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\begin_layout Title
Erratum: Geometry of discrete quantum computing
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\begin_layout Author
Andrew J. Hanson
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, Gerardo Ortiz
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, Amr Sabry
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, Yu-Tsung Tai
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\begin_layout Address

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 School of Informatics and Computing, Indiana University, Bloomington, IN 47405, U.S.A
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\begin_layout Address

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 Department of Physics, Indiana University, Bloomington, IN 47405, U.S.A
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\begin_layout Address

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 Department of Mathematics, Indiana University, Bloomington, IN 47405, U.S.A
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\begin_layout Email
ortizg@indiana.edu
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Section
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5.4 of our paper
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 requires a clarification and a correction.
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Clarification: Unentangled vs.
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product states.
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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
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(27) reports its purity to be 1
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. When using finite Galois fields 
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, for particular choices of 
\begin_inset Formula $p$
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, it is possible for equation
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(27) to produce a purity of 1 for some entangled states. For example, the normalized entangled state 
\begin_inset Formula $\ket{\Psi} = \ket{011}+\left(2+\rmi\right)\ket{100}+\ket{101}+\ket{110}$
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 has 
\begin_inset Formula $P_{\fh}=1$
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 for 
\begin_inset Formula $p=7$
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. In addition, the process of determining whether a 
\emph on
given
\emph default
 state 
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 is a product state may depend on 
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.
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Thus, in finite fields, the simplest way to calculate the number of unentangled states is to disregard equation
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(27) and count the number of product states. This is exactly how the counting in section
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5.4 was done, but the paper did not point out that counting relying only on equation
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(27) might not lead to the same result.
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\begin_layout Paragraph*
Correction: Maximally entangled states.
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We present below a new version of section
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5.4 that correctly counts maximally entangled states. The rest of the article is independent of this revision.
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\begin_layout Subsection
Maximal entanglement
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\begin_layout Standard
Equation
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(27) for 
\begin_inset Formula $P_{\fh}$
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 includes a normalization factor 
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. In the discrete case, this normalization factor is undefined when 
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. Equation
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(27) also includes a summation of 
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 terms. In the discrete case, certainly when 
\begin_inset Formula $p ~|~ n$
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 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
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(27).
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For maximally entangled states, the purity calculation in conventional quantum mechanics using equation
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(27) produces 0. Given the above observations, in a discrete field, equation
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(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 $P_{\fh}=0$
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 for 
\begin_inset Formula $p=3$
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, 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}
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which avoids the normalization factor and simply checks that each summand is 0.
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We now implement these procedures to enumerate the maximally entangled states for the specific cases for 
\begin_inset Formula $n = 2 ,3 $
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 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 $n=2$
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, 
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 are 
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(p+1)p^{2}(p-1)^{2}=\{144,14\,112,145\,200,2339\,280,\ldots\}\ ,
\]
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and for general 
\begin_inset Formula $n$
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, 
\begin_inset Formula \[(p+1)p^{n}(p-1)^{n}\ .\]
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The irreducible state counts are reduced by 
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, giving 
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p^{2}(p-1)^{2}=\{36,1764,12\,100,116\,964,\ldots\}\ ,
\]
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and in general for 
\begin_inset Formula $n$
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-qubits, there are 
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 instances of pure product states.
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Performing the computation using equation (
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), we find the numbers of maximally entangled states for two qubits to be 
\begin_inset Formula \[
p\left(p^2-1\right)\left(p+1\right)=\left\{ 96,2688,15\,840,136\,800,\ldots\right\} \ .
\]
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The irreducible state counts for maximal entanglement are reduced by 
\begin_inset Formula $\left(p+1\right)$
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, giving, for 
\begin_inset Formula $n=2$
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, 
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p\left(p^{2}-1\right)=\left\{ 24,336,1320,6840,\ldots\right\} \ .
\]
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For three qubits, there are 
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 (total) and 
\begin_inset Formula $p^{3}\left(p^{4}-1\right)$
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 (irreducible) instances of pure maximally entangled states, while the general formula for 4-qubit states remains unclear.
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Therefore, the ratio of maximally entangled to product states is 
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\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} \ ,
\]
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for 
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 and 
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, respectively.
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We would like to thank John Gardiner for pointing out the need for this correction.
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References
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