text:
\section{Introduction}
Quantum entanglement is an incredibly rich phenomenon relevant to a wide range of fields from condensed matter physics to quantum information science to particle physics. In quantum information theory entanglement is regarded as a resource for the tasks of quantum communication and computation~\cite{Wilde13,Preskill}.

The structure of multipartite entanglement is far more complex than that in bipartite systems. There are various inequivalent entanglement classes~\cite{DVC00,VDMV02}. There are also such peculiar properties  as  monogamy relations~\cite{TerhalMon04,CKW00,KW04}  exhibited by some correlations of particles.

Genuine multipartite entanglement~(GME)~\cite{GME99} is an extreme form of the described multipartite phenomenon. Pure GME states are entangled with respect to any bipartition of a compound system, the property that has recently found applications in the protocols of quantum information processing~(see, e.~g., \cite{YeCh06,MP08,MEO18}).

An interesting direction of research is construction and characterization of genuinely entangled subspaces~(GESs) --- subspaces consisting entirely of genuinely multipartite entangled states. To our knowledge, this concept was first introduced in Ref.~\cite{DemAugWit18}, where some approaches to construction of GESs were also proposed. GESs can be interesting in connection with generating mixed GME  states --- any state with support on a GES is genuinely entangled. It is believed that GESs will be useful in quantum error correction~\cite{HuGra20} and quantum cryptography~\cite{SheSrik19}.

The research into the new methods of constructing GESs was continued in Refs.~\cite{DemAugQut19,WaChenZhGu19,AgHalBa19,DemAug20}. In Refs.~\cite{DemAugQut19,WaChenZhGu19,AgHalBa19} the construction was based mainly on the unextendible product bases~(UPB) method~\cite{BDMSST99,DMSST03} and its variations. The approach in Ref.~\cite{DemAug20} was aimed at construction of GESs of maximal possible dimension and relied on the characterization of some bipartite entangled subspaces.

The motivation of the present paper is to develop an alternative approach to the construction of GES which is not based on UPB. At the center of our considerations is the correspondence between quantum channels and subspaces of a tensor product Hilbert space~(see Section~\ref{sec::prel}). In many cases such technique allows us to obtain  families of GESs with simple parameterization, including those of maximal possible dimension. Another aspect is detecting entanglement of mixed states and estimating entanglement measures for them. It is known that there is an entanglement witness associated with the projection on an entangled bipartite subspace~\cite{Sar08}. The extension of this witness to the GME case was obtained in Ref~\cite{DemAugWit18}. In connection with the~(bipartite) witness also lower bounds on some entanglement measures for bipartite mixed states can be derived~\cite{KVAnt20}. We aim to extend these bounds to the GME case and use the examples of  GESs we construct to illustrate their application. It was shown that computation of the convex-roof entanglement measures is NP-complete~\cite{Huang14}, so, in genneral, efficient algorithms or even closed mathematical expressions for them are impossible unless $\mathrm P = \mathrm{NP}$. In this regard such bounds play an important role in the theory of entanglement.

The paper is organized as follows. In Section~\ref{sec::prel} we provide the necessary definitions and theoretical background. The methods of constructing GESs and their application to concrete three- and four-partite systems are presented in Section~\ref{sec::const}. In Section~\ref{sec::EM} we provide estimation of entanglement of some constructed subspaces and talk about lower bounds on entanglement measures for mixed states connected with overlap  with a given GES. We conclude in Section~\ref{sec::conc} and discuss possible developments of the present line of research.

\section{Preliminaries}\label{sec::prel}
\subsection{Genuine multipartite entanglement}

Consider $n$-partite quantum states in the finite dimensional tensor product Hilbert space $H_1\otimes\ldots\otimes H_n$. A pure $n$-partite state $\ket{\psi}$ is called \emph{fully separable} if it can be written as a tensor product of states for every subsystem,~i.~e.
\begin{equation}
    \ket{\psi}=\ket{\phi}_1\otimes\ldots\otimes\ket{\phi}_n.
\end{equation}
States that are not fully separable are said to be \emph{entangled}.

A pure $n$-partite state $\ket{\psi}$ is called \emph{biseparable} if it can be written as a tensor product
\begin{equation}
    \ket{\psi} = \ket{\psi}_A\otimes\ket{\psi}_{\bar{A}}
\end{equation}
with respect to some bipartition $A|\bar A$ where $A$ denotes a particular subset of subsystems and $\bar A$ denotes its complement. A multipartite pure state is called \emph{genuinely multipartite entanged} if it is not biseparable with respect to any bipartition.

There is generalization of these concepts to mixed states. A mixed  multipartite state is called \emph{biseparable} if it can be decomposed into a convex sum of biseparable pure states~(note that different terms of the sum can be biseparable with respect to different bipartitions). Otherwise it is called \emph{genuinely multipartite entanged}.

A subspace of a multipartite Hilbert space consisting entirely of entangled pure states is called \emph{completely entangled}~(CES). The examples of CESs are known~\cite{Parth04,Bhat06,CMW08}. The natural generalization of this notion is \emph{genuinely multipartite entangled subspaces}~(GESs) -- those composed entirely of genuinely entangled pure states.

\subsection{Entanglement measures of states and subspaces}
For the purpose of quantifying entanglement  many entanglement measures were introduced, initially for bipartite systems~\cite{PlenVir07,Hor09,GuhnToth09}. Among them -- the concurrence~\cite{BenVinSmol96,Woot98,BDH02}, the negativity~\cite{VidWer02,LCOK03} and the geometric measure of entanglement~\cite{BarnLin01,WeiGold03}.

The \emph{concurrence} of a pure bipartite state $\psi$ is defined by
\begin{equation}\label{purconc}
    C(\psi) = \sqrt{2\left(1-\Tr{\rho_A^2}\right)},
\end{equation}
where $\rho=\dyad{\psi}$ and $\rho_A = \mathrm{Tr}_B\{\rho\}$.

For mixed states $\rho$ the concurrence is given by the convex roof construction, the minimum average concurrence taken over all ensemble decompositions of $\rho$:
\begin{equation}\label{concconv}
    C(\rho) = \min_{\{(p_j,\,\psi_j)\}}\,\sum_j\,p_j\,C(\psi_j).
\end{equation}

The \emph{negativity} of $\rho$ is defined as
\begin{equation}\label{negdef}
    N(\rho) = \frac12(\norm{\rho^{T_B}}_1 - 1),
\end{equation}
where $\rho^{T_B}$ is the partial transpose of $\rho$ with respect to party $B$, and $\norm{A}_1 = \Tr{\sqrt{A^{\dagger}A}}$ is the trace norm of $A$.

From the definition of $N$ it is seen that entanglement of states with a positive partial transpose~(PPT states) is \emph{not detected} by this measure.

The \emph{convex-roof extended negativity}~(CREN) is given by
\begin{equation}\label{cren}
    N^{\mathrm{CREN}}(\rho) = \min_{\{(p_j,\,\psi_j)\}}\,\sum_j\,p_j\,N(\psi_j).
\end{equation}

The two measures defined via convex roof are able to detect \emph{all} entangled states.

Given the Schmidt decomposition $\ket{\psi} = \sum_i\,\sqrt{\lambda_i}\ket{i}\otimes\ket{i}$ of a bipartite pure state the geometric measure of entanglement is defined by
\begin{equation}\label{geom}
    G(\psi) = 1 - \max_i\{\lambda_i\}.
\end{equation}
This measure is also extended to mixed states by the convex roof construction.

Bipartite entanglement measures can be generalized to measures of genuine multipartite entanglement~\cite{Guhnetall20}. Given a bipartite entanglement measure $E$, the corresponding GME measure for a pure multipartite state $\ket{\psi}$ is defined as
\begin{equation}\label{geomGME}
    E_{GME}(\psi) = \min_A E_A(\psi),
\end{equation}
where the minimum is taken over all possible bipartitions $A|\bar A$ of a multipartite system and $E_A$ denotes the bipartite entanglement measure with respect to bipartition $A|\bar A$. The measure $E_{GME}$ is extended to mixed states by the convex roof construction
\begin{equation}\label{cfGME}
    E_{GME}(\rho) = \min_{\{(p_j,\,\psi_j)\}}\,\sum_j\,p_j\,E_{GME}(\psi_j),
\end{equation}
where, as usual, the minimization runs over all possible ensemble decompositions $\rho = \sum_j p_j\dyad{\psi_j}$.

There are several ways of quantifying entanglement of a subspace. In the present paper we will use the approach suggested in Refs.~\cite{GourWall07},\cite{DemAugQut19}: the entanglement of a subspace $W$ measured by $E$ is defined by
\begin{equation}
   E(W) =  \min_{\ket{\psi}\in W}E(\psi).
\end{equation}
In place of $E$ here we will use the geometric measure of genuine entanglement, $G_{GME}$, which can be calculated via eqs.~(\ref{geom}) and (\ref{geomGME}).

\subsection{Quantum channels and entangled subspaces}

Let $H$ be a finite dimensional Hilbert space, and $\mathcal B(H)$~-- a collection of linear operators on $H$. Given two finite dimensional Hilbert spaces $H_A$ and $H_B$,  a \emph{quantum channel} is a linear, completely positive and trace-preserving map between $\mathcal B(H_A)$ and $\mathcal B(H_B)$ \cite{Wilde13}. By definition, a quantum channel maps density operators to density operators.

Action of a quantum channel $\mathrm\Phi\colon\,\mathcal B(H_A)\rightarrow\mathcal B(H_B)$ on an operator $C \in\mathcal B(H_A) $, by Choi's theorem\cite{Choi75}, can be represented as
\begin{equation}\label{Kraus}
    \mathrm\Phi(C) = \sum_{i=1}^N\,K_i C K_i^{\dagger},
\end{equation}
where $N\leqslant\mathrm{dim}(H_A)\,\mathrm{dim}(H_B)$, and the operators $K_i\colon\,H_A\rightarrow H_B$ satisfy the trace-preserving property
\begin{equation}\label{TrP}
    \sum_{i=1}^N\, K_i^{\dagger}K_i = I,
\end{equation}
where $I$ -- the identity operator, which in this case acts on $H_A$. Representation~(\ref{Kraus}) is often referred to as the Kraus decomposition of a quantum channel with Kraus operators $\{K_i\}$.

There is a one-to-one correspondence between channels and linear subspaces of composite Hilbert spaces\cite{AubSz17}. Given a set of three Hilbert spaces $(H_A,\, H_B,\, H_C)$, a subspace $W$ of the composite space $H_B\otimes H_C$, and an isometry $V\,\colon\, H_A\rightarrow H_B\otimes H_C$ whose range is $W$, the corresponding quantum channel $\mathrm\Phi\colon\,\mathcal B(H_A)\rightarrow\mathcal B(H_B)$ can be defined by
\begin{equation}\label{IsoRep}
    \mathrm\Phi(\rho) = \mathrm{Tr}_{H_C} (V\rho V^{\dagger}).
\end{equation}
Conversely, due to Stinespring's dilation theorem~\cite{Stine55}, any quantum channel $\mathrm\Phi\colon\,\mathcal B(H_A)\rightarrow\mathcal B(H_B)$ can be represented by eq.~(\ref{IsoRep}) for some subspace $W\subset H_{B}\otimes H_{C}$.

Given some Kraus representation~(\ref{Kraus}) of a quantum channel $\mathrm\Phi\colon\,\mathcal B(H_A)\rightarrow\mathcal B(H_B)$, the isometry~$V\,\colon\, H_A\rightarrow H_B\otimes H_C$ can be defined by the action on a vector state $\ket{\psi}\in H_A$ in the following way:
\begin{equation}\label{IsoK}
    V\ket{\psi} = \sum_{i=1}^N\,K_i\ket{\psi}\otimes\ket{i},
\end{equation}
where $\{\ket{i}\}$ -- an orthonormal basis in $H_C$, and $\mathrm{dim}(H_C) = N$. Choosing some orthonormal basis $\{{\ket{\mu}}\}$ in $H_A$, via eq.~(\ref{IsoK}) we obtain an orthonormal system of vectors $\{V\ket{\mu}\}$ spanning the subspace $W\subset H_{B}\otimes H_{C}$.

Fig.~\ref{fig:channel} provides the diagrammatic representation of eq.~(\ref{IsoRep}). We adopt tensor diagram notation for quantum processes from Ref.~\cite{CoeKis17}. Refs.~\cite{BiamonteEtAll15}, \cite{Biamonte19} are also excellent introductions into diagrammatic reasoning in quantum information theory.

\begin{figure}[b]
\includegraphics[scale=0.35]{stine
\caption{\label{fig:channel} A representation of quantum channel $\mathrm\Phi$ acting on a pure state $\ket{\phi}\in H_A$: at first, isometry $V$ takes the state  to $H_B\otimes H_C$, then subsystem $C$ is traced out. The result is some density operator acting on $H_B$.}
\end{figure}

The entanglement of the subspace $W\subset H_{B}\otimes H_{C}$ depends on the output characteristics of the corresponding quantum channel such as the maximal output norm \cite{Holev00}. Let $\mathcal D(H)$ denote the set of density operators in $\mathcal B(H)$. \emph{The maximal output norm} of a channel $\mathrm\Phi\colon\,\mathcal B(H_A)\rightarrow\mathcal B(H_B)$  is defined by
\begin{equation}\label{OutNorm}
    \nu_p(\mathrm\Phi) = \sup_{\rho\in\mathcal D(H_A)}\norm{\mathrm\Phi(\rho)}_p,\,\,p>1,
\end{equation}
where $\norm{\rho}_p = (\mathrm{Tr}(\abs{\rho}^p))^{1/p}$ is the standard $p$-norm. Since $p$-norm is convex, it will take its maximum on the extremal states, and the supremum in eq.~(\ref{OutNorm}) can be taken over pure input states. We will mostly use the Frobenius norm~($p = 2$) and refer to
$(\nu_2(\mathrm\Phi))^2$ as the \emph{output purity} of a channel $\mathrm\Phi$.

Obviously, a subspace $W\subset H_{B}\otimes H_{C}$ is completely entangled if and only if the output purity of the corresponding quantum channel is strictly less than one. In this case the isometry $V$ will transform the vector states of $H_A$ into bipartite entangled states of the subspace $W\subset H_{B}\otimes H_{C}$, and there will not be separable states in $W$. Therefore, we can construct bipartite completely entangled subspaces finding examples of quantum channels with output purity less than one along with the corresponding isometries. Choosing some orthonormal basis $\{\ket{i}\}$ in $H_A$, we obtain the orthonormal basis $\{V\ket{i}\}$ of the entangled subspace $W$ via the isometry $V$. As we will see, the most interesting cases,  examples of entangled subspaces of maximal dimension, can be obtained by constructing dimension changing channels $\mathrm\Phi\colon\,\mathcal B(H_A)\rightarrow\mathcal B(H_B)$ with $\mathrm{dim}(H_A)\ne\mathrm{dim}(H_B)$. Furthermore, such channels can be used in construction of genuinely entangled  subspaces.

\begin{figure}[b]
\includegraphics[scale=0.35]{isolowdim1
\caption{\label{fig:iso23} An isometry $V$ acting on  subsystem $B$ of  a pure bipartite entangled state $\ket{\psi_{AB}}$. The resulting pure state is tripartite, with subsystems $A$, $C$, $D$.}
\end{figure}

\section{\label{sec::const}Construction of entangled subspaces}

In this section we describe several methods of constructing completely and genuinely entangled subspaces with the use of quantum channels. We begin with the construction of  genuinely entangled subspaces from entangled subspaces of a bipartite Hilbert space.

\subsection{\label{sub::constch}GES from bipartite CES}

This method was inspired by tensor diagrams. Given a pure bipartite entangled state $\ket{\psi}_{AB}$, we can act on one of its two parties, $B$, with an isometry $V\,\colon\, H_B\rightarrow H_C\otimes H_D$ corresponding to some quantum channel $\mathrm\Phi\colon\,\mathcal B(H_B)\rightarrow\mathcal B(H_C)$, as it is illustrated on Fig.~\ref{fig:iso23}. When will the resulting state be entangled for any bipartite cut? It is convenient to consider the initial state $\ket{\psi}_{AB}$ in the Schmidt form:
\begin{equation}
    \ket{\psi}_{AB} = \sum_i\,\sqrt{p_i}\,\ket{i}_A\otimes\ket{i'}_B,
\end{equation}
where there are at least two nonzero Schmidt coefficients $\sqrt{p_i}$.
The resulting state
\begin{eqnarray}
    \ket{\chi}_{ACD} &=& (I\otimes V)\ket{\psi}_{AB}\nonumber\\
     & &= \sum_i\,\sqrt{p_i}\,\ket{i}_A\otimes V\ket{i'}_B
\end{eqnarray}
is surely entangled across bipartition $A|CD$: tracing out, for example, subsystem $CD$, we obtain a mixed state on $A$:
\begin{equation}\label{CESbi}
    \mathrm{Tr}_{CD}\left(\dyad{\chi}_{ACD}\right) = \sum_i\,p_i \dyad{i}_A.
\end{equation}

Now consider bipartition $C|AD$. Tracing out subsystem $A$, then $D$, we obtain:
\begin{eqnarray}\label{chbi}
\mathrm{Tr}_{AD}\left(\dyad{\chi}_{ACD}\right) &&\nonumber\\
&&=\sum_i\,p_i\mathrm{Tr}_D(V\dyad{i'}_B V^{\dagger})\nonumber\\
&&=\sum_i\,p_i\mathrm{\Phi}(\dyad{i'}_B)\label{partC},
\end{eqnarray}
where the second equality is due to eq.~(\ref{IsoRep}). For an arbitrary channel $\mathrm{\Phi}$ this state may be pure -- for example, we can choose a channel mapping all states to some pure state $\dyad{\phi}_C$. In order to guarantee mixedness, we can choose a channel with \emph{output purity less than one} -- in this case each term in eq.~(\ref{partC}) will be a mixed state, and the resulting state will be a convex combination of such mixed density operators. Entanglement across bipartition $D|AC$ is analyzed similarly. Therefore, we've established that genuinely entangled tripartite states can be obtained from entangled bipartite ones by applying to  one of the parties the isometry corresponding to a quantum channel with output purity less than 1.

Now, applying such an isometry to each state in a completely entangled subspace of a bipartite Hilbert space, we obtain a genuinely entangled subspace of a tripartite Hilbert space, and, since isometry preserves inner products, the orthonormal system of vectors spanning the latter is obtained by action of $V$ on the orthonormal system spanning the former.

As an illustration of this approach we construct a four-dimensional genuinely entangled $3\otimes3\otimes3$ subspace from a completely entangled $3\otimes3$ subspace of the same dimension. Here the bipartite $3\otimes3$ subspace is taken to be spanned by the following orthonormal system of vectors:

\begin{subequations}\label{33ex}
\begin{equation}\label{33a}
    \ket{\psi_1} = \sqrt{\lambda_1} \ket{0}\otimes\ket{0} + \sqrt{1-\lambda_1}\ket{2}\otimes\ket{2},
\end{equation}
\begin{equation}
    \ket{\psi_2} = \sqrt{\lambda_2} \ket{1}\otimes\ket{0} + \sqrt{1-\lambda_2}\ket{0}\otimes\ket{1},
\end{equation}
\begin{equation}
    \ket{\psi_3} = \sqrt{\lambda_3} \ket{2}\otimes\ket{0} + \sqrt{1-\lambda_3}\ket{1}\otimes\ket{2},
\end{equation}
\begin{equation}\label{33d}
    \ket{\psi_4} = \sqrt{\lambda_4} \ket{2}\otimes\ket{1} + \sqrt{1-\lambda_4}\ket{0}\otimes\ket{2},
\end{equation}
\end{subequations}
where $0 < \lambda_i < 1,\:i = 1,\,2,\,3,\,4$.

In fact, eq.~(\ref{33ex}) determines a family of completely entangled subspaces parameterized by $\{\lambda_i\}$. They can be constructed with the use of specific dimension changing quantum channels~(see Appendix~\ref{app:33}).

Now, given a concrete $3\otimes3$ entangled subspace, we choose an isometry to act with on subsystem $B$, the second qutrit. As was shown above, the range of the isometry should be a completely entangled subspace, or, equivalently, the corresponding quantum channel should have output purity less than one. Such a role can play, for example,  the isometry $V\,\colon\, H_B\rightarrow H_C\otimes H_D$ which maps orthonormal basis in $H_B$ to vectors spanning the antisymmetric subspace of $H_C\otimes H_D$:
\begin{subequations}
\begin{equation}
    V\ket{0} = \frac1{\sqrt2}(\ket{0}\otimes\ket{1} - \ket{1}\otimes\ket{0}),
\end{equation}
\begin{equation}
    V\ket{1} = \frac1{\sqrt2}(\ket{0}\otimes\ket{2} - \ket{2}\otimes\ket{0}),
\end{equation}
\begin{equation}
    V\ket{2} = \frac1{\sqrt2}(\ket{1}\otimes\ket{2} - \ket{2}\otimes\ket{1}).
\end{equation}
\end{subequations}
Acting with $(I\otimes V)$ on the basis~(\ref{33a}-\ref{33d}), we obtain the orthonormal system of vectors spanning  a four-dimensional genuinely entangled subspace:
\begin{subequations}
\begin{eqnarray}
\ket{\phi_1} = \sqrt{\frac{\lambda_1}2}(\ket{0}\otimes\ket{0}\otimes\ket{1} - \ket{0}\otimes\ket{1}\otimes\ket{0})\nonumber\\
+ \sqrt{\frac{1 -\lambda_1}2}(\ket{2}\otimes\ket{1}\otimes\ket{2} - \ket{2}\otimes\ket{2}\otimes\ket{1}),
\end{eqnarray}
\begin{eqnarray}
\ket{\phi_2} = \sqrt{\frac{\lambda_2}2}(\ket{1}\otimes\ket{0}\otimes\ket{1} - \ket{1}\otimes\ket{1}\otimes\ket{0})\nonumber\\
+ \sqrt{\frac{1 -\lambda_2}2}(\ket{0}\otimes\ket{0}\otimes\ket{2} - \ket{0}\otimes\ket{2}\otimes\ket{0}),
\end{eqnarray}
\begin{eqnarray}
\ket{\phi_3} = \sqrt{\frac{\lambda_3}2}(\ket{2}\otimes\ket{0}\otimes\ket{1} - \ket{2}\otimes\ket{1}\otimes\ket{0})\nonumber\\
+ \sqrt{\frac{1 -\lambda_3}2}(\ket{1}\otimes\ket{1}\otimes\ket{2} - \ket{1}\otimes\ket{2}\otimes\ket{1}),
\end{eqnarray}
\begin{eqnarray}
\ket{\phi_4} = \sqrt{\frac{\lambda_4}2}(\ket{2}\otimes\ket{0}\otimes\ket{2} - \ket{2}\otimes\ket{2}\otimes\ket{0})\nonumber\\
+ \sqrt{\frac{1 -\lambda_4}2}(\ket{0}\otimes\ket{1}\otimes\ket{2} - \ket{0}\otimes\ket{2}\otimes\ket{1}).
\end{eqnarray}
\end{subequations}

Another example will be given in Section~\ref{sec::EM} in the context of estimating entanglement of the constructed subspaces.

The disadvantage of the described approach is that we cannot construct genuinely entangled subspaces of maximal dimension -- the resulting subspace has the same dimension as the initial bipartite one. In general, according to Parthasarathy's result \cite{Parth04}, the maximal dimension of completely entangled subspaces of a $d_1\otimes d_2$ Hilbert space is
\begin{equation}\label{MaxDim}
d_1 d_2 - (d_1 + d_2) + 1,
\end{equation}
and so this is the upper bound for the dimension of  genuinely entangled $d_1\otimes d'_2\otimes d'_3$ subspaces generated with this approach ($d'_2$, $d'_3$ -- output dimensions of the isometry used for the construction).

\subsection{\label{sec::mxdim}Constructing GES of maximal dimension}

Our next approach aims at constructing genuinely entangled subspaces of maximal possible dimension.
Given a pure state $\ket{\psi}_{ABC}$ on a tripartite $d\otimes d\otimes d$ Hilbert space, we consider its bipartition $AB|C$ where subsystems $A$ and $B$ are gathered into one subsystem $AB$ described by a Hilbert space of dimension $d^2$~(see Fig.~\ref{fig:gather}).

\begin{figure}[t]
\includegraphics[scale=0.35]{gather1
\caption{\label{fig:gather} Gathering subsystems of a tripartite pure state $\ket{\psi}_{ABC}$ into bipartition $AB|C$. The dimensions of the corresponding Hilbert spaces are shown.}
\end{figure}

\begin{figure}[t]
\includegraphics[scale=0.35]{isogen1
\caption{\label{fig:isogen}Steps of constructing a genuinely entangled subspace: the isometry $V$ maps a Hilbert space $H_E$ to a completely entangled subspace of $H_A\otimes H_B$, then subsystem $A$ is factorized into $A_1$ and $A_2$.}
\end{figure}

According to Ref.~\cite{Parth04}, the maximal dimension of subspaces, which are completely entangled over bipartition $AB|C$, is $d^2\times d - (d^2 +d) +1$. The same being said about the other two bipartitions, $BC|A$ and $AC|B$, this number is the maximal dimension of genuinely entangled $d\otimes d\otimes d$ subspaces.

Our strategy is first to construct a maximal dimensional completely entangled  subspace of a tensor product Hilbert space $H_A\otimes H_B$ with $\mathrm{dim}(H_A) = d^2$ and $\mathrm{dim}(H_B) = d$, then \emph{factorize} the first subsystem $A$ into two $d$-dimensional subsystems $A_1$ and $A_2$, i.~e., perform an  operation somewhat inverse to the one shown on Fig.~\ref{fig:gather}. There are many ways to do such a factorization, and  for our purpose it should be chosen in such a way that the resulting tripartite states are entangled  across bipartitions $A_1|A_2B$ and $A_2|A_1B$ (the entanglement across $A_1A_2|B$ is preserved under such an operation).

According to the above plan, we first construct an isometry $V\colon\,H_E\rightarrow H_A\otimes H_B$, where $H_E$ -- a Hilbert space of dimension $d^2\times d - (d^2 +d) +1$. If this isometry corresponds, via Stinespring's dilation, to a quantum channel with output purity less than one, then its range will be a completely entangled subspace of $H_A\otimes H_B$ with maximal dimension. Next, we apply  a proper factorization of $H_A$ into $H_{A_1}\otimes H_{A_2}$ and obtain a genuinely entangled tripartite subspace of maximal possible dimension, $d^2\times d - (d^2 +d) +1$~(see also Fig.~\ref{fig:isogen}).

In order to find the desired isometry, we analyze the expression for the output purity of a quantum channel $\mathrm\Phi\colon\,\mathcal B(H_E)\rightarrow\mathcal B(H_A)$ in terms of its Kraus operators. As it was mentioned above, it will suffice to consider the action of $\mathrm\Phi$ only on pure states, and for an arbitrary pure state $\ket{\psi}\in H_E$ we have:
\begin{eqnarray}
\mathrm{Tr}(\mathrm\Phi (\dyad{\psi})^2)&&\nonumber\\  =&&\sum_{i,\,j}\mathrm{Tr}\left(K_i\dyad{\psi}K_i^{\dagger}K_j\dyad{\psi}K_j^{\dagger}\right)\nonumber\\
=&&\sum_{i,\,j}\mathrm{Tr}\left(\dyad{\psi}K_i^{\dagger}K_j\dyad{\psi}K_j^{\dagger}K_i\right)\nonumber\\
=&&\sum_{i,\,j}\abs{\bra{\psi}K_i^{\dagger}K_j\ket{\psi}}^2\nonumber\\
\leqslant&&\sum_{i,\,j}\bra{\psi}K_i^{\dagger}K_i\ket{\psi}\bra{\psi}K_j^{\dagger}K_j\ket{\psi}\nonumber\\
&&=1\label{KrT},
\end{eqnarray}
where the Kraus decomposition~(\ref{Kraus}) was used; the inequality follows from the Cauchy-Schwarz inequality; the last equality follows from eq.~(\ref{TrP}) averaged over $\ket{\psi}$ and squared.

It is known that the Cauchy-Schwarz inequality for two vectors holds strictly, without equality if and only if they are linearly independent. Therefore, to guarantee that the trace in Eq.~(\ref{KrT}) is strictly less than one, we need to search for Kraus operators $\{K_i\}$ satisfying the following conditions:
\begin{enumerate}
    \item The eigenvalues of $K_i^{\dagger}K_i$, for all $i$, are strictly less than one~(otherwise, the trace in Eq.~(\ref{KrT}) will evaluate to $1$ on the corresponding eigenvector)):
    \begin{equation}\label{Kc1}
        \norm{K_i^{\dagger}K_i} < 1.
    \end{equation}
    \item For \emph{any} $\ket{\psi}\in H_E$ there is at least one pair $\left(K_i,\,K_j\right)$ of distinct Kraus operators such that the vectors $K_i\ket{\psi}$ and $K_j\ket{\psi}$ are not proportional to each other.
\end{enumerate}

Having found the proper Kraus operators $\{K_i\}$, we construct the corresponding isometry $V$ via eq.~(\ref{IsoK}). The final step is to do an appropriate factorization of the fist subsystem into two ones, and this procedure significantly depends on the structure of the completely entangled subspace determined by the isometry $V$.

We illustrate the approach with several examples.

\subsubsection{GES in three-qubit systems}

Now we consider a system of $3$ qubits whose pure states are described by vectors in a $2\otimes 2\otimes 2$ Hilbert space. According to the described above procedure, we gather the first and the second subsystems into one and try to construct a completely entangled $4\otimes 2$ subspace of maximal dimension $4\times 2 - (4 + 2) + 1 = 3$. The corresponding isometry $V\colon\,H_E\rightarrow H_A\otimes H_B$, with $\mathrm{dim}(H_E) = 3$, $\mathrm{dim}(H_A) = 4$, $\mathrm{dim}(H_B) = 2$, can be expressed by eq.~(\ref{IsoK}) in terms of Kraus operators $K_i\colon\,H_E\rightarrow H_A$ of a quantum channel with output purity less than one.

There are two Kraus operators in this particular case, and, in search for $K_1,\,K_2$, we can consider the simplest situation when each $K_i^{\dagger}K_i$ is diagonal, with some numbers $\lambda^{(i)}_1,\,\lambda^{(i)}_2,\,\lambda^{(i)}_3$ on the main diagonal, such that
\begin{eqnarray}\label{lam22}
    &0 < \lambda^{(i)}_j < 1,\quad  \lambda^{(1)}_j + \lambda^{(2)}_j = 1&,\nonumber\\
    &i=1,\,2;\,j=1,\,2,\,3.&
\end{eqnarray}
Conditions~(\ref{TrP}) and~(\ref{Kc1}) are then satisfied. Such a choice suggests the following structure of $\{K_i\}$ in the form of singular value decomposition:
\begin{equation}\label{K42}
 K_i = W_i   \begin{pmatrix}
    \sqrt{\lambda^{(i)}_1} &          0      & 0\\
          0       &   \sqrt{\lambda^{(i)}_2}  & 0\\
          0       &      0          & \sqrt{\lambda^{(i)}_3}\\
          0       &      0          &   0
    \end{pmatrix},
\end{equation}
where there is some freedom in the choice of  unitary $4\times 4$ matrices $W_i$, $i=1,\,2$.

Trying to keep everything as simple as possible, we can set $W_1 = I$, and choose $W_2$ to be some permutation matrix, for example, the one corresponding to a permutation \begin{equation*}\left(\begin{array}{cc}
     1\, 2\, 3\, 4  \\
     4\, 1\, 2\, 3
\end{array}\right)
\end{equation*}
Now let us choose an arbitrary vector state $\ket{\phi}\in H_E$ with components $(\phi_1,\,\phi_2,\,\phi_3)$ and, using eq.~(\ref{K42}) and our particular choice of $W_1,\,W_2$,  write out the corresponding vectors $K_1\ket{\phi}$
and $K_2\ket{\phi}$ as columns of a matrix:
\begin{equation}
    \begin{pmatrix}
    \sqrt{\lambda^{(1)}_1}\phi_1 & \sqrt{\lambda^{(2)}_2}\phi_2\\
    \sqrt{\lambda^{(1)}_2}\phi_2 & \sqrt{\lambda^{(2)}_3}\phi_3\\
    \sqrt{\lambda^{(1)}_3}\phi_3 & 0\\
    0 & \sqrt{\lambda^{(2)}_1}\phi_1
    \end{pmatrix}.
\end{equation}
It is easily seen that these vectors are linearly independent for any nonzero $\ket{\phi}$: all $2\times 2$ minors of the matrix evaluate to $0$ simultaneously if and only if \mbox{$\phi_1 = \phi_2 = \phi_3 = 0$}. Therefore, we have shown that operators $K_1,\,K_2$ are the Kraus operators of a channel with output purity less than one. Reconstructing the associated isometry $V\colon\,H_E\rightarrow H_A\otimes H_B$ with the use of eq.~(\ref{IsoK}) and acting with it on an orthonormal basis in $H_E$, we obtain an orthonormal system of vectors spanning a completely entangled subspace of $H_A\otimes H_B$:
\begin{subequations}\label{ces22}
\begin{equation}
    \sqrt{\lambda^{(1)}_1}\ket{0}\otimes\ket{0} + \sqrt{\lambda^{(2)}_1}\ket{3}\otimes\ket{1},
\end{equation}
\begin{equation}\label{2q2}
    \sqrt{\lambda^{(1)}_2}\ket{1}\otimes\ket{0} + \sqrt{\lambda^{(2)}_2}\ket{0}\otimes\ket{1},
\end{equation}
\begin{equation}
    \sqrt{\lambda^{(1)}_3}\ket{2}\otimes\ket{0} + \sqrt{\lambda^{(2)}_3}\ket{1}\otimes\ket{1}.
\end{equation}
\end{subequations}
Next, we try to convert this subspace into a genuinely entangled tripartite one by choosing a proper way of factorization of the first subsystem $A$ of dimension $4$ into subsystems $A_1$ and $A_2$, each with dimension $2$. A naive approach would be to choose the following correspondence between the bases of $H_A$ and $H_{A_1}\otimes H_{A_2}$:
\begin{eqnarray}\label{schq}
\ket{0}\qquad&\rightarrow&\qquad\ket{0}\otimes\ket{0},\nonumber\\
\ket{1}\qquad&\rightarrow&\qquad\ket{0}\otimes\ket{1},\nonumber\\
\ket{2}\qquad&\rightarrow&\qquad\ket{1}\otimes\ket{0},\nonumber\\
\ket{3}\qquad&\rightarrow&\qquad\ket{1}\otimes\ket{1},
\end{eqnarray}
but, obviously, after such a replacement, the spanning vector~(\ref{2q2}) itself turns into a separable one. Any permutation of scheme~(\ref{schq}) doesn't help either.

Our second approach is to act on subsystem $A$ with some simple unitary transformation $U_A$ and then factorize $A$ according to scheme~(\ref{schq}). $U_A$  is a local transformation with respect to bipartition $A_1A_2|B$ and, therefore, it preserves entanglement here. On the other hand, $U_A$ can change the situation across bipartitions $A_1|A_2B$ and $A_2|A_1B$ and create entanglement there~(see also Fig.~\ref{fig:isounitq}). We choose this transformation to be
\begin{equation}\label{uniq}
    U_A = \begin{pmatrix}
    1 & 0 & 0 & 0\\
    0 & \frac1{\sqrt{2}} & \frac1{\sqrt{2}} & 0\\
    0 & \frac1{\sqrt{2}} & -\frac1{\sqrt{2}} & 0\\
    0 & 0 & 0 & 1
    \end{pmatrix},
\end{equation}
thus mixing $\ket{1}_A$ and $\ket{2}_A$.

\begin{figure}[b]
\includegraphics[scale=0.35]{isounitq1
\caption{\label{fig:isounitq} Construction of a genuinely entangled $2\otimes 2\otimes 2$ subspace. A properly chosen unitary $U$ creates entanglement across bipartitions $A_1|A_2B$ and $A_2|A_1B$ while preserving entanglement across $A_1A_2|B$. Line splitting denotes factorization of subsystem $A$ according to scheme~(\ref{schq}). }
\end{figure}

Combining unitary transformation~(\ref{uniq}) and factorization~(\ref{schq}), from the basis~(\ref{ces22}) we obtain an orthonormal system of vectors in $H_{A_1}\otimes H_{A_2}\otimes H_B$:
\begin{subequations}\label{ges22}
\begin{eqnarray}
\ket{\psi_1} &=& \sqrt{\lambda_1}\ket{0}\otimes\ket{0}\otimes\ket{0}\nonumber\\
& &+ \sqrt{1 - \lambda_1}\ket{1}\otimes\ket{1}\otimes\ket{1},\\
\ket{\psi_2} &=& \sqrt{\frac{\lambda_2}{2}}\left(\ket{0}\otimes\ket{1} + \ket{1}\otimes\ket{0}\right)\otimes\ket{0}\nonumber\\
& &+ \sqrt{1 - \lambda_2}\ket{0}\otimes\ket{0}\otimes\ket{1},\\
\ket{\psi_3} &=& \sqrt{\frac{\lambda_3}{2}}\left(\ket{0}\otimes\ket{1} - \ket{1}\otimes\ket{0}\right)\otimes\ket{0}\nonumber\\
&+& \sqrt{\frac{1 - \lambda_3}{2}}\left(\ket{0}\otimes\ket{1} + \ket{1}\otimes\ket{0}\right)\otimes\ket{1},
\end{eqnarray}
\end{subequations}
where eq.~(\ref{lam22}) is taken into account and the redundant superscrits of lambdas are now omitted.

We already know that the subspace spanned by vectors~(\ref{ges22}) is completely entangled across bipartition $A_1A_2|B$. The entanglement across other bipartitions, for example, $A_1|A_2B$, can be certified in a standard way --  vectors~(\ref{ges22}) are decomposed accordingly
\begin{equation}
    \ket{\psi_{\mu}} = \sum_{i,\,j,\,k}\,a^{(\mu)}_{i,\,jk}\ket{i}_{A_1}\otimes\ket{jk}_{A_2B},\quad\mu = 1,\,2,\,3,
\end{equation}
and it is analyzed at what complex values $\beta_{\mu}$ the linear combination of matrices
\begin{equation}
    \sum_{\mu=1}^3\,\beta_{\mu}a^{(\mu)}
\end{equation}
has rank less than two, i.~e., all minors of order two evaluate to zero. If there is only trivial solution, $\beta_{\mu}=0\quad\forall\,\mu$, then the subspace is entangled across the cut. This can be done by hand, or, even better, with a computer algebra system supporting the Groebner basis algorithm~\cite{Buch70}.

Using the described procedure, it can be verified that the subspace determined by eq.~(\ref{ges22}) is completely entangled across bipartitions $A_1|A_2B$ and $A_2|A_1B$ and, therefore, is genuinely entangled.

\begin{figure}[b]
\includegraphics[scale=0.35]{fq1
\caption{\label{fig:fq} Construction of a genuinely entangled $2\otimes2\otimes2\otimes2$ subspace. A properly chosen unitaries $U_1$ and $U_2$ create entanglement  across all bipartitions except $A_1A_2|B_1B_2$. Line splittings denote factorizations of subsystems $A$ and $B$ according to scheme~(\ref{schq}). }
\end{figure}

\subsubsection{GES in four-qubit systems}

Pure states of a four-qubit system are described by vectors in a $2\otimes2\otimes2\otimes2$ Hilbert space. Let $A_1$, $A_2$, $B_1$, $B_2$ denote the $4$ subsystems. Joining $3$ subsystems, say $A_2$, $B_1$, $B_2$, into one, we consider the bipartition $A_1|A_2B_1B_2$ and  vector states in a $2\otimes8$ Hilbert space, where the maximal dimension of CES  is equal to $2\times8-(2+8)+1=7$. Bipartitions of another type, such as $A_1A_2|B_1B_2$, yield the maximal dimension of CES equal to $4\times4-(4+4)+1=9$. Therefore, the maximal dimension of GES in a $2\otimes2\otimes2\otimes2$ Hilbert space is upper-bounded by $7$.

To consruct such GES, we first construct a $7$-dimensional CES in a $4\otimes4$ Hilbert space, then factorize each four-dimensional subsystem into two two-dimensional ones.

To obtain the CES, we search for the isometry $V\colon\,H_E\rightarrow H_A\otimes H_B$, with the input dimension $\mathrm{dim}(H_E) = 7$ and the output dimensions $\mathrm{dim}(H_A) = 4$, $\mathrm{dim}(H_B) = 4$. In addition, the range of $V$  must be completely entangled.

The subsystems $A$ and $B$ are then factorized into $A_1,\,A_2$ and $B_1,\,B_2$, respectively, and each factorization is done according to scheme~(\ref{schq}). This is still not sufficient for obtaining a GES, and the process must be accompanied with applying some unitary transformations $U_1$ and $U_2$ to the parties $A$ and $B$. $U_1$ and $U_2$ are aimed at creating entanglement across all bipartitions except $A_1A_2|B_1B_2$ where entanglement is guaranteed by the structure of the isometry $V$. The whole procedure is presented on Fig.~\ref{fig:fq}.

The $7$-dimensional CES of a $4\otimes4$ Hilbert space can be obtained by the  approach used in the previous examples. By our construction~(see Appendix~\ref{app:44} for the details), it is spanned by the orthonormal system
\begin{subequations}\label{sfs}
\begin{equation}
    \sqrt{\lambda_1}\ket{0}\otimes\ket{0} + \sqrt{1-\lambda_1}\ket{2}\otimes\ket{1},
\end{equation}
\begin{equation}
    \sqrt{\lambda_2}\ket{1}\otimes\ket{2} + \sqrt{1-\lambda_2}\ket{3}\otimes\ket{3},
\end{equation}
\begin{equation}
    \sqrt{\lambda_3}\ket{0}\otimes\ket{1} + \sqrt{1-\lambda_3}\ket{3}\otimes\ket{2},
\end{equation}
\begin{equation}
    \sqrt{\lambda_4}\ket{3}\otimes\ket{0} + \sqrt{1-\lambda_4}\ket{2}\otimes\ket{2},
\end{equation}
\begin{equation}
    \sqrt{\lambda_5}\ket{1}\otimes\ket{1} + \sqrt{1-\lambda_5}\ket{0}\otimes\ket{3},
\end{equation}
\begin{equation}
    \sqrt{\lambda_6}\ket{1}\otimes\ket{0} + \sqrt{1-\lambda_6}\ket{2}\otimes\ket{3},
\end{equation}
\begin{equation}
    \sqrt{\lambda_7}\ket{2}\otimes\ket{0} + \sqrt{1-\lambda_7}\ket{3}\otimes\ket{1},
\end{equation}
\end{subequations}
where $0<\lambda_i<1,\:i=1,\ldots,7$.

The factorization by scheme~(\ref{schq}) of each of the two subsystems inevitably creates separable vectors from those in eq.~(\ref{sfs}), and so we need some entangling unitaries $U_1$ and $U_2$ to apply to $A$ and $B$, respectively. There are $7$ possible bipartitions of a four-partite system, and, choosing $U_1$ and $U_2$, we need to check entanglement in $6$ of them. Entanglement across bipartition $A_1A_2|B_1B_2$ is provided by the structure of the isometry $V$. Of course, such verification can be most effectively done with the use of computer algebra and the Groebner basis algorithm.

Due to the computational complexity of the problem, we were able to calculate only with concrete values for $\lambda_i$ from eqs.~(\ref{sfs}). At first we set $\lambda_i = 1/2$, $i=1,\ldots,7$. In addition, unlike the example with $3$ qubits, we couldn't come up with simple unitaries $U_1$ and $U_2$ mixing only $2$-$3$ basis states. From Ref.~\cite{Tremain2011} we took several standard unitaries, $3$- and $4$-dimensional, and on the basis of them constructed two $4$-dimensional unitaries for mixing basis states:
\begin{equation}
    Q=\frac12\begin{pmatrix}
    -1 & 1  &  1 &  1\\
     1 & -1 &  1 &  1\\
     1 &  1 & -1 &  1\\
     1 &  1 &  1 & -1
    \end{pmatrix},
\end{equation}
\begin{equation}
    T = \begin{pmatrix}
    -1/3 & 2/3 & 0 & 2/3\\
    2/3 & -1/3 & 0 & 2/3\\
    0 & 0 & 1 & 0\\
    2/3 & 2/3 & 0 & -1/3
    \end{pmatrix}.
\end{equation}
Setting then
\begin{equation}
    U_1 = QT,\qquad U_2 = T
\end{equation}
did the job.

Combining these transformations with factorization~(\ref{schq}), we obtain two new schemes of factorization. For subsystem $A$ it is defined by
\begin{eqnarray}\label{subA}
\ket{0}_A\quad&\rightarrow&\quad\frac16(5\ket{0}\otimes\ket{0} - \ket{0}\otimes\ket{1}\nonumber\\
&+&3\ket{1}\otimes\ket{0} - \ket{1}\otimes\ket{1}),\nonumber\\
\ket{1}_A\quad&\rightarrow&\quad\frac16(-\ket{0}\otimes\ket{0} + 5\ket{0}\otimes\ket{1}\nonumber\\
&+&3\ket{1}\otimes\ket{0} - \ket{1}\otimes\ket{1}),\nonumber\\
\ket{2}_A\quad&\rightarrow&\quad\frac12(\ket{0}\otimes\ket{0} + \ket{0}\otimes\ket{1}\nonumber\\
&-&\ket{1}\otimes\ket{0} + \ket{1}\otimes\ket{1}),\nonumber\\
\ket{3}_A\quad&\rightarrow&\quad\frac16(-\ket{0}\otimes\ket{0} - \ket{0}\otimes\ket{1}\nonumber\\
&+&3\ket{1}\otimes\ket{0} + 5\ket{1}\otimes\ket{1}),
\end{eqnarray}
for subsystem $B$ it is given by
\begin{eqnarray}\label{subB}
\ket{0}_B\quad&\rightarrow&\quad\frac13(-\ket{0}\otimes\ket{0} + 2\ket{0}\otimes\ket{1}\nonumber\\
&&+ 2\ket{1}\otimes\ket{1}),\nonumber\\
\ket{1}_B\quad&\rightarrow&\quad\frac13(2\ket{0}\otimes\ket{0} - \ket{0}\otimes\ket{1}\nonumber\\
&&+ 2\ket{1}\otimes\ket{1}),\nonumber\\
\ket{2}_B\quad&\rightarrow&\quad\ket{1}\otimes\ket{0},\nonumber\\
\ket{3}_B\quad&\rightarrow&\quad\frac13(2\ket{0}\otimes\ket{0} + 2\ket{0}\otimes\ket{1}\nonumber\\
&&- \ket{1}\otimes\ket{1}).
\end{eqnarray}

Now, to obtain a four-qubit GES, one should apply substitutions~(\ref{subA}) and (\ref{subB}) in eqs.~(\ref{sfs}) for the basis states of the first and the second subsystems, respectively, and set all $\lambda_i$ to $1/2$.

Actually, we've done calculations for several different sets of values of $\lambda_i$ and each time obtained a GES.
We can conjecture that eqs.~(\ref{sfs}) together with eqs.~(\ref{subA}) and (\ref{subB}) determine a GES for \emph{any} set of values of $\lambda_i$.

\subsubsection{\label{subsub:333}GES in three-qutrit systems}

Now we construct an example of GES of maximal dimension in a $3\otimes 3\otimes 3$ Hilbert space. The bound on the  dimension in this case is the same as the bound on the dimension  of a $9\otimes 3$ CES: $9\times3 - (9 + 3) + 1 = 16$. Consequently, we need to construct an isometry $V\colon\,H_E\rightarrow H_A\otimes H_B$, with $\mathrm{dim}(H_E) = 16$, $\mathrm{dim}(H_A) = 9$, $\mathrm{dim}(H_B) = 3$, and with a completely entangled range.

Our first step is to construct the Kraus operators $K_i\colon\,H_E\rightarrow H_A,\,i=1,\,2,\,3$, which in this case are represented by  matrices with dimensions $9\times16$. As in the previous examples, we first  consider matrices corresponding to $K_i^{\dagger}K_i$, which should satisfy eqs.~(\ref{TrP}) and~(\ref{Kc1}). Any attempt to make them all be purely diagonal immediately fails:  we can write out the diagonals of $K_1^{\dagger}K_1$, $K_2^{\dagger}K_2$, $K_3^{\dagger}K_3$ as columns of a matrix and, to satisfy eqs.~(\ref{TrP}) and~(\ref{Kc1}), in each row of this matrix there must be at least two nonzero values adding up to $1$. Therefore, the total number of these values in all rows must be  not less than $16\times 2=32$. On the other hand, being the eigenvalues of $K_i^{\dagger}K_i$, these numbers are the squares of singular values of $K_i$. There can be at most $9$ nonzero singular values for each $K_i$ because it is represented by a $9\times 16$ matrix. In total, for all three $K_i$ there can be at most $3\times 9 = 27$ nonzero singular values, so we have a contradiction which shows that $K_i^{\dagger}K_i$ cannot be purely diagonal.

To come around the described problem, we introduce a block-diagonal form for $K_i^{\dagger}K_i$. For each matrix representing $K_i^{\dagger}K_i$ we consider the area consisting of three diagonals: the main diagonal and the two adjacent ones below and above. This area is divided into two parts down the diagonals: the fist part entirely consists of $2\times 2$ blocks and the second one is purely diagonal -- it has nonzero values only on the main diagonal. The $2\times 2$ blocks placed on the same positions in $K_1^{\dagger}K_1$, $K_2^{\dagger}K_2$, $K_3^{\dagger}K_3$ should add up to a $2\times 2$ identity matrix to satisfy eq.~(\ref{TrP}). To satisfy also eq.~(\ref{Kc1}), they should have eigenvalues strictly less than $1$. As we've shown above, for each $K_i^{\dagger}K_i$ we only have $9$ eigenvalues at our disposal to cover the diagonals, so we need to spare them. This can be achieved by choosing all $2\times 2$ blocks to have only $1$ nonzero eigenvalue. In this case each such block will cover two positions on the diagonal, but it will take only one eigenvalue. Having placed $5$ such blocks down the diagonals in each $K_i^{\dagger}K_i$, we will have $9 - 5 = 4$ eigenvalues left at our disposal. We can then put these eigenvalues down the main diagonal, thus covering the remaining positions. If we put the described three-diagonal areas of $K_1^{\dagger}K_1$, $K_2^{\dagger}K_2$, $K_3^{\dagger}K_3$ in front of each other in the form of columns, the picture will look like this:
\begin{equation}\label{Kstruct}
    \begin{matrix}
    K_1^{\dagger}K_1\colon & & & &   K_2^{\dagger}K_2\colon  & & & & K_3^{\dagger}K_3\colon\\
    \\
    P_1^{(1)}& & & & P_1^{(2)} & & & & P_1^{(3)}\\
    P_2^{(1)}& & & & P_2^{(2)} & & & & P_2^{(3)}\\
    P_3^{(1)}& & & & P_3^{(2)} & & & & P_3^{(3)}\\
    P_4^{(1)}& & & & P_4^{(2)} & & & & P_4^{(3)}\\
    P_5^{(1)}& & & & P_5^{(2)} & & & & P_5^{(3)}\\
    \lambda_1^{(1)} & & & & 0 & & & & \lambda_1^{(3)}\\
    0 & & & & \lambda_1^{(2)} & & & & \lambda_2^{(3)}\\
    \lambda_2^{(1)} & & & & \lambda_2^{(2)} & & & & 0\\
    \lambda_3^{(1)} & & & & 0 & & & & \lambda_3^{(3)}\\
     0 & & & & \lambda_3^{(2)} & & & & \lambda_4^{(3)}\\
     \lambda_4^{(1)} & & & & \lambda_4^{(2)} & & & & 0

    \end{matrix},
\end{equation}
where lambda's in each row  are positive and add up to $1$, and $P_i^{(j)}$ -- $2\times 2$ matrices having only $1$ nonzero~(positive) eigenvalue and adding up to a $2\times2$ identity matrix:
\begin{equation}\label{povm}
    P_i^{(1)} + P_i^{(2)} + P_i^{(3)} = I,\quad i=1,\dots,5.
\end{equation}

Any other arrangement of lambda's is acceptable as long as there are at least two of them adding up to $1$ in each row. Here we've placed them in some checkerboard style.

The number of $2\times 2$ blocks used in each $K_i^{\dagger}K_i$ is optimal: if we had used $4$ blocks instead, then we would have had $9-4=5$ free lambda's to cover $16 - 4\times2 = 8$ positions on the diagonal, which is impossible even with the use of such a sparse checkerboard disposition.

The operators satisfying eq.~(\ref{povm}) for each fixed $i$ constitute a POVM in a two-dimensional Hilbert space. One particular choice of $P^{(k)}$~\cite{Preskill}  can be defined by
\begin{eqnarray}\label{PrP}
P^{(1)} = A\dyad{u},\: P^{(2)}=A\dyad{v}&&\nonumber\\
P^{(3)} = I - A(\dyad{u} + \dyad{v}),&&
\end{eqnarray}
where $A$ -- a constant and $\ket{u}$, $\ket{v}$ -- vectors parameterized by
\begin{eqnarray}
    &\ket{u}=\left(\begin{array}{c}
         \cos{\alpha}  \\
         \sin{\alpha}
    \end{array}
    \right),\quad\ket{v}=\left(\begin{array}{c}
         \sin{\alpha}  \\
         \cos{\alpha}
    \end{array}
    \right)&,\nonumber\\
    &0 < \alpha < \pi/4.&
\end{eqnarray}

It can be easily verified that when
\begin{equation}\label{Ac}
    A = \frac1{1+\sin{2\alpha}},
\end{equation}
the operator $P^{(3)}$ has only one nonzero eigenvalue, and thus the identity operator $I$ is decomposed into the sum of three rank one operators $P^{(k)}$.

By construction~(\ref{Kstruct}), there are five triples of operators $\left(P_i^{(1)},\,P_i^{(2)},\,P_i^{(3)}\right)$, $i = 1,\ldots,5$, and each is defined by its own parameter $\alpha_i$ according to eqs.~(\ref{PrP})~-~(\ref{Ac}). From now on we will denote such triples as $\left(P_{\alpha_i}^{(1)},\,P_{\alpha_i}^{(2)},\,P_{\alpha_i}^{(3)}\right)$, $i = 1,\ldots,5$, thus stressing out their dependence on $\alpha_i$.

Having found proper $K_i^{\dagger}K_i$ with the use of eqs.~(\ref{Kstruct})~-~(\ref{Ac}), we can construct operators $K_i$ themselves taking advantage of the singular value decomposition:
\begin{equation}\label{svd333}
    K_i = W_i\,\Sigma_i\,V_i,\quad i=1,\,2,\,3,
\end{equation}
where $W_i$ is a $9\times9$ unitary; $\Sigma_i$ is a $9\times16$ matrix consisting of $\sqrt{\lambda^{(i)}_j}$ and square roots of the eigenvalues of the operators $P^{(i)}_j$; $V_i$~is~a $16\times 16$ unitary which depends on $\alpha_j$, $j=1,\ldots,5$~(see Appendix~\ref{app:333} for details).

There is  freedom in choosing the unitaries $W_i$, which we set to  some permutation matrices.

When we set particular values for $\lambda^{(i)}_j$ and $\alpha_j$, the procedure of construction of a $9\otimes3$ CES reduces to the search of appropriate permutations $W_i$. According to the general approach, we choose an arbitrary vector state $\ket{\phi}\in H_E$ with components $\phi_j$, $j=1,\ldots,16$, and write the corresponding vectors $K_i\ket{\phi}$, $i=1,\,2,\,3$,
as columns of a matrix. If there is only trivial solution $\{\phi_j\}$, for which all $2\times2$ minors of the matrix evaluate to zero, then at least two of these vectors are linearly independent, and we have a CES. We use a computer algebra system and the Groebner basis algorithm for such an analysis. When Groebner basis reduces to $\{1\}$, there is only trivial solution. If it is not the case, we look at the basis, identify  the minors which  cause  problem and  adjust the permutations $W_i$ accordingly~(usually it takes one or two elementary transpositions). After that we compute the Groebner basis again and, if necessary, repeat the procedure until we obtain a CES.

We were not able to calculate in abstract variables here due to high dimensionality of the problem and the complexity of the polynomial system for minors, so we  made a particular choice of $\lambda^{(i)}_j$ and $\alpha_j$ for scheme~(\ref{Kstruct})~(see Appendix~\ref{app:333}, eq.~(\ref{InSet})). For such initial setting we came up with the proper permutations $W_i$~(see Appendix~\ref{app:333}, eq.~(\ref{WP})) in several iterations of the described above procedure. Calculating $K_i$ from eq.~(\ref{svd333}) and constructing the corresponding isometry with the use of eq.~(\ref{IsoK}), we obtained an orthonormal system of vectors spanning a $9\otimes3$ CES~(see Appendix~\ref{app:333}, eq.~(\ref{Sp333})). Next, we factorized the $9$-dimensional subsystem into two $3$-dimensional ones according to some particular scheme~(see Appendix~\ref{app:333}, eq.~(\ref{sch3})). Finally, after the analysis of the two new bipartitions, we introduced a simple unitary transformation acting on the two subsystems and creating entanglement~(see Appendix~\ref{app:333}, eq.~(\ref{U3})). In this way we obtained a $3\otimes3\otimes3$ GES.

\section{\label{sec::EM}Entanglement of GES and the associated bounds on entanglement measures}

\subsection{Controlling genuine entanglement of constructed subspaces}

Let us return to the first approach to constructing GES described in Subsection~\ref{sub::constch}. Such subspaces are obtained from CES of a bipartite Hilbert space by the action of some isometry on one party. The entanglement of GES can be estimated if the characteristics of both the initial CES and the channel corresponding to the isometry are known. Across one bipartition it is determined by the entanglement of the initial CES~(see eq.~(\ref{CESbi}) for bipartition $A|CD$), across the other two -- by the output norm of the quantum channel~(see eq.~(\ref{chbi}) for bipartition $C|AD$). The maximal output norm of the channel will be an upper bound on the norms of the reduced density operators of the states in GES across these two bipartitions because the supremum in eq.~(\ref{OutNorm}) is achieved on pure states.

As an illustration, we take the initial CES to be the antisymmetric subspace $W_-$ of a $3\otimes3$ Hilbert space, spanned by the three vectors
\begin{eqnarray}\label{asym}
    &\ket{\psi}_{ij} =  (\ket{i}\otimes\ket{j} - \ket{j}\otimes\ket{i})/\sqrt2&,\nonumber\\
    &i,\,j = 0,\ldots,2,\quad i<j.&
\end{eqnarray}
It is known~\cite{Vid02},\cite{KVAnt20} that the maximal first Schmidt coefficient  over all states in this subspace is given by
\begin{equation}\label{supsch}
    \sup_{\psi_{AB}\in W_-}\norm{\mathrm{Tr}_B\dyad{\psi_{AB}}}^{1/2} = 1/\sqrt2.
\end{equation}

Consider the Holevo-Werner channel~\cite{HolWern02} acting on $d\times d$ density matrices as
\begin{eqnarray}\label{HW}
\mathrm{\Phi}(\rho) = \frac1{d-1}\left(I - \rho^T\right)\nonumber\\
= \sum_{i<j}K_{ij}\rho K_{ij}^{\dagger},
\end{eqnarray}
where
\begin{eqnarray}\label{KrHW}
    K_{ij} =\frac1{\sqrt{d-1}}\left(|i\rangle\langle j| - |j\rangle\langle i|\right),\nonumber\\
    i,\,j = 0,\ldots,d-1,\quad i<j,
\end{eqnarray}
and $\rho^T$ denotes the matrix transpose with respect to the computational basis.

The maximal output norm of the Holevo-Werner channel can be easily obtained from the first representation in eq.~(\ref{HW}) and is given by
\begin{equation}\label{HWn}
    \nu_p(\mathrm\Phi) = (d-1)^{-(1-1/p)}
\end{equation}

In our case, $d=3$, the channel can be represented by the $3$ Kraus operators given by eq.~(\ref{KrHW}). The associated isometry $V$, obtained from  eq.~(\ref{IsoK}), acts on the computational basis as
\begin{eqnarray}
V\ket{0}&=& \frac1{\sqrt2}\left[-\ket1\otimes\ket0 - \ket2\otimes\ket1\right],\nonumber\\
V\ket{1}&=& \frac1{\sqrt2}\left[\ket0\otimes\ket0 - \ket2\otimes\ket2\right],\nonumber\\
V\ket{2}&=& \frac1{\sqrt2}\left[\ket0\otimes\ket1 + \ket1\otimes\ket2\right].
\end{eqnarray}
Acting with this isometry on the second qutrit of the vectors in eq.~(\ref{asym}), we obtain an orthonormal system spanning a GES of a $3\otimes3\otimes3$ Hilbert space:
\begin{subequations}\label{HWges}
\begin{eqnarray}
\ket{\phi_1} = \frac12(\ket{0}\otimes\ket{0}\otimes\ket{0} - \ket{0}\otimes\ket{2}\otimes\ket{2}\nonumber\\
+ \ket{1}\otimes\ket{1}\otimes\ket{0} + \ket{1}\otimes\ket{2}\otimes\ket{1}),
\end{eqnarray}
\begin{eqnarray}
\ket{\phi_2} = \frac12(\ket{0}\otimes\ket{0}\otimes\ket{1} + \ket{0}\otimes\ket{1}\otimes\ket{2}\nonumber\\
+ \ket{2}\otimes\ket{1}\otimes\ket{0} + \ket{2}\otimes\ket{2}\otimes\ket{1}),
\end{eqnarray}
\begin{eqnarray}
\ket{\phi_3} = \frac12(\ket{1}\otimes\ket{0}\otimes\ket{1} + \ket{1}\otimes\ket{1}\otimes\ket{2}\nonumber\\
- \ket{2}\otimes\ket{0}\otimes\ket{0} + \ket{2}\otimes\ket{2}\otimes\ket{2}).
\end{eqnarray}
\end{subequations}

Let $S$ denote this subspace. Consider the geometric measure $G_{GME}$ of genuine entanglement of $S$. The three qutrits are referred to in the same way as on Figure~\ref{fig:iso23}: $A$, $C$ and $D$. The maximal first Schmidt coefficient over all states in GES across bipartition $A|CD$ is determined by that of the initial antisymmetric bipartite subspace  given by eq.~(\ref{supsch}). The maximal first Schmidt coefficient across bipartitions $C|AD$ and $D|AC$ will be upper bounded by the output norm $\nu_{\infty}(\mathrm\Phi)^{1/2}$ of the Holevo-Werner channel, which, by eq.~(\ref{HWn}), also equals to $1/\sqrt2$. Combining the results across all bipartitions, with the use of eqs.~(\ref{geom}), (\ref{geomGME}) we obtain the lower bound on the geometric measure of $S$:
\begin{equation}\label{Gb}
    G_{GME}(S)\geqslant 1/2.
\end{equation}

\subsection{Lower bounds on GME concurrence and negativity of mixed states}

Obtained in Ref.~\cite{KVAnt20}  lower bounds on the concurrence and the convex-roof extended negativity of an arbitrary bipartite mixed state $\rho$ are determined by its overlap with some completely entangled subspace $W$ of a $d\times d$ Hilbert space. For the concurrence the bound reads
\begin{equation}
    C(\rho)\,\geqslant\,\max\left(\sqrt{\frac{2}{d(d-1)}}\,\frac{\Tr{\rho\,\Pi_W} - \bar\lambda^{\scriptscriptstyle W}_1}{\bar\lambda^{\scriptscriptstyle W}_1},\,0\right),
\end{equation}
where $\Pi_W$ -- an orthogonal projector onto $W$, $\bar\lambda^{\scriptscriptstyle W}_1$ -- the supremum of the largest Schmidt coefficient squared taken over all vector states in the subspace $W$.
The bound on the negativity is given by
\begin{equation}
    N^{\mathrm{CREN}}(\rho)\,\geqslant\,\max\left(\frac{\Tr{\rho\,\Pi_W} - \bar\lambda^{\scriptscriptstyle W}_1}{2\bar\lambda^{\scriptscriptstyle W}_1},\,0\right).
\end{equation}

The underlying separability criterion was derived earlier in Ref.~\cite{Sar08} with the use of the theory of entanglement witnesses.~(see also Ref.~\cite{DemAugWit18} for the generalization to the GME case).

With the use of eqs.~(\ref{geom})-(\ref{cfGME}) these inequalities can be straightforwardly extended to the GME case:
\begin{eqnarray}
    C_{GME}(\rho)\,&&\geqslant\,\sqrt{\frac{2}{d(d-1)}}\frac1{1 - G_{GME}(W)}\nonumber\\
    \times&&\max(\Tr{\rho\,\Pi_W} + G_{GME}(W) - 1,\,0),\label{Cgme}\\
    N_{GME}(\rho)\,&&\geqslant\,\frac1{2(1 - G_{GME}(W))}\nonumber\\
    \times&&\max(\Tr{\rho\,\Pi_W} + G_{GME}(W) - 1,\,0)\label{Ngme},
\end{eqnarray}
where now $W$ is a genuinely entangled subspace. Therefore, the two GME measures can be estimated for a mixed state $\rho$ if the value $G_{GME}(W)$ or some good lower bound on it can be determined.

As an example, with the GES $S$ from eq.~(\ref{HWges}), for a  mixed state $\rho$  the lower bound on the negativity is given by
\begin{equation}
    N_{GME}(\rho)\,\geqslant\,\max(\Tr{\rho\,\Pi_S} - 1/2,\,0),
\end{equation}
where eqs.~(\ref{Gb}) and (\ref{Ngme}) were used.

Eqs.~(\ref{Cgme}) and (\ref{Ngme})~(or the underlying separability criterion) are also convenient for estimating the robustness of genuine multipartite entanglement under  mixing with noise. Consider a state $\rho$ with support in some GES $W$ being mixed with noise described by a density operator $N$. After mixing the state is given by the convex sum
\begin{equation}\label{NoisyS}
    (1-p)\rho + p N,\quad p\in[0;1].
\end{equation}
By eqs.~(\ref{Cgme}) and (\ref{Ngme}), this state will still be genuinely entangled if the lower bound on the entanglement measures is positive, i.~e.,
\begin{equation}
    \mathrm{Tr}\{(1-p)\rho\Pi_W + p N\Pi_W\} + G_{GME}(W) > 1.
\end{equation}
Taking into account that $\mathrm{Tr}\{\rho\Pi_W\}=1$~($\rho$ has support in $W$), we obtain the upper bound
\begin{equation}
    p < \frac{G_{GME}(W)}{1 - \mathrm{Tr}\{N\Pi_W\}}
\end{equation}
saying that under such proportions of the noise $N$ the state in eq.~(\ref{NoisyS}) is guaranteed to be genuinely entangled.

Let $H$ denote the Hilbert space of the described states. If only the white noise is considered,
\begin{equation}
N = \frac1{\mathrm{dim}(H)}I,
\end{equation}
then the bound on $p$ reads
\begin{equation}\label{wnrob}
    p < \frac{G_{GME}(W)}{1 - \mathrm{dim}(W)/\mathrm{dim}(H)}.
\end{equation}
As an example, for the subspace $S$ given by eq.~(\ref{HWges}) the robustness of entanglement under the white noise  can be estimated with the use of eq.~(\ref{Gb}). It is given by $p < 9/16=0.5625$.

Following Ref~\cite{KVAnt20}, it is also interesting to analyze the case when the expression for $N$ is unknown  and only the spectrum of the noise  is given~(\emph{robustness from spectrum}). The immediate generalization of eq.~(55) in Ref.~\cite{KVAnt20} to the GME case is given by
\begin{equation}\label{SpN}
    p < G_{GME}(W)/\norm{N}_{\scriptstyle{(\mathrm{codim}(W))}},
\end{equation}
where $\mathrm{codim}(W) = \mathrm{dim}(H)-\mathrm{dim}(W)$ and $\norm{N}_{(k)}$ is the sum of $k$ largest eigenvalues of $N$~(\emph{the $k$-th Ky Fan norm}~\cite{Bhatia97, HornJohn13}).

Eq.~(\ref{SpN}) may be useful in analysis of quantum states under the influence of random noise where some results on eigenvalues of random matrices can be applied.

\section{\label{sec::conc}Discussions}

Several methods of constructing GESs have been presented. In the first one tripartite GESs are obtained from bipartite CESs by applying an isometry corresponding to a quantum channel with the maximal output norm strictly less than $1$. On the one hand, the dimensions of the obtained subspaces are not high and bounded by those of the initial CESs. On the other hand, the method is pretty simple and allows us to control the  measures of entanglement of the constructed subspaces on condition that we have control over the CESs and the output characteristics of the channel being used. Highly entangled GESs can be generated in this manner. Quite possibly, other characteristics, such as distillability of entanglement across every bipartition, can be controlled in a similar way. In principle, the whole procedure can be applied to obtain $n+1$-partite GESs from $n$-partite ones. It is also interesting to ask whether this method can be realized in experimental setting: the isometry, for example, can be extended in some way to a unitary which may correspond to some physical operation. These questions will be addressed in the forthcoming work.

The second method is based on the direct construction of certain quantum channels with the subsequent factorization of some compound systems into smaller ones. The method can generate GESs of maximal possible dimension but it gets computationally hard with the increase of the number of parties and the dimensionalities of their systems. We were able to come up with families of maximal GESs for three-qubit systems and, to some extent, for four-qubit and three-qutrit systems where we can only conjecture that we are dealing with  GESs in a wide range of the values of the parameters~(we checked that they are GESs for various sets of the parameters).

One may benefit from these parameterizations in the analysis of various properties of entangled states and subspaces. As an example, starting from a maximal GES for a system of $3$ qubits given by eqs.~(\ref{ges22}), one can easily find another maximal GES orthogonal to the first one:
\begin{subequations}
\begin{eqnarray}
\ket{\psi_1} &=& \sqrt{1 - \lambda_1}\ket{0}\otimes\ket{0}\otimes\ket{0}\nonumber\\
& &- \sqrt{\lambda_1}\ket{1}\otimes\ket{1}\otimes\ket{1},\nonumber\\
\ket{\psi_2} &=& \sqrt{\frac{1-\lambda_2}{2}}\left(\ket{0}\otimes\ket{1} + \ket{1}\otimes\ket{0}\right)\otimes\ket{0}\nonumber\\
& &- \sqrt{\lambda_2}\ket{0}\otimes\ket{0}\otimes\ket{1},\nonumber\\
\ket{\psi_3} &=& \sqrt{\frac{1-\lambda_3}{2}}\left(\ket{0}\otimes\ket{1} - \ket{1}\otimes\ket{0}\right)\otimes\ket{0}\nonumber\\
&-& \sqrt{\frac{\lambda_3}{2}}\left(\ket{0}\otimes\ket{1} + \ket{1}\otimes\ket{0}\right)\otimes\ket{1}.\nonumber
\end{eqnarray}
\end{subequations}
These two GESs span in total $6$ dimensions out of 8 in a tripartite Hilbert space~(but note that their direct sum can't be genuinely entangled). Their orthogonal complement, a $2$-dimensional subspace, contains separable vectors, for example, $\ket{1}\otimes\ket{1}\otimes\ket{0}$.  Given an arbitrary mixed state of $3$ qubits, one can optimize over the parameters $\lambda_i$ and find the GES of one of the two types having maximal overlap with the state. If the state is genuinely entangled, it can be expected that the overlap will be significant. With the geometric measure of the optimal GES obtained, for example, numerically, one could then use eqs.~(\ref{Cgme}), (\ref{Ngme}) to detect genuine entanglement of the state or to get estimates for its entanglement measures. The same can be said in relation to the four-qubit subspaces determined by eqs.~(\ref{sfs}), (\ref{subA}), (\ref{subB}), in which case the two orthogonal GESs will span together $14$ dimensions out of $16$ possible. The numerical analysis of the effectiveness of such an approach  will be conducted elsewhere.

It seems that there is some trade-off between the geometric measure of entanglement of a GES and its dimension. It would be interesting to find examples of GESs optimal with respect to the robustness of entanglement of their states under mixing with the white noise, an estimate for which is given in eq.~(\ref{wnrob}).

\begin{acknowledgments}
The author thanks Otfried G{\"u}hne for many fruitful discussions and valuable comments.
This work was supported by Lomonosov Moscow State University.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


We at Union Dental Group invite you to come visit our location and meet our team. We always enjoy seeing new smiles around here! To give you a preview of what our office is like, we've collected feedback from our current patients and written it down here for you to read.


At pureLiFi, we are continually innovating and progressing our technology. The below products are early examples of our technology.
LiFi is developing quickly in our labs, and we're happy to discuss this with interested firms.
The Li-1st marked the introduction of a groundbreaking wireless communication technology. This system became the world's first LiFi technology available on the market.


Tax Payment Plans Category Archives — New York Tax Attorney Blog Published by New York Tax Lawyers — Timothy S. Hart Law Group, P.C.
When a taxpayer can not pay the IRS a lump sum payoff for their tax debt, they need to create a payment plan with the IRS. There are a number of options to consider when the payment plan is established. Obviously, it is almost always best to create the lowest monthly payment plan amount as possible and then pay more to get the balance down quickly, but not be at risk of defaulting the payment plan if money is tight. Therefore, there a number of factors to consider to get the correct balance of paying down the debt quickly, and not having a future issue. As with other payment plans that you might establish with a creditor, the amount you will pay will depend on the amount you owe, the amount you can afford to pay, and what assets and income you have.
As an overview, the IRS has many programs that allow for payments of taxes over time. The payment plans are within the discretion of the IRS, which means they decide the amount per month they will accept. Often it takes a lot of hard work to get the IRS to agree on an affordable number. The IRS does not give much flexibility with the monthly payment being late once the payment plan is established. In some rare instances, the payments can fluctuate if the person or business has seasonal income, or income on a non-recurring basis, such a lawyer who gets paid on contingency basis.
There are three common ways that you can make payments under the payment plan.


Taiwan and mainland China are currently experiencing a period of official conciliation. Economic relations continue to deepen: the 2013 Cross-Straits Agreement on Trade in Services (Liang'an fuwu maoyi xieyi 兩岸服務貿易協議), although controversial in Taiwan, is opening up new areas of exchange that include finance, telecommunications and entertainment. Ever-greater numbers of mainland tourists visit Taiwan, and senior officials from Taiwan's ruling Nationalist Party are going to Beijing. The result is a change in the tone of the debate on each side about the relationship.
The histories of Taiwan and mainland China diverged at the end of the nineteenth century, when Taiwan became a colony of Japan. In 1945, it joined the Republic of China, and in 1949 the Nationalist government of the Republic, fleeing from the Communists, relocated to Taipei. The end of the Qing dynasty, the chaos of the Republican era and then the founding of the People's Republic of China in 1949, meant the Mainland followed a different path of modernisation, both economically and politically.
Taiwan, having missed the excesses and disasters of Maoism, has offered a model for modernisation as an 'Asian Tiger', reaching an apotheosis with democratisation in the late 1980s. But as China has undergone its own period of hyper-growth, becoming the world's second largest economy, Taiwan has been reframed as an eastern island hinterland of a new superpower.
Often, over the last century and more, writers and commentators on each side have seen in the other an alternative set of historical possibilities. Tackling themes of progress, tradition, modernisation and civilisation, they have addressed social and political problems in their own experience through the example of the alternative other.
In May 2012, the noted mainland Chinese writer, blogger, Internet celebrity and occasional racing car driver Han Han wrote about a visit to Taiwan that addressed the topic of civility in Taiwan society as a way of commenting on contemporary mainland life. His visit had been high profile and he was part of a group that included technology business figures who met with Taiwanese President Ma Ying-jeou in the Presidential Palace.
In his essay titled 'Winds of the Pacific' (Taipingyangde feng 太平洋的风) Han Han (notably avoiding the official title 'President') says that it was not his meeting with 'Mr' Ma that left the strongest impression. He begins sentimentally with a description of his flight touching down in Taipei, when he hears a ringtone of Sylvia Chang singing 'Playing in the Snow' (Xixue 戲雪) — a song that evokes the exodus of Chinese Nationalist refugees who fled Communism by relocating to Taiwan in 1948–1949. His speaks about the Taiwan of his imagination — a place that is populated by well-known filmmakers like Hou Hsiao-hsien and the late Edward Yang, and the Republican-era intellectuals who followed the Nationalists to the island such as Liang Shih-chiu, Lin Yu-tang and Hu Shih.
Yes, I want to thank Hong Kong and Taiwan. They have sheltered China's civilisation, preserving the beauty of the nation and saving its roots from calamity. They do have things that can be criticised. We have a Ritz-Carlton and a Peninsula Hotel, and Gucci and LV, and the wife of a county head in China is richer than their highest officials. For one of our major film productions they could make twenty or thirty of theirs and they could never afford our Expo or Olympic Games. But when walking on the streets of Taiwan, talking to those taxi drivers, food stallholders and people on the street, I feel no sense of pride at all.
Through Taiwan as a counter-example, Han Han captures key anxieties of mainland Chinese intellectuals of a cultural and social life upturned first by Maoism and then by marketisation. Taiwan offers a point of reference from which Han Han can excoriate the failures of a whole generation in China and the breakdown of social relations under China's Communist experiment. But at the same time, Han Han recognises the wealth and power that China has achieved in his lifetime, and speaks from a position within it. Taiwan is a backwater when measured against the scale of consumerism and global mega-events possible in today's China.
Lung Ying-tai, Taiwan's first Minister for Culture.
Han Han's piece attracted widespread Internet and media comment, including an acknowledgement in a speech by President Ma himself. Not long after Han Han visited Taiwan, the essayist and cultural critic Lung Ying-tai became Taiwan's first Minister for Culture, heading up a new ministry created out of the old Council for Cultural Affairs and the Government Information Office. On a visit to Zhongshan University in Guangzhou to give a public lecture, Lung was asked about Han Han's essay. In her response, she celebrated Han Han has having the kind of voice a healthy society needs. She also said that Taiwan demonstrated that Chinese culture at its heart is one of compassion. Lung said that for people like Han Han, the last sixty years on the Mainland has been an experience of political struggle that has left a spiritual wound, and that the past two decades of economic hyper-development has left Mainlanders hard-pressed to find a sense of personal worth. She said that Han Han was expressing a longing that the compassion within Chinese culture could once again be a source for his spiritual life.
In her comments in Guangzhou, Lung also compared Han Han's essay with her own public critiques of Chinese culture written over many decades in the context of authoritarianism in Taiwan and later of that in mainland China. She said writing her celebrated 1985 volume of essays, Wild Fire (Yehuoji 野火集) had caused a furious controversy in Taiwan because it exposed many social taboos of the period. 'The skin of society was fragile, and just one smack inflamed it and made it bleed'.
In Wild Fire, Lung Ying-tai had excoriated the Chinese (by which she meant the people of Taiwan) for their apparent apathy in the face of political oppression, corruption and environmental degradation after nearly forty years of martial law. She appealed to a similar sense of civility as Han Han but also to the civic action required to maintain it: 'You and I are the same ordinary people, with the same needs: a peaceful environment, clean neighbourhoods, orderly streets, civilised and considerate communities'. At the time, expressing such views was risky and controversial in Taiwan and the essays were read with excitement by a receptive public. They also predate the legitimisation of Taiwanese identity politics, in which speaking about the Taiwanese as Chinese has become controversial in its own right.
Both Han Han and Lung Ying-tai speak about the issue of civility being at the heart of the crisis of Chinese society. Both explore the meaning of the term civility as a way of speaking about personal dignity, social relations and the everyday, and, in a broader sense, as an assessment of the state of the polity and its capacity for self-reflection.
Not everyone was so impressed with Han Han, however. Young bloggers in Taiwan picked up a patronising subtext in his equation of civility and weakness: by contrasting the civility of the everyday in Taiwan to China's contemporary wealth and power, he had made Taiwan out to be a quaint and rustic backwater.
A typical response came from the Taiwanese blogger Lin Shu Shu. Lin wrote a post entitled 'A cultivated citizen or traditional Chinese virtues: Han Han misunderstands Taiwan' in which he credited colonisation by Japan for shaping modern Taiwan, both directly through its modernising institutions and indirectly from the organised opposition to it from the Taiwanese. Lin identifies political activists like Jiang Wei-shui — founder of the Taiwan Cultural Association (Taiwan wenhua xiehui 臺灣文化協會) in 1921 and the short-lived Taiwan People's Party (Taiwan Minzhongdang 臺灣民眾黨) in 1927 — as introducing notions such as the rule of law and democratic political representation that led to democratisation in the 1980s.
Lin asserts that virtue is not unique to Chinese culture or civilisation. Chinese culture and civility flourish on Taiwan because of the rule of law. Lin argues that: 'only after China completely eliminates despotism will its full splendour be realised'. In other words, Taiwan is more civilised because it is modern, not because it has maintained traditional Chinese culture — that alone does not guarantee either virtue or civility.


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text:
Indelible: A Sean McPherson Novel, Book 1
by Laurie Buchanan Laurie Buchanan
Paperback-$16.95 NOOK Book-$9.49 View All Available Formats & Editions
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When a sniper killed his partner, Sean McPherson was injured in the ambush. Now an ex-cop, he takes a job at a writing retreat in the Pacific Northwest. At Pines&Quill, he hopes to heal and put his life back together in the Zen-like capacity of groundskeeper and all-around handyman.
Sniper, Jason Hughes, blames McPherson for the loss of more than ten million dollars' worth of heroin—and he wants revenge.
In the guise of a New York City limo driver working on a sizzling tell-all memoir, Hughes arrives at Pines&Quill along with three other writers in residence: a bohemian psychic taking a break from grueling work as a forensic intuitive, a bitter divorcée who wants to open herself to a new life, and a vibrant and resilient potter navigating life from a wheelchair.
With conflicting agendas, uncertain loyalties, and romantic entanglements at play, Hughes finds it difficult to get McPherson in his sights. Gradually, he forms a different plan, one that threatens the lives of everyone at the retreat.
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A cross between Dr. Doolittle, Nanny McPhee, and a type-A Buddhist, Laurie Buchanan is an active listener, observer of details, payer of attention, reader and writer of books, kindness enthusiast, and an unabashed optimist. A former holistic health practitioner and transformational life coach, she holds a doctorate in holistic health with an emphasis in energy medicine. Her first two books, Note to Self: A Seven-Step Path to Gratitude and Growth, and The Business of Being: Soul Purpose In and Out of the Workplace, are nonfiction titles designed to motivate, inspire, and transform. Laurie lives in the Pacific Northwest with her pilot-husband, Len, and their Irish Wolfhound, Willa. She enjoys yoga, long walks, bicycling, photography, and travel. To learn more, please visit her website at www.lauriebuchanan.com. She resides in Boise, Idaho.
2021 15th Annual National Indie Excellence® Awards: Winner, Crime Fiction/Suspense
2021 Eric Hoffer Awards: Finalist
2021 PenCraft Award for Literary Excellence, Suspense
2021 Firebird Book Award, Suspense, Thriller
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"Indelible is for the reader who loves a fast-paced page-turner; for the reader who likes some sweet romance mixed in with riveting suspense; and for the reader who enjoys getting deeply involved with each character in a well-written book. Indelible is for the writer who fantasizes about an enchanting writing retreat like Indelible's Pines & Quill."
—Pamela S. Wright, author of The Right Wrong Man and Twin Desires
"Indelible is a delightful read! The perfect book for an escape on a rainy afternoon. The literary hooks Laurie weaves into the end of each chapter, make Indelible a novel hard to put down. Laurie's vivid character descriptions remind me of the Nancy Drew books I loved to read so many years ago. Indelible makes any reader want to stay at Pines & Quill! I'm looking forward to reading the next Sean McPherson novel!"
—Ann Margaret Johns, author of Clarkston's Curse: One Child's Quest to Explain the Series of Tragedies in Her Hometown
"Secrets lie deep behind the pages of this gripping story. Indelible offers readers an expertly drawn cast of characters in an equally dramatic setting. Watching the next move will keep you on the edge of your seat."
—Sherry Biscoe, author of Mists of Garibaldi-Tales of the Supernatural and The Man In Number 7
"Early in Indelible, I realized I was in the hands of a gifted storyteller. Through flawed characters and a vivid setting, tension mounts in this fast-paced thriller, leading to a riveting climax."
—Greg Renz, author of Beneath the Flames
"Smart and sassy, Laurie Buchanan's Indelible is a fast-paced mystery with deftly drawn characters and a palpable sense of place. I couldn't put it down!"
—Ashley E. Sweeney, author of Answer Creek
"Indelible hits a truly sweet spot in an idyllic setting at a writing retreat in the Pacific Northwest. Just when you get comfortable, it turns from cozy to thriller. This is a fantastic debut of what will surely be an addictive series."
—Shelley Blanton-Stroud, author of Copy Boy: A Novel
"Like a killer in the night, Laurie Buchanan's descriptive prose lures the unsuspecting reader into this chilling mystery. She has created a complex cast of characters in this ripping yarn, and her final reveal leaves the reader hungry for more. I look forward to the next installment!"
—Diane Demetre, author of The Diana Daniels Mysteries
"Buchanan creates characters I couldn't help but care for and a setting so beautiful I'd love to visit. The story sucked me right in, and the suspense kept me reading. I can't wait for my next chance to visit Pines & Quill!"
—Elizabeth Anne Wood, author of Bound: A Daughter, a Domme, and an End-of-Life Story
"Buchanan has given us a new hero in Sean McPherson, a man struggling with his past who is touching and sensitive, yet ready to hunt down evil to defend those who need him. Amid a cast of engaging characters, the suspense builds in a crescendo along the Pacific Northwest coast."
—Sharon Dukett, author of No Rules: A Memoir
"The pace is a roller coaster ride. The tension builds from the first page and doesn't let up until the very end."
—Chris Norbury, author of Straight River and Castle Danger
"If you're an adrenaline junkie like I am, you'll love this book!"
—Gail Cushman, author of the Breaking Barrier and Wrinkly Bits series
"Hold on to your seat: the ride is fast and furious. Be ready to stay up all night."
—Tower Lowe, author of No Way Out
"Fast-paced and with plenty of plot twists, you're sure to be swept up into the story."
—Debbie Herbert, USA Today best-selling author of southern suspense
"Unputdownable! Buchanan reels the reader in with descriptions of the setting—a writers' retreat in the breath-taking natural preserves of Washington State. Each of the writer-guests has a personal agenda and style, and the characters are relatable. The tension of the plot contrasts starkly with the peacefulness of the surroundings, so the reader is led repeatedly through crisis and calm, building up to the climax of the book. Simply captivating!"
—Saralyn Richard, author of A Palette for Love and Murder
"Author Laurie Buchanan has written entertaining and believable characters in her debut novel, Indelible. The investigator McPherson is likable and will carry the series along with a wide fan base! A fun read!"
—Jennifer Greer, author of A Desperate Place: A McKenna and Riggs Novel
"Indelible is a fantastic start to what one hopes is a long-running series. Buchanan has created a fascinating thriller set in a writers' retreat in the Pacific Northwest. Atmospheric and delicious, Buchanan treats her readers' senses while taking them on a roller-coaster of emotion. Cleverly balancing humor, romance, and a healthy dose of thrill, Buchanan delivers a perfect mystery. You won't be able to put it down!"
—Michelle Cox, author of the Henrietta and Inspector Howard series


text:
Erdoğan's conversion of Hagia Sophia into a mosque and the danger of war
By Ulaş Ateşci and Alex Lantier
Turkish President Recep Tayyip Erdoğan's decree Friday turning the Hagia Sophia in Istanbul into a working mosque is an Islamist populist gesture that will accelerate the drive to war across the Mediterranean and the Middle East.
The history of this unique, nearly 1,500-year old basilica is of cultural significance to billions of people in many countries and cultures. It is not so much a Turkish monument as a world monument that the course of history has entrusted to the Turkish people for safekeeping. Amid the growing ethnic and sectarian bloodshed caused by decades of imperialist war in the Middle East, the attempt to claim the Hagia Sophia for Islam alone cannot fail to have unforeseen consequences.
The Hagia Sophia was completed in 537 during the reign of Roman emperor Justinian I, as a Christian church in what was then Constantinople. UNESCO, which has placed it on the World Heritage List, calls it one of the "unique architectural masterpieces of Byzantine [art] … designed by Anthemios of Tralles and Isidoros of Miletus in 532-537."
This breathtaking building was a center of the Eastern Orthodox Church until the 15th century, except for a brief period in the 13th century when Crusaders sacked Constantinople and turned the Hagia Sophia into a Roman Catholic cathedral. In 1453, Ottoman Sultan Mehmet II conquered the city, turning the Hagia Sophia into a mosque. The Ottoman Empire added minarets to the Hagia Sophia, as well as antique Hellenistic art drawn from across Turkey.
The fall of the Ottoman Empire and the establishment in 1923 of an independent Turkish Republic, in a war fought with Soviet aid against invading British, French, Italian, Greek and Armenian armies, changed the Hagia Sophia's status. In 1935, President Mustafa Kemal Atatürk and his cabinet converted the Hagia Sophia into a museum. UNESCO designated it a World Heritage Site in 1985. In the period since 2007, over 30 million people have visited the Hagia Sophia.
Since the 1935 decree turned the Hagia Sophia into a museum, Islamists and certain Turkish nationalists have called for reversal of this decision and to turn it back into a mosque.
Significantly, the risk of international conflict posed by such communal and nationalist appeals was acknowledged by Erdoğan himself barely a year ago. Just before the March 2019 local elections, he dismissed the call to convert the Hagia Sophia into a mosque as a "provocation." Referring to the thousands of mosques scattered in non-Muslim-majority countries across the world, he asked: "Do they think what would happen to those mosques? … I will not be duped."
Fifteen months later, Erdoğan has resorted to this provocation, and it has not taken long for the first denunciations to begin pouring in. The Russian Orthodox Church called the Hagia Sophia's conversion into a mosque "unacceptable," while Catholic Pope Francis declared himself "very distressed." Indian media, incensed by Erdoğan's criticisms of anti-Muslim communal riots in India, are demanding that New Delhi intervene to criticize Erdoğan.
The Greek government, a nominal NATO "ally" of Turkey, which is locked in a bitter dispute with Turkey over Cyprus and oil drilling rights in the Mediterranean, also condemned the decision. Athens warned that "it does not only affect relations between Turkey and Greece, but its relations with the European Union."
Underlying the shift in the Erdoğan's policy are intractable international crises and class conflicts for which the Turkish bourgeoisie has no solutions. Three decades since the Stalinist regime's dissolution of the Soviet Union removed the main political and military obstacle to US-led wars in the Middle East, Turkey is surrounded by wars and bitter commercial and geopolitical conflicts that threaten to escalate into an all-out military conflagration.
Erdoğan's decision to join US-led wars in Libya and Syria in 2011 has ended in a bloody debacle; it now finds itself waging proxy wars in both these countries against its nominal allies. Turkey is backing the Libyan Government of National Accord (GNA) against the French- and Russian-backed militias of warlord Khalifa Haftar, and clashing with Greece in the Mediterranean. As part of its ongoing war against US-backed Kurdish nationalist groups, it has repeatedly invaded the north of Syria, whose government is supported by Russia.
At the same time, the COVID-19 pandemic is vastly intensifying international class conflict and fueling the Erdoğan government's collapse in the polls to the lowest level since 2002, to about 30 percent. According to trade union estimates, the pandemic has led to at least 11 million job losses in Turkey, raising the number of unemployed to over 17 million, a historic record. The Turkish bourgeoisie are watching with concern and dismay the growing strikes and protests by workers for safe working conditions and against austerity across America, Europe and internationally.
Especially after mass multi-racial and multi-ethnic protests internationally against the US police murder of George Floyd, they are terrified of growing mass political opposition.
Erdoğan's turn to using the Hagia Sophia to incite religious and communal sentiment aims to suppress these class contradictions, divide the working class along religious and national lines, and promote nationalism amid an escalating spiral towards war between the major powers.
Precisely because this is a class policy, it cannot be opposed by supporting Erdoğan's opponents within the Turkish bourgeois establishment. The decision to turn the Hagia Sophia back into a mosque is supported not only by Erdoğan's Islamist Justice and Development Party (AKP) and its Nationalist Movement Party (MHP) ally but also by its bourgeois opponents, including the Kemalist Republican People's Party (CHP) and its far-right ally, the Good Party—parties that imperialist governments and media present as the enlightened "alternative" to Erdoğan.
While CHP leader Kemal Kılıçdaroğlu's reaction to the Hagia Sophia decision was to state, "Open it, if you want it to open," Muharrem İnce, former presidential candidate of the CHP-led opposition against Erdoğan in the 2018 presidential election, declared his support for the decision. İnce said he would be among worshippers in the first prayer in the Hagia Sophia on July 24. It is no coincidence that these parties also fully support Erdoğan's war policies across the region in the interests of the Turkish ruling class.
The Hagia Sophia question provides its own confirmation of Leon Trotsky's Theory of Permanent Revolution. In the final analysis, democratic rights and human culture cannot be defended, and ethnic and religious divisions in countries of belated capitalist development cannot be overcome, under the leadership of the bourgeoisie. The defense of democratic rights and the overcoming of ethnic and religious conflict require a turn to the working class, mobilized on an international and socialist program against capitalism and imperialist war.
None of these struggles can be advanced without a direct struggle against imperialism, nationalism and war. Expressions of concern over the Hagia Sophia from governments and religious bodies in imperialist countries of America and Europe are false to the core. Their manifestly hypocritical criticisms only play into the hands of the Turkish bourgeoisie's promotion of nationalism—amid explosive tensions between Turkey and not only Russia, but also its ostensible NATO "allies" including Washington, Paris and Athens in Syria, Libya and the eastern Mediterranean.
While French Foreign Minister Jean-Yves Le Drian declared France "deplores" Ankara's decision, which European Union (EU) foreign policy chief Josep Borrell called "regrettable," US Secretary of State Mike Pompeo issued a statement on July 1 urging Erdoğan "to continue to maintain the Hagia Sophia as a museum" and adding that he "views a change in the status of the Hagia Sophia as diminishing the legacy of this remarkable building ..."
The imperialist ruling classes have overseen for decades the plundering of priceless cultural sites across the Middle East, and their empty invocations of concern today deserve nothing but contempt. The Iraqi National Museum was looted after the illegal US-led invasion of country in 2003. The NATO proxy war in Syria, launched by Washington and the European powers, led to the destruction of cultural sites dating back millennia, such as the old city of Palmyra.
While the Trump administration criticizes Erdoğan's transformation of the Hagia Sophia into a mosque, it supported the granting of full control over Jerusalem and the Temple Mount/Al Aqsa mosque to the Israeli state.
Imperialist governments' obscene looting of the globe with wars, bank bailouts, tax cuts and other hand-outs to the global financial aristocracy goes hand-in-hand with disregard for art and culture in their own countries. As Le Drian hypocritically expresses his concern over the Hagia Sophia, he said nothing of the fate of Notre Dame cathedral in Paris. His own government spent so little on fire security during the catherdral's renovation that a fire destroyed the roof and spire and very nearly the entire building, which is closed for years to come.
The Erdoğan government's transformation of the Hagia Sophia into a mosque is an exposure not only of the Turkish bourgeoisie. It is a warning that amid an intensifying spiral of war and a deadly global pandemic, the capitalist class around the world is engaged in a relentless promotion of nationalism and religious politics that, if it is not stopped, can only end in even greater disasters.
The social force that can and must be mobilized against this growing drive to war is the international working class. Already before the COVID-19 pandemic, mass strikes in the United States and "yellow vest" protests in France—together with mass protests from Algeria and Lebanon to those in Iraq and Iran over the US murder of Iranian General Soleimani—signaled a new upsurge of the class struggle. The critical question is to unify these growing struggles of the international working class by arming them with a socialist and anti-imperialist perspective.
This is the only way to stop the drive to war, growing incitement of religious and national divisions by the bourgeoisie, and to defend the common cultural and historical heritage of humanity.
After the Turkish elections: How the pseudo-left rallied behind the CHP
[2 October 2019]
The burning of Notre-Dame cathedral in Paris
[17 April 2019]
Reports reveal scale of destruction of Syria's world historic heritage
[29 December 2014]