Algorithmic aspects of semistability of quiver representations

Here, we present the formal definition of a quiver representation. We follow the terminologies in [DW17, BFGO+19]. Let $Q=(Q_{0},Q_{1})$ be a quiver with a vertex set $Q_{0}$ and an arc set $Q_{1}$. In this paper, we consider only acyclic quivers except in Section 6. For each arc $a\in Q_{1}$, we denote the tail and head of $a$ by $ta$ and $ha$, respectively. A representation $V$ of $Q$ consists of complex vector spaces $V(i)$ for vertex $i\in Q_{0}$ and linear maps $V(a):V(ta)\to V(ha)$ for arc $a\in Q_{1}$. A subrepresentation $W$ of $V$ is a representation of the same quiver such that $W(i)\leq V(i)$ for $i\in Q_{0}$, and $W(a)=V(a)|_{W(ta)}$ and $\operatorname{im}W(a)\leq W(ha)$ for $a\in Q_{1}$. The vector of dimensions of $V(i)$ is called the dimension vector of the representation, denoted by $\operatorname{\underline{dim}}V$. We call the vector space of all representations of $Q$ with the dimension vector $\alpha$ the representation space of $Q$ with dimension vector $\alpha$, which we denote by $\operatorname{Rep}(Q,\alpha)$. After fixing the dimension vector $\alpha$ and a basis of each $V(i)$, we can represent $V(a)$ as an $\alpha(ha)\times\alpha(ta)$ matrix. Therefore, the representation space can be identified as

$\displaystyle\operatorname{Rep}(Q,\alpha)=\bigoplus_{a\in Q_{1}}\operatorname{Mat}(\alpha(ha),\alpha(ta)),$

(1.1)

where $\operatorname{Mat}(m,n)$ denotes the space of $m\times n$ complex matrices.

Fix a dimension vector $\alpha$. Let

$\displaystyle\operatorname{GL}(Q,\alpha)\coloneqq\prod_{i\in Q_{0}}\operatorname{GL}(\alpha(i)),$

(1.2)

where $\operatorname{GL}(n)$ denotes the general linear group of degree $n$. Then, $\operatorname{GL}(Q,\alpha)$ acts on the representation space by a change of basis:

$\displaystyle g\cdot V\coloneqq(g_{ha}V(a)g_{ta}^{-1})_{a\in Q_{1}}.$

(1.3)

Note that this is a left action, i.e., $(gh)\cdot V=g\cdot(h\cdot V)$ for $g,h\in\operatorname{GL}(Q,\alpha)$. We say that a representation $V$ is semistable under the $\operatorname{GL}(Q,\alpha)$-action if the orbit closure of $V$ does not contain the origin, i.e.,

$\displaystyle\inf_{g\in\operatorname{GL}(Q,\alpha)}\sum_{a\in Q_{1}}\lVert g_{ha}V(a)g_{ta}^{-1}\rVert_{\mathrm{F}}^{2}>0.$

(1.4)

Otherwise, $V$ is said to be unstable. The set of all unstable representations is called the null-cone of the $\operatorname{GL}(Q,\alpha)$-action. It is easy to see that any representation is unstable under the $\operatorname{GL}(Q,\alpha)$-action if $Q$ is acyclic.¹¹1The readers may wonder whether we can check the semistability (under $\operatorname{GL}(Q,\alpha)$-action) of quiver representations of cyclic quivers. We will address this point later.

However, the semistability of quivers under subgroups of $\operatorname{GL}(Q,\alpha)$ turns out to be more intricate. Let $\sigma\in\mathbb{Z}^{Q_{0}}$ be an integer vector on $Q_{0}$, which we call a weight. Let $\chi_{\sigma}$ be the corresponding multiplicative character of $\operatorname{GL}(Q,\alpha)$, i.e.,

$$\chi_{\sigma}(g)=\prod_{i\in Q_{0}}\det(g_{i})^{\sigma(i)}.$$

Note that $\chi_{\sigma}$ is a one-dimensional representation of $\operatorname{GL}(Q,\alpha)$; $\operatorname{GL}(Q,\alpha)$ acts on $\mathbb{C}$ by $g\cdot x\coloneqq\chi_{\sigma}(g)x$. A representation $V$ is said to be $\sigma$-semistable if the orbit closure of $(V,1)\in\operatorname{Rep}(Q,\alpha)\oplus\mathbb{C}$ under the $\operatorname{GL}(Q,\alpha)$ action does not contain the origin, i.e.,

$\displaystyle\inf_{g\in\operatorname{GL}(Q,\alpha)}\left(\sum_{a\in Q_{1}}\lVert g_{ha}V(a)g_{ta}^{-1}\rVert_{\mathrm{F}}^{2}+\lvert\chi_{\sigma}(g)\rvert^{2}\right)>0.$

(1.5)

It turns out that checking the $\sigma$-semistability of a quiver representation includes operator scaling (noncommutative rank computation) and the membership problem of the BL polytopes. We will see these examples in the following sections.

Our first result is a deterministic algorithm that, given a quiver representation $V$ and weight $\sigma$, decides whether the representation is $\sigma$-semistable in time polynomial in the bit complexity of $V$ and absolute values of the entries of $\sigma$. Let $\alpha(Q_{0})\coloneqq\sum_{i\in Q_{0}}\alpha(i)$.

This improves the previous result [Hus21] which runs in time polynomial in the number of paths in $Q$, which can be exponential in the size of $Q$. Furthermore, if the absolute value of the entries of $\sigma$ is constant, our algorithm runs in polynomial time. This includes the known result for operator scaling [GGOW19].

[Kin94] showed the following characterization of $\sigma$-semistability, which is known as King’s criterion. Let $\sigma(\alpha)\coloneqq\sum_{i\in Q_{0}}\sigma(i)\alpha(i)$ for a dimension vector $\alpha$. Then, a representation $V$ is $\sigma$-semistable if and only if $\sigma(\operatorname{\underline{dim}}V)=0$ and $\sigma(\operatorname{\underline{dim}}W)\leq 0$ for any subrepresentation $W$ of $V$.

King’s criterion is a common generalization of the noncommutative rank (nc-rank) computation and membership problem of BL polytopes.

We study the following optimization problem: given a quiver representation $V$ and weight $\sigma$, find a subrepresentation $W$ of $V$ that maximizes $\sigma(\operatorname{\underline{dim}}W)$. In the case of the nc-rank, such a subrepresentation corresponds to a subspace $U$ that maximizes $\dim U-\dim(\sum_{a=1}^{m}A_{i}U)$. Such a subspace is called a shrunk subspace and can be regarded as a certificate of the nc-rank [IQS18, FSG23]. In the case of the BL polytopes, the problem corresponds to separation for the BL polytope [GGOW18].

King’s criterion can be regarded as maximizing a modular function over the modular lattice of subrepresentations. For any subrepresentations $W_{1},W_{2}$ of $V$, define the subrepresentations $W_{1}+W_{2}$ and $W_{1}\cap W_{2}$ as follows. For each $i\in Q_{0}$,

$\displaystyle(W_{1}+W_{2})(i)\coloneqq W_{1}(i)+W_{2}(i),\quad(W_{1}\cap W_{2})(i)\coloneqq W_{1}(i)\cap W_{2}(i),$

(1.6)

where the addition and intersection on the right-hand side are those of the vector spaces. Furthermore, the linear map of $a\in Q_{1}$ in $W_{1}+W_{2}$ (resp. $W_{1}\cap W_{2}$) is defined as the restriction of $V(a)$ to $(W_{1}+W_{2})(ta)$ (resp. $(W_{1}\cap W_{2})(ta)$). Then, $W_{1}+W_{2}$ and $W_{1}\cap W_{2}$ are indeed subrepresentations of $V$. Thus, the subrepresentations of $V$ form a modular lattice. Furthermore, the function $f(W)\coloneqq\sigma(\operatorname{\underline{dim}}W)$ is a modular function, i.e., for any subrepresentations $W_{1},W_{2}$ of $V$,

$\displaystyle f(W_{1})+f(W_{2})=f(W_{1}+W_{2})+f(W_{1}\cap W_{2}).$

(1.7)

Thus, subrepresentations maximizing $f$ form a sublattice, and there is a unique inclusion-wise minimum maximizer. Our second result is a deterministic algorithm to find such a maximizer of King’s criterion.

King’s criterion was originally proved using the Hilbert-Mumford criterion (see, e.g., [DW17, Sections 9.6 and 9.8]), a fundamental result in the GIT. We provide an alternative elementary proof in Appendix A for the sake of completeness.

We use the algorithm for finding the maximizers of King’s criterion to devise an algorithm for finding the Harder-Narasimhan (HN) filtration [HN75, HA02] of a quiver representation. Roughly speaking, the HN-filtration decomposes a quiver representation into the direct sum of smaller representations.

More precisely, let $\sigma\in\mathbb{Z}^{Q_{0}}$ be a weight and $\tau\in\mathbb{Z}_{+}^{Q_{0}}$ a strictly monotone weight, i.e., a nonnegative weight such that $\tau(\operatorname{\underline{dim}}W)>0$ if $W\neq\{0\}$. Here, $\{0\}$ denotes the zero representation, which is the representation whose dimension vector is the zero vector. We define the slope of a quiver nonzero representation $V$ as $\mu(V)=\sigma(\operatorname{\underline{dim}}V)/\tau(\operatorname{\underline{dim}}V)$. We say that $V$ is $\mu$-semistable²²2It is also called $(\sigma:\tau)$-semistability in the literature. if $\mu(W)\leq\mu(V)$ for any nonzero subrepresentation $W$ of $V$. The HN-filtration theorem states that for any quiver representation $V$, there exists a unique filtration $\{0\}=W_{0}<W_{1}<\cdots<W_{k}=V$ such that (i) $\mu(W_{i}/W_{i-1})>\mu(W_{i+1}/W_{i})$ for $i\in[k-1]$ and (ii) $W_{i}/W_{i-1}$ is $\mu$-semistable. Here, $W_{i}<W_{i+1}$ means that $W_{i}$ is a subrepresentation of $W_{i+1}$ with $W_{i}\neq W_{i+1}$, and $W_{i}/W_{i-1}$ is a representation of $Q$ such that $(W_{i}/W_{i-1})(j)$ is the quotient space $W_{i}(j)/W_{i-1}(j)$ for $j\in Q_{0}$ and $(W_{i}/W_{i-1})(a)$ is the corresponding quotient linear map of $W_{i}(a)$ for $a\in Q_{1}$. We note that semistability with respect to a slope can be reduced to that for a weight; see Lemma 4.4.

Our third result is a deterministic algorithm for finding the HN-filtration.

This result improves a recent result [Che24] which runs in time polynomial in the number of paths in $Q$.

Recently, [HS24] introduced the coarse Dulmage-Mendelsohn (DM) decomposition of a linear matrix, generalizing the classic DM-decomposition of a bipartite graph. They showed that a natural gradient flow of operator scaling converges to the coarse DM-decomposition. However, their result did not provide an efficient algorithm to compute the coarse DM-decomposition because the gradient flow may take exponential time to converge. We show that the coarse DM-decomposition is indeed a special case of the HN-filtration for the generalized Kronecker quiver. Since the absolute values of the weights involved for this special case are polynomially bounded, our algorithm finds the coarse DM-decomposition in polynomial time; see Section 4.3.

Motivated by King’s criterion, we investigate a polyhedral cone that is the set of $\sigma\in\mathbb{R}^{Q_{0}}$ satisfying $\sigma(\operatorname{\underline{dim}}V)=0$ and $\sigma(\operatorname{\underline{dim}}W)\leq 0$ for any subrepresentation $W$ of $V$. Since the number of distinct $\operatorname{\underline{dim}}W$ is finite, the above linear system is also finite and hence defines a polyhedral cone. We call the polyhedral cone the King cone of a quiver representation $V$. Interestingly, the King cone is related to the cone of feasible flows in network-flow problems.

Let us first consider the easiest case. If $V$ is a quiver representation with $\dim V(i)=1$ for all $i\in Q_{0}$, then King’s criterion characterizes the existence of a nonnegative flow $\varphi$ on the support quiver with the boundary condition $\partial\varphi=\sigma$. To state it more precisely, we introduce notation on flows. Let $Q=(Q_{0},Q_{1})$ be a quiver (or a directed graph). For a vertex subset $X\subseteq Q_{0}$, let $\operatorname{Out}(X)$ denote the set of outgoing arcs from $X$, i.e., $\operatorname{Out}(X)\coloneqq\{a=(i,j):i\in X,\ j\in Q_{0}\setminus X\}$. Similarly, let $\operatorname{In}(X)$ denote the set of incoming arcs to $X$. If $X=\{i\}$, we abbreviate $\operatorname{Out}(\{i\})$ and $\operatorname{In}(\{i\})$ as $\operatorname{Out}(i)$ and $\operatorname{In}(i)$, respectively. If $\operatorname{Out}(X)=\emptyset$, then $X$ is called a lower set of $Q$. For a flow $\varphi\in\mathbb{R}^{Q_{1}}$ on $Q$, its boundary $\partial\varphi\in\mathbb{R}^{Q_{0}}$ is defined by $\partial\varphi(i)\coloneqq\sum_{a\in\operatorname{Out}(i)}\varphi(a)-\sum_{a^{\prime}\in\operatorname{In}(i)}\varphi(a^{\prime})$ for $i\in Q_{0}$.

Let us return to the $\sigma$-semistability of a representation $V$ with $\dim V(i)=1$ for all $i\in Q_{0}$. In this case, $V(a)\in\mathbb{C}$ for each arc $a\in Q_{1}$, and a subrepresentation $W$ of $V$ can be identified with a vertex subset $X\subseteq Q_{0}$. By the definition of a subrepresentation, if $i\in X$ and $W(a)\neq 0$ for $a=(i,j)\in Q_{1}$, then $j\in X$. This implies that $X$ is a lower set in the support quiver of $V$, namely, the subquiver of $Q$ whose arcs are $a\in Q_{1}$ with $V(a)\neq 0$. Therefore, King’s criterion is equivalent to the purely combinatorial condition that $\sigma(Q_{0})=0$ and $\sigma(X)\leq 0$ for each lower set $X$ of $Q$, which characterizes the existence of a nonnegative flow $\varphi$ on the support quiver with the boundary condition $\partial\varphi=\sigma$ by Gale’s theorem [Gal57] (see, e.g., [KV18, Theorem 9.2]).

By generalizing the above observation, we show that if $V$ is a rank-one representation of $Q$, i.e., $V(a)$ is a rank-one matrix for each $a\in Q_{1}$, then

• •

  King’s criterion can be rephrased as a purely combinatorial condition with respect to the linear matroids arising from the rank-one matrices $V(a)$ of $Q$, and

• •

  the rephrased condition above can be further viewed as the feasibility condition of a network flow-type problem called submodular flow.

That is, the King cone is representable as the feasibility of a certain instance of the submodular flow problem. This enables us to decide the $\sigma$-semistability for rank-one representations in strongly polynomial time.

This theorem recovers the following well-known results when applied to the generalized Kronecker quiver and a star quiver.

• •

  A rank-one linear matrix $\sum_{k=1}^{m}x_{k}v_{k}f_{k}$ is (nc-)nonsingular (where $v_{k}$ is a column vector and $f_{k}$ a row vector) if and only if the linear matroids of $(f_{k}:k\in[m])$ and $(v_{k}:k\in[m])$ have a common base [Lov89].

• •

  If each linear operator $B_{i}=f_{i}$ is of rank-one for $i\in[m]$ (where $f_{i}$ is a row vector), the BL polytope coincides with the base polytope of the linear matroid of $(f_{i}:i\in[m])$ [Bar98].

Thus far, we have considered the $\sigma$-semistability of acyclic quivers. As a complementary result, we show that the semistability of cyclic quivers under the $\operatorname{GL}(Q,\alpha)$-action can be efficiently reduced to noncommutative polynomial identity testing. In particular, we show that the polynomial can be represented by an algebraic branching program (ABP). This yields a deterministic algorithm for deciding the semistability of general quivers because noncommutative polynomial identity testing for ABP can be conducted in deterministic polynomial time [RS05]; see Section 6. Note that [BFGO+19] devised another deterministic algorithm for the problem with their framework of noncommutative optimization, which is built upon deep results in various areas of mathematics. See also the discussion in related work.

In this subsection, we outline our techniques.

Existing studies on algorithms for quiver semistability have focused on bipartite quivers [CK21, CD21, CI22, FM23]. We remark that semistability in bipartite quivers is essentially operator scaling with a block structure; see Section 2.4. In bipartite quivers, the number of paths is equal to the number of arcs; hence, a weak running time was sufficient in the previous studies. [Hus21, Che24] studied general acyclic quivers. They first used the reduction of [DM17] to the generalized Kronecker quiver and applied the nc-rank algorithm [IQS18] in a black box manner. Hence, their algorithm runs in time polynomial in the number of paths, which can be exponential.

Several polyhedral cones associated with quiver representations have been studied in the literature [CCK22, VW23]. The moment cone of a quiver $Q$ and a dimension vector $\alpha$ is the polyhedral cone generated by the highest weights of the representations of $Q$ with the dimension vector $\alpha$. The membership of the moment cone can be decided in strongly polynomial time for bipartite quivers [CCK22] and even general acyclic quivers [VW23]. Another polyhedral cone is the conic hull of weight $\sigma\in\mathbb{Z}^{Q_{0}}$ such that there exists a nonzero semi-invariant polynomial in $\operatorname{Rep}(Q,\alpha)$ with weight $\sigma$. The membership problem of this cone is called the generic semistability problem [CCK22]. By definition, $\sigma$ is in this cone if and only if there exists a generic $\sigma$-semistable representation of $Q$, hence the name. To the best of our knowledge, the generic semistability problem remains open for general acyclic quivers.

The semistability of quiver representations is a special case of semistability in the GIT. In the most abstract setting, GIT studies group actions on algebraic varieties. We say that a point in the variety is semistable if its orbit closure does not contain the origin. [BFGO+19] proposed a framework of noncommutative optimization to devise algorithms for GIT problems in the general setting. Although noncommutative optimization is a broad and general framework, it does not always provide efficient algorithms for all GIT problems. Currently, most of the known tractable problems originate from a family of operator scaling problems, which are also contained in the semistability of quiver representations. Furthermore, it is built upon deep results in various areas of mathematics, such as algebraic geometry, Lie algebra, and representation theory, rendering it difficult for non-experts to understand. Another conceptual contribution of this paper is identifying the semistability of quiver representations as a useful subclass of GIT. The semistability of quiver representations is rich enough to capture various interesting problems in the literature while also supporting the design efficient algorithms using elementary techniques.

The rest of this paper is organized as follows. Section 2 introduces the necessary background and notation of quiver representations, operator scaling, and noncommutative rank. Section 3 presents our algorithms for deciding the $\sigma$-semistability and finding maximizers of King’s criterion. Section 4 presents our HN-filtration algorithm and its application to the coarse DM-decomposition. Section 5 investigates the King cone of rank-one representations and shows the reduction to submodular flow. Section 6 describes our reduction from the semistability of general quivers to polynomial identity testing of noncommutative ABPs.

A linear map $\Phi:\operatorname{Mat}(n)\to\operatorname{Mat}(m)$ is said to be completely positive, or CP for short, if $\Phi(X)=\sum_{\ell=1}^{k}A_{\ell}XA_{\ell}^{\dagger}$ for some $A_{\ell}\in\operatorname{Mat}(m,n)$. These $A_{\ell}$ are called the Kraus operators of $\Phi$. The dual map $\Phi^{*}:\operatorname{Mat}(m)\to\operatorname{Mat}(n)$ of $\Phi$ is defined by $\Phi^{*}(X)=\sum_{\ell=1}^{k}A_{\ell}^{\dagger}XA_{\ell}$. For $(g,h)\in\operatorname{GL}(m)\times\operatorname{GL}(n)$, we define the scaling $\Phi_{g,h}$ of $\Phi$ by

$\displaystyle\Phi_{g,h}(X)\coloneqq g\Phi(h^{\dagger}Xh)g^{\dagger}=\sum_{\ell=1}^{k}(gA_{\ell}h^{\dagger})X(gA_{\ell}h^{\dagger})^{\dagger}.$

(2.1)

If $m=n$, the CP map is said to be square.

Let $\Phi:\operatorname{Mat}(n)\to\operatorname{Mat}(n)$ be a square CP map. Let $\operatorname{\mathsf{ds}}(\Phi)\coloneqq\lVert\Phi(I)-I\rVert_{\mathrm{F}}^{2}+\lVert\Phi^{*}(I)-I\rVert_{\mathrm{F}}^{2}$, where $\lVert\cdot\rVert_{\mathrm{F}}$ denotes the Frobenius norm. Then, $\Phi$ is said to be approximately scalable if for any $\varepsilon>0$, there exists $(g,h)\in\operatorname{GL}(n)\times\operatorname{GL}(n)$ such that $\operatorname{\mathsf{ds}}(\Phi_{g,h})\leq\varepsilon$. The goal of the operator scaling problem is to decide whether a given CP map is approximately scalable or not.

Operator scaling is closely related to the noncommutative rank (nc-rank) of linear matrices. An $m\times n$ symbolic matrix $A$ is called a linear matrix if $A=\sum_{\ell=1}^{k}x_{\ell}A_{\ell}$ for indeterminates $x_{\ell}$ and matrices $A_{\ell}\in\operatorname{Mat}(m,n)$. Sometimes, it is more convenient to see a linear matrix as a matrix space spanned by $A_{1},\dots,A_{k}$. We denote the corresponding matrix space of a linear matrix $A$ by $\mathcal{A}$. For a subspace $U\leq\mathbb{C}^{n}$, let $\mathcal{A}U\coloneqq\langle\{Au:A\in\mathcal{A},\ u\in U\}\rangle,$ which is a subspace of $\mathbb{C}^{m}$. The nc-rank of a linear matrix $A$ (denoted by $\operatorname{nc-rank}A$) is defined by

$\displaystyle\operatorname{nc-rank}A\coloneqq\min\{n+\dim\mathcal{A}U-\dim U:U\leq\mathbb{C}^{n}\}.$

(2.2)

A square linear matrix $A$ is said to be nc-nonsingular if $\operatorname{nc-rank}A=n$. Informally speaking, $\operatorname{nc-rank}A$ is the rank of $A$, where the indeterminates $x_{i}$ are pairwise noncommutative, i.e., $x_{i}x_{j}\neq x_{j}x_{i}$ for $i\neq j$. See [Coh95, FR04] for more details. A pair $(L,R)$ of subspaces $L\leq\mathbb{C}^{m}$ and $R\leq\mathbb{C}^{n}$ is called an independent subspace if $L\cap\mathcal{A}R=\{0\}$. Over the complex field, $(L,R)$ is independent if and only if $\operatorname{tr}(\Pi_{L}\Phi(\Pi_{R}))=0$, where $\Pi_{L}$ denotes the orthogonal projection matrix onto $L$. Then,

$\displaystyle\operatorname{nc-rank}A=m+n-\max\{\dim L+\dim R:\text{$(L,R)$ an independent subspace}\}.$

(2.3)

An independent subspace $(L,R)$ is said to be maximum if $\dim L+\dim R$ is maximum. In particular, a square linear matrix $A$ is nc-nonsingular if and only if $\dim L+\dim R\leq n$ for any independent subspace $(L,R)$. Gurvits’ theorem [Gur04] states that a square CP map $\Phi$ with the Kraus operator $A_{\ell}$ ($\ell\in[k]$) is approximately scalable if and only if the linear matrix $\sum_{\ell=1}^{k}x_{\ell}A_{\ell}$ is nc-nonsingular.

Shrunk subspaces are minimizers of the objective function in nc-rank:

$\displaystyle f(U)=n+\dim\mathcal{A}U-\dim U.$

(2.4)

It is easy to see that $f$ is submodular, i.e., $f(U)+f(V)\geq f(U\cap V)+f(U+V)$ for any subspaces $U,V\leq\mathbb{C}^{n}$. Therefore, the shrunk subspaces form a sublattice of $\mathbb{C}^{n}$, i.e., if $U$ and $V$ are shrunk subspaces, then $U\cap V$ and $U+V$ are also shrunk subspaces. Hence, there exists a unique shrunk subspace with minimum dimension, which we call the minimum shrunk subspace. A shrunk subspace can be regarded as a certificate of the value of the nc-rank. In the GIT perspective, shrunk subspaces correspond to one-parameter subgroups that bring the matrix tuple $(A_{1},\dots,A_{k})$ to the origin. The minimum shrunk subspace is particularly important because it has a rational basis with polynomial bit complexity if $(A_{1},\dots,A_{k})$ has Gaussian integer entries [IQS18].

Several algorithms can find the minimum shrunk subspace in polynomial time [IQS18, FSG23]. We use the algorithm in [FSG23], which is based on a modified operator Sinkhorn iteration. We use it as a black box and consider only the square case for simplicity. The details of the algorithm can be referred from [FSG23].

Operator scaling with specified marginals is a generalization of operator scaling. Let $(b^{+},b^{-})\in\mathbb{R}^{n}_{+}\times\mathbb{R}^{m}_{+}$ be a pair of nonincreasing nonnegative vectors, which we call the target marginals. We say that a CP map $\Phi:\operatorname{Mat}(n)\to\operatorname{Mat}(m)$ is approximately scalable to the target marginals $(b^{+},b^{-})$ if there exist nonsingular upper triangular matrices $(g,h)\in\mathrm{B}(m)\times\mathrm{B}(n)$ such that

$\displaystyle\lVert\Phi_{g,h}(I)-\operatorname{Diag}(b^{-})\rVert_{\operatorname{tr}}\leq\varepsilon,\quad\lVert\Phi_{g,h}^{*}(I)-\operatorname{Diag}(b^{+})\rVert_{\operatorname{tr}}\leq\varepsilon.$

(2.5)

Such target marginals are said to be feasible. We define $\Delta b^{+}\in\mathbb{R}_{+}^{n}$ as $\Delta b^{+}_{j}=b^{+}_{j}-b^{+}_{j+1}$ for $j\in[n]$, where we conventionally define $b^{+}_{n+1}\coloneqq 0$. Similarly, we define $\Delta b^{-}\in\mathbb{R}_{+}^{m}$. Let $F^{+}_{j}=\langle\mathbf{e}_{1},\dots,\mathbf{e}_{j}\rangle$ be the standard flag of $\mathbb{C}^{n}$ for $j\in[n]$. Similarly, we define $F^{-}_{i}$ for $i\in[m]$. The following theorem characterizes the set of feasible marginals by a certain linear system.