Maximally entangled states.
\end_layout
\begin_layout Standard
We present below a new version of section
\begin_inset space ~
\end_inset
5.4 that correctly counts maximally entangled states. The rest of the article is independent of this revision.
\end_layout
\begin_layout Subsection
Maximal entanglement
\end_layout
\begin_layout Standard
Equation
\begin_inset space ~
\end_inset
(27) for
\begin_inset Formula $P_{\fh}$
\end_inset
includes a normalization factor
\begin_inset Formula $\frac{1}{n}$
\end_inset
. In the discrete case, this normalization factor is undefined when
\begin_inset Formula $p ~|~ n$
\end_inset
. Equation
\begin_inset space ~
\end_inset
(27) also includes a summation of
\begin_inset Formula $n$
\end_inset
terms. In the discrete case, certainly when
\begin_inset Formula $p ~|~ n$
\end_inset
but also in other cases, this summation may vanish in the field even if the individual summands are non-zero. These anomalies are irrelevant for the classification of unentangled states as this computation is performed by directly checking the possibility of direct decomposition into product states, disregarding equation
\begin_inset space ~
\end_inset
(27).
\end_layout
\begin_layout Standard
For maximally entangled states, the purity calculation in conventional quantum mechanics using equation
\begin_inset space ~
\end_inset
(27) produces 0. Given the above observations, in a discrete field, equation
\begin_inset space ~
\end_inset
(27) may be undefined or may report a purity of 0 even for partially entangled states. For example, the normalized 5-qubit state
\begin_inset Formula $\ket{\Psi}=\left(1-\rmi\right)\left(\ket{00}+\ket{11}\right)\otimes\ket{000}$
\end_inset
has
\begin_inset Formula $P_{\fh}=0$
\end_inset
for
\begin_inset Formula $p=3$
\end_inset
, and is not maximally entangled because only the first two qubits are entangled. In the discrete case, we therefore check for maximally entangled states using the following equations,
\begin_inset Formula \begin{equation}
\forall j,\forall\mu,\braket{\sigma_{\mu}^{j}}^2 =0\textrm{ ,}\label{eq:john}
\end{equation}
\end_inset
which avoids the normalization factor and simply checks that each summand is 0.
\end_layout
\begin_layout Standard
We now implement these procedures to enumerate the maximally entangled states for the specific cases for
\begin_inset Formula $n = 2 ,3 $
\end_inset
and compare these to the counts for product states. We can verify explicitly that the numbers of unit-norm product states for
\begin_inset Formula $n=2$
\end_inset
,
\begin_inset Formula $p=\{3,7,11,19,\ldots\}$
\end_inset
are
\begin_inset Formula \[
(p+1)p^{2}(p-1)^{2}=\{144,14\,112,145\,200,2339\,280,\ldots\}\ ,
\]
\end_inset
and for general
\begin_inset Formula $n$
\end_inset
,
\begin_inset Formula \[(p+1)p^{n}(p-1)^{n}\ .\]
\end_inset
\end_layout
\begin_layout Standard
The irreducible state counts are reduced by
\begin_inset Formula $(p+1)$
\end_inset
, giving
\begin_inset Formula \[
p^{2}(p-1)^{2}=\{36,1764,12\,100,116\,964,\ldots\}\ ,
\]
\end_inset
and in general for
\begin_inset Formula $n$
\end_inset
-qubits, there are
\begin_inset Formula $p^{n}\left(p-1\right)^{n}$
\end_inset
instances of pure product states.
\end_layout
\begin_layout Standard
Performing the computation using equation (
\begin_inset CommandInset ref
LatexCommand ref
reference "eq:john"
plural "false"
caps "false"
noprefix "false"
\end_inset
), we find the numbers of maximally entangled states for two qubits to be
\begin_inset Formula \[
p\left(p^2-1\right)\left(p+1\right)=\left\{ 96,2688,15\,840,136\,800,\ldots\right\} \ .
\]
\end_inset
The irreducible state counts for maximal entanglement are reduced by
\begin_inset Formula $\left(p+1\right)$
\end_inset
, giving, for
\begin_inset Formula $n=2$
\end_inset
,
\begin_inset Formula \[
p\left(p^{2}-1\right)=\left\{ 24,336,1320,6840,\ldots\right\} \ .
\]
\end_inset
For three qubits, there are
\begin_inset Formula $p^{3}\left(p^{4}-1\right)\left(p+1\right)$
\end_inset
(total) and
\begin_inset Formula $p^{3}\left(p^{4}-1\right)$
\end_inset
(irreducible) instances of pure maximally entangled states, while the general formula for 4-qubit states remains unclear.
\end_layout
\begin_layout Standard
Therefore, the ratio of maximally entangled to product states is
\begin_inset Formula \[
\frac{\textbf{Max entangled}}{\textbf{Product}}=\frac{p+1}{p\left(p-1\right)} ,\, \frac{\left(p^2+1\right)\left(p+1\right)}{\left(p-1\right)^2} \ ,
\]
\end_inset
for
\begin_inset Formula $n=2$
\end_inset
and
\begin_inset Formula $3$
\end_inset
, respectively.
\end_layout
\begin_layout Standard
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
\backslash
ack
\end_layout
\end_inset
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
{}
\end_layout
\end_inset
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
{
\end_layout
\end_inset
We would like to thank John Gardiner for pointing out the need for this correction.
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
}
\end_layout
\end_inset
\end_layout
\begin_layout Section*
References
\end_layout
\begin_layout Standard
\begin_inset ERT
status collapsed
\begin_layout Plain Layout
{}
\end_layout
\end_inset
\end_layout
\begin_layout Standard
\begin_inset CommandInset bibtex
LatexCommand bibtex
bibfiles "gardinerbib"
options "iopart-num"
\end_inset
\end_layout
\end_body
\end_document