k-Positivity and Schmidt number under orthogonal group symmetries

Abstract

In this paper, we present a new application of group theory to develop a systematical approach to efficiently compute the Schmidt numbers. The Schmidt number is a natural quantification of entanglement in quantum information theory, but computing its exact value is generally a challenging task even for very concrete examples. We exhibit a complete characterization of all orthogonally covariant k-positive maps. This result generalizes earlier results by Tomiyama (Linear Algebra Appl 69:169–177, 1985). Furthermore, we optimize duality relations between k-positivity and Schmidt numbers under group symmetries. This new approach enables us to transfer the results of k-positivity to the computation of the Schmidt numbers of all orthogonally invariant quantum states.

1 Introduction

The study of positivity has a rich historical background with significant applications in operator algebra and quantum theory. One prominent application is a characterization of quantum channels, which describe quantum evolutions in open quantum systems interacting with their environments. The mathematical structure of quantum channels is provided by completely positive trace-preserving (CPTP) linear maps in the Schrödinger picture. In this paper, we will focus on finite-dimensional quantum systems. Note that, if $d=\min \left\{ \textrm{dim}(H_A),\textrm{dim}(H_B)\right\}$, then a linear map $\mathcal {L}:B(H_A)\rightarrow B(H_B)$ is CP if and only if $\mathcal {L}$ is d-positive, i.e.

$$\begin{aligned} \textrm{id}_{d}\otimes \mathcal {L}: M_d\otimes B(H_A)\rightarrow M_d\otimes B(H_B) \end{aligned}$$

(1.1)

is positive, where $M_d$ is the set of all linear operators acting on $\mathbb {C}^d$. An intermediate concept between positivity and complete positivity is the so-called k-positivity ($1\le k\le d$), which means that

$$\begin{aligned} \textrm{id}_{k}\otimes \mathcal {L}: M_k\otimes B(H_A)\rightarrow M_k\otimes B(H_B) \end{aligned}$$

(1.2)

is positive. While extensive research was conducted on k-positivity [6, 7, 9, 19, 22, 35, 36], it is hard to expect an efficient method to verify whether a given linear map is k-positive.

On the other hand, quantum entanglement is a fundamental phenomenon in quantum science and serves as a crucial resource in quantum information processing [18]. Quantifying quantum entanglement is a central issue in this field, and it has been revealed that k-positivity is closely linked to quantum entanglement via the so-called Schmidt number [34]. Specifically, the Schmidt number of a quantum state $\rho$ is greater than k if and only if there exists a certain k-positive linear map $\mathcal {L}$ such that $({\text {id}}\otimes \mathcal {L})(\rho )$ is not positive semidefinite. In this context, k-positive maps can be considered Schmidt number witnesses [29]. While various attempts have been made to obtain lower and upper bounds for the Schmidt numbers [3, 6, 11, 19, 24, 27, 34, 39], accurate computations still pose significant challenges. To the best of our knowledge, there are very few explicit examples where the Schmidt numbers can be precisely calculated in high-dimensional systems. Some known examples are the isotropic states [16, 34], the Werner states [23, 38], and recently in [3].

One crucial progress of this paper is a new framework for quantitative analysis of quantum entanglement generalizing the methodologies proposed in [26]. The main focus of [26] was to develop a universal framework to study the problem of whether a given state is entangled or not under compact group symmetry. In this paper, we extend their framework to cover more general questions of the Schmidt numbers (Theorem 3.6). Indeed, we prove that a much smaller set of k-positive maps is sufficient as detectors to compute the Schmidt numbers under compact group symmetries. Furthermore, our abstract approach not only establishes the duality between k-positive maps and the Schmidt numbers but also provides more general duality results between mapping cones (Theorem 3.3).

The generalized duality results enable us to analyze k-positivity and the Schmidt number for quantum objects under the standard orthogonal group symmetries [37] (Theorems 4.1, 4.4, and 4.9). Specifically, we provide a complete characterization of k-positivity of all linear maps of the form

$$\begin{aligned} \small \mathcal {L}^{(d)}_{a,b}(Z)=(1-a-b)\frac{\text {Tr}(Z)}{d}I_d+aZ+bZ^{\top } \end{aligned}$$

(1.3)

and apply our duality results to compute the Schmidt numbers of all quantum states of the form

$$\begin{aligned} \small \rho ^{(d)}_{a,b}=\frac{1-a-b}{d^2}I_{d^2} +\frac{a}{d}\sum _{i,j=1}^d |ii\rangle \langle jj|+\frac{b}{d}\sum _{i,j=1}^d |ij\rangle \langle ji|. \end{aligned}$$

(1.4)

Note that [34, 35] cover special cases $\mathcal {L}^{(d)}_{a,0}$, $\mathcal {L}^{(d)}_{0,b}$, and $\rho ^{(d)}_{a,0}$ where all these subclasses are parametrized as one-dimensional spaces. To the best of our knowledge, our computations provide the first example of the complete characterization of Schmidt numbers in a non-trivial class parameterized by at least two real variables (in arbitrarily high-dimensional settings).

To visualize the full characterization of the k-positivity of $\mathcal {L}^{(d)}_{p,q}$ and the Schmidt number of $\rho ^{(d)}_{a,b}$, let us denote by $\mathcal {P}^{(d)}_k$ the set of k-positive linear maps $\mathcal {L}^{(d)}_{p,q}$, and by $\mathcal {S}^{(d)}_k$ the set of quantum states $\rho ^{(d)}_{a,b}$ whose Schmidt number is less than or equal to k. Our main results reveal that the geometic structures of $\mathcal {P}^{(d)}_k$ and $\mathcal {S}^{(d)}_k$ are categorized into four distinct cases: (1) $k=1$, (2) $1<k\le \frac{d}{2}$, (3) $\frac{d}{2}<k<d$, (4) $k=d$. For a special case $d=4$, we provide a visual representation of the convex sets $\mathcal {P}^{(4)}_k$ and $\mathcal {S}^{(4)}_k$ ($1\le k\le 4$) in Fig. 1 below. Note that the regions are highly non-trivial to describe since conics are necessary for both $\mathcal {P}^{(4)}_3$ and $\mathcal {S}^{(4)}_3$.

Fig. 1

Regions of the k-positive maps $\mathcal {L}_{p,q}^{(4)}$ and quantum states $\rho ^{(4)}_{a,b}$ of the Schmidt number k

2 Preliminaries

2.1 Positive maps and Schmidt number

Let us fix some notations that are frequently used throughout this paper. First of all, we will use bracket notation from physics. All Hilbert space H will be assumed to be finite-dimensional equipped with an inner product $\langle \cdot |\cdot \rangle$ and an orthonormal basis $\left\{ e_i\right\} _{i=1}^d$. For $v,w\in H$, we denote by $|v\rangle$ an operator mapping $\lambda \in \mathbb {C}$ to $\lambda v\in H$, and by $\langle w |$ a linear functional mapping $\xi \in H$ to $\langle w | \xi \rangle \in \mathbb {C}$. Then $|v\rangle \langle w|$ is a rank one operator which maps $\xi \in H$ to $\langle w|\xi \rangle |v\rangle \in H$. In particular, we write $|i\rangle :=|e_i\rangle$ and $|ij\rangle :=|i\rangle \otimes |j\rangle$. Then $\left\{ |i\rangle \langle j|\right\} _{i,j=1}^d$ forms matrix units in B(H). An operator $\rho \in B(H)$ is called a quantum state if it is positive and ${\text {Tr}}(\rho )=1$. We denote by $\textbf{D}(H)$ the set of all quantum states.

For every vector $\xi \in H_A\otimes H_B$, there exists a Schmidt decomposition [25] $|\xi \rangle =\sum _{i=1}^k{\lambda _i}|v_i\rangle \otimes |w_i\rangle$ where $\lambda _1\ge \cdots \ge \lambda _k>0$, and $\left\{ v_i\right\} _{i=1}^k$ and $\left\{ w_i\right\} _{i=1}^k$ are orthonormal subsets in $H_A$ and $H_B$, respectively. The numbers k and $\left\{ \lambda _i\right\} _{i=1}^k$ are uniquely determined, and we call k the Schmidt rank of $\xi$ and write $\textrm{SR}(|\xi \rangle )=k$. Now we denote by $\textbf{P}_{AB}$ the set of positive operators on $H_A\otimes H_B$ and consider the following subsets

$$\begin{aligned} \textbf{Sch}_{k,AB}:=\textrm{conv}\left\{ |\xi \rangle \langle \xi |: \textrm{SR}(|\xi \rangle )\le k\right\} \end{aligned}$$

for any natural number k (or simply write $\textbf{P}$ and $\textbf{Sch}_k$ when these cause no confusion). Then the Schmidt number of a positive operator $X\in \textbf{P}_{AB}$ is defined as the smallest natural number k such that $X\in \textbf{Sch}_k$, and we write $\textrm{SN}(X)=k$. Note that $\textrm{SN}(X)\le \min (d_A,d_B)$ and $\textbf{Sch}_k=\textbf{P}_{AB}$ for all $k\ge \min (d_A,d_B)$. We employ another notation $\textbf{SEP}_{AB}:=\textbf{Sch}_{1,AB}$ for the positive operators of the Schmidt number 1, and $X\in \textbf{P}_{AB}$ is called separable if $X\in \textbf{SEP}_{AB}$ and entangled otherwise.

Let us denote by $B(B(H_A),B(H_B))$ the set of all linear maps from $B(H_A)$ into $B(H_B)$ and by $B^h(B(H_A),B(H_B))$ the set of all Hermitian preserving maps, i.e., $\mathcal {L}\in B(B(H_A),B(H_B))$ with $\mathcal {L}(Z)^*=\mathcal {L}(Z^*)$ for $Z\in B(H_A)$. We also denote by $\mathcal {POS}_{AB}\subset B^{h}(B(H_A),B(H_B))$ the cone of positive maps from $B(H_A)$ into $B(H_B)$. A list of subclasses of positive maps of our interest is the following:

• $\mathcal {POS}_{k,AB}$, the set of k-positive maps (note that $\mathcal {POS}_1=\mathcal {POS}$),

• $\mathcal{C}\mathcal{P}_{AB}$, the set of completely positive (CP) maps,

• $\mathcal{S}\mathcal{P}_{k,AB}:=\textrm{conv}\left\{ \textrm{Ad}_K: K\in B(H_B,H_A), \textrm{rank}(K)\le k\right\}$, the set of k-superpositive maps [33], where $\textrm{Ad}_K(X):=KXK^*$,

• $\mathcal{E}\mathcal{B}_{AB}:=\mathcal{S}\mathcal{P}_1$, the set of entanglement-breaking (EB) maps [20].

Note that we have a nested chain of the subclasses as follows.

$$\begin{aligned} \mathcal {POS}\supsetneq \mathcal {POS}_2\supsetneq \cdots \supsetneq \mathcal {POS}_{\min (d_A,d_B)}=\mathcal{C}\mathcal{P}&\nonumber \\ =\;\mathcal{S}\mathcal{P}_{\min (d_A,d_B)}&\supsetneq \cdots \supsetneq \mathcal{S}\mathcal{P}_2\supsetneq \mathcal{E}\mathcal{B}. \end{aligned}$$

(2.1)

On the other hand, linear maps acting on quantum systems are standardly identified with bipartite operators via the so-called Choi-Jamiołkowski correspondence [5, 21]. For $\mathcal {L}\in B(B(H_A),B(H_B))$, the (normalized) Choi matrix $C_{\mathcal {L}}\in B(H_A\otimes H_B)$ is defined by

$$\begin{aligned} C_{\mathcal {L}}:=({\text {id}}_A\otimes \mathcal {L})(|\Omega _A\rangle \langle \Omega _A|)=\frac{1}{d_A}\sum _{i,j=1}^{d_A}|i\rangle \langle j|\otimes \mathcal {L}(|i\rangle \langle j|), \end{aligned}$$

where $\displaystyle |\Omega _A\rangle :=\frac{1}{\sqrt{d_A}}\sum \nolimits _{j=1}^{d_A} |jj\rangle$ is called the maximally entangled vector in $H_A\otimes H_A$. It is known that [5, 20, 31, 33]

• $\mathcal {L}$ is Hermitian preserving if and only if $C_{\mathcal {L}}$ is Hermitian,

• $\mathcal {L}$ is k-positive if and only if $C_{\mathcal {L}}\in \textbf{BP}_{k, AB}$, the set of k-block positive operators (that is, $\langle \xi |C_{\mathcal {L}}|\xi \rangle \ge 0$ for all $\xi \in H_A\otimes H_B$ such that $\textrm{SR}(|\xi \rangle )\le k$),

• $\mathcal {L}$ is CP if and only if $C_{\mathcal {L}}\in \textbf{P}_{AB}$,

• $\mathcal {L}$ is k-superpositive if and only if $C_{\mathcal {L}}\in \textbf{Sch}_{k,AB}$,

• $\mathcal {L}$ is EB if and only if $C_\mathcal {L}\in \textbf{SEP}_{AB}$.

We define the adjoint linear map $\mathcal {L}^*\in B(B(H_B),B(H_A))$ with respect to the Hilbert–Schmidt inner product, i.e.,

$$\begin{aligned} \textrm{Tr}(\mathcal {L}(Z)^*W)=\textrm{Tr}(Z^* \mathcal {L}^*(W)),\;\; Z,W\in B(H_A). \end{aligned}$$

Recall that the adjoint operation $\mathcal {L}\mapsto \mathcal {L}^*$ preserves all the above properties, i.e., k-positivity, complete positivity, and k-superpositivity.

2.2 Mapping cones and duality

Let us briefly recall several notions in convex analysis and the theory of mapping cones. First, $B^{h}(B(H_A),B(H_B))$ is a real vector space equipped with an inner product

$$\begin{aligned} \langle \Phi ,\Psi \rangle :={\text {Tr}}(C_{\Phi }^*C_{\Psi })={\text {Tr}}(C_{\Phi }C_{\Psi }). \end{aligned}$$

(2.2)

For a subset $\mathcal {K}\subset B^h(B(H_A),B(H_B))$, the dual cone $\mathcal {K}^{\circ }$ of $\mathcal {K}$ is defined by

$$\begin{aligned} \mathcal {K}^{\circ }:=\left\{ \Phi \in B^{h}(B(H_A),B(H_B)):\langle \Phi ,\Psi \rangle \ge 0\; \forall \, \Psi \in \mathcal {K}\right\} . \end{aligned}$$

(2.3)

It is well-known in convex analysis [28] that $\mathcal {K}^{\circ \circ }$ is the smallest closed convex cone containing $\mathcal {K}$. In particular, $\mathcal {K}$ is a closed convex cone if and only if $\mathcal {K}^{\circ \circ }=\mathcal {K}$.

A closed convex cone $\mathcal {K}\subset \mathcal {POS}_{AB}$ is called a mapping cone [30, 32] if it is invariant under the compositions by CP maps, i.e.,

$$\begin{aligned} \mathcal{C}\mathcal{P}_{BB}\circ \mathcal {K}\circ \mathcal{C}\mathcal{P}_{AA}\subset \mathcal {K}, \end{aligned}$$

(2.4)

where $\mathcal {K}_1\circ \mathcal {K}_2:=\left\{ \Phi \circ \Psi : \Phi \in \mathcal {K}_1,\; \Psi \in \mathcal {K}_2\right\}$. Since the identity maps ${\text {id}}_A,{\text {id}}_B$ are CP maps, Eq. (2.4) is equivalent to $\mathcal{C}\mathcal{P}_{BB}\circ \mathcal {K}\circ \mathcal{C}\mathcal{P}_{AA}= \mathcal {K}$.

If $\mathcal {K}$ is a nonzero mapping cone in $\mathcal {POS}_{AB}$, then the associated $K^{\circ }$ and $K^*:=\left\{ \mathcal {L}^*:\mathcal {L}\in \mathcal {K}\right\}$ are mapping cones in $\mathcal {POS}_{AB}$ and $\mathcal {POS}_{BA}$, respectively [14, 30]. Moreover, all the classes

$$\begin{aligned} \mathcal {POS}, \mathcal {POS}_k, \mathcal{C}\mathcal{P}, \mathcal{S}\mathcal{P}_k, \mathcal{E}\mathcal{B}\end{aligned}$$

(2.5)

introduced in Sect. 2.1 are mapping cones. Note that there are many other mapping cones important in quantum information theory, such as the cone of PPT (Positive Partial Transpose) maps, decomposable maps, and recently k-entanglement breaking maps [8, 12, 31]. Now, for a mapping cone $\mathcal {K}$, we can consider both the Choi correspondence $C_{\mathcal {K}}$ and the dual cone $\mathcal {K}^{\circ }$. Some important examples are exhibited in Table 1 below.

Table 1 Mapping cones, Choi correspondences, and dual cones

Let us explain more direct connections between the Choi correspondences $C_{\mathcal {K}}$ and the dual cones $\mathcal {K}^{\circ }$. First of all, a natural pairing between Hermitian operators $X\in B^{h}(H_{AB})$ and Hermitian-preserving linear maps $\mathcal {L}\in B^{h}(B(H_A),B(H_B))$ is given by

$$\begin{aligned} \langle X,\mathcal {L}\rangle :={\text {Tr}}(C_{\mathcal {L}}X)=\langle \Omega _A|({\text {id}}_A\otimes \mathcal {L}^*)(X)|\Omega _A\rangle . \end{aligned}$$

(2.6)

Then an extended form of the famous Horodecki criterion is given as follows with respect to the pairing (2.6) above.

Proposition 2.1

[14, Proposition 4.1] Suppose that a nonzero closed convex cone $\mathcal {K}\subset B^h(B(H_A),B(H_B))$ satisfies $\mathcal {K}\circ \mathcal{C}\mathcal{P}_{AA}\subset \mathcal {K}$.

• (1) The following are equivalent for $\mathcal {L}\in B(B(H_A),B(H_B))$:

  • (a) $\mathcal {L}\in \mathcal {K}$,

  • (b) $({\text {id}}_A\otimes \mathcal {L}^*)(X)\in \textbf{P}_{AA}$ for every $X\in C_{\mathcal {K}^{\circ }}$.

  • (c) $\langle X,\mathcal {L}\rangle \ge 0$ for every $X\in C_{\mathcal {K}^{\circ }}$.

• (2) The following are equivalent for $X\in B(H_A\otimes H_B)$:

  • (a) $X\in C_{\mathcal {K}}$,

  • (b) $({\text {id}}_A\otimes \mathcal {L}^*)(X)\in \textbf{P}_{AA}$ for every $\mathcal {L}\in \mathcal {K}^{\circ }$,

  • (c) $\langle X,\mathcal {L}\rangle \ge 0$ for every $\mathcal {L}\in \mathcal {K}^{\circ }$.

Note that Proposition 2.1 can be applied for arbitrary mapping cone $\mathcal {K}$. For example, the Horodecki criterion is for a special case $C_{\mathcal {K}}=\textbf{SEP}$ and $\mathcal {K}^{\circ }=\mathcal {POS}$, and it was used to study separability of quantum states [17]. Furthermore, Proposition 2.1 has been applied to study quantum states with upper bounds on the Schmidt numbers [34], decomposable maps [31], k-positive maps [13, 34], and k-superpositive maps [33].

Very recently, Park et al. [26] proposed an optimized form of Proposition 2.1 under compact group symmetries for a special case where $C_{\mathcal {K}}=\mathcal {POS}$ and $\mathcal {K}^{\circ }=\textbf{SEP}$. Furthermore, they introduced a concrete application to analyze separability of quantum states. One of the main contributions of this paper is to extend their results for general mapping cones $\mathcal {K}$ under compact group symmetries. See Sect. 3 and Corollary 3.5 for more details.

2.3 Compact group symmetries and twirling operations

Let $\pi :G\rightarrow \mathcal {U}(H)$, $\pi _A:G\rightarrow \mathcal {U}(H_A)$ and $\pi _B:G\rightarrow \mathcal {U}(H_B)$ be (finite-dimensional) unitary representations of a compact Hausdorff group G. An operator $X\in B(H)$ is called $\pi$-invariant if

$$\begin{aligned} \pi (x)X= X \pi (x) \end{aligned}$$

(2.7)

for all $x\in G$, and a linear map $\mathcal {L}:B(H_A)\rightarrow B(H_B)$ is called $(\pi _A,\pi _B)$-covariant if

$$\begin{aligned} \mathcal {L}(\pi _A(x)Z\pi _A(x)^*)=\pi _B(x)\mathcal {L}(Z)\pi _B(x)^* \end{aligned}$$

(2.8)

for all $x\in G$ and $Z\in B(H_A)$. We denote by $\textrm{Inv}(\pi )$ the set of all $\pi$-invariant operators in B(H), and by $\textrm{Cov}(\pi _A,\pi _B)$ the set of all $(\pi _A,\pi _B)$-covariant maps in $B(B(H_A),B(H_B))$. A unitary representation $\pi$ is called irreducible if $\textrm{Inv}(\pi )=\mathbb {C}\cdot I_d$. If $\pi$ is irreducible, so is the contragredient representation $\overline{\pi }:G\rightarrow \mathcal {U}(H)$. Here, $\overline{\pi }(x)=\overline{\pi (x)}$ for $x\in G$. For unitary representations $\pi _A:G\rightarrow \mathcal {U}(H_A)$ and $\pi _B:G\rightarrow \mathcal {U}(H_B)$, the tensor representation $\pi _A\otimes \pi _B:G\rightarrow \mathcal {U}(H_A\otimes H_B)$ is given by $(\pi _A\otimes \pi _B)(x)=\pi _A(x)\otimes \pi _B(x)$ for $x\in G$.

We are interested in two types of standard averaging techniques. The $\pi$-twirling operation on B(H) is defined by

$$\begin{aligned} \mathcal {T}_{\pi }X:=\int _G[\textrm{Ad}_{\pi (x)}(X) ]\textrm{d}x=\int _{G}[\pi (x)X\pi (x)^*]\textrm{d}x \end{aligned}$$

(2.9)

for any operator $X\in B(H)$ and a unitary representation $\pi$ of G. Moreover, the $(\pi _A,\pi _B)$-twirling operation on $B(B(H_A),B(H_B))$ is defined by

$$\begin{aligned} \mathcal {T}_{\pi _A,\pi _B}\Phi :=\int _G[\textrm{Ad}_{\pi _B(x^{-1})}\circ \Phi \circ \textrm{Ad}_{\pi _A(x)}]\textrm{d}x \end{aligned}$$

(2.10)

for any linear maps $\Phi \in B(B(H_A),B(H_B))$ and unitary representations $\pi _A,\pi _B$ of G. Here, $\textrm{d}x$ is the (normalized) Haar measure of G.

Let us collect some useful properties of the twirling operations for later uses. First of all, $\mathcal {T}_{\pi }$ is a conditional expectation onto the $*$-subalgebra $\textrm{Inv}(\pi )$ of B(H) in the sense that $\mathcal {T}_{\pi }\circ \mathcal {T}_{\pi }=\mathcal {T}_{\pi }$ and the range of $\mathcal {T}_{\pi }$ is $\textrm{Inv}(\pi )$. Similarly, $\mathcal {T}_{\pi _A,\pi _B}$ can be seen as a projection onto the space $\textrm{Cov}(\pi _A,\pi _B)$. These two operations satisfy the following properties.

Proposition 2.2

[26] Let $\pi :G\rightarrow \mathcal {U}(H)$, $\pi _A:G\rightarrow \mathcal {U}(H_A)$ and $\pi _B:G\rightarrow \mathcal {U}(H_B)$ be unitary representations of G. Then we have the following.

• (1) $\textrm{Tr}((\mathcal {T}_{\pi }Z)^*\,W)=\textrm{Tr}(Z^* (\mathcal {T}_{\pi }W))$ for any $Z,W\in B(H)$.

• (2) $\left( \mathcal {T}_{\pi _A,\pi _B}\Phi \right) ^*=\mathcal {T}_{\pi _B,\pi _A}(\Phi ^*)$ for any $\Phi \in B(B(H_A), B(H_B))$.

• (3) $C_{(\mathcal {T}_{\pi _A,\pi _B}\Phi )}=\mathcal {T}_{\overline{\pi _A}\otimes {\pi _B}}\left( C_{\Phi }\right)$ for any $\Phi \in B(B(H_A), B(H_B))$.

Since invariant operators and covariant linear maps are the images of the twirling projections, Proposition 2.2 (2) and (3) imply the following conclusions.

Corollary 2.3

[26] Let $X\in B(H_A\otimes H_B)$ be a bipartite operator and $\Phi :B(H_A)\rightarrow B(H_B)$ be a linear map. Then

• (1) $\Phi \in \textrm{Cov}(\pi _A,\pi _B)$ if and only if $\Phi ^*\in \textrm{Cov}(\pi _B,\pi _A)$.

• (2) $\Phi \in \textrm{Cov}(\pi _A,\pi _B)$ if and only if $C_{\Phi }\in \textrm{Inv}(\overline{\pi _A}\otimes \pi _B)$.

3 Mapping cones with compact group symmetry

From now on, we describe how standard duality results between mapping cones can be naturally carried over into our framework of compact group symmetry. Such a strategy of optimizing the duality relations under compact group symmetries was pursued in [26] for a special mapping cone $\mathcal {K}=\mathcal{E}\mathcal{B}$, where we have $C_{\mathcal {K}}=\textbf{SEP}$ and $\mathcal {K}^{\circ }=\mathcal {POS}$. The authors applied the optimized Horodecki criterion to study separability of invariant quantum states under the standard symmetries of the signed symmetric group (or the hyperoctahedral group). In this section, we prove that their approach covers not only for the special $\mathcal {K}=\mathcal{E}\mathcal{B}$, but also for general mapping cones $\mathcal {K}$ under compact group symmetries. Then, in Sect. 4, we apply the generalized results to characterize k-positivity of all orthogonally invariant quantum states.

Lemma 3.1

Let $\pi _A,\pi _B$ be two unitary representations of G.

• (1) For $\Phi ,\Psi \in B(B(H_A),B(H_B))$, we have

  $$\begin{aligned} \langle \mathcal {T}_{\pi _A,\pi _B}\Phi ,\Psi \rangle =\langle \Phi ,\mathcal {T}_{\pi _A,\pi _B}\Psi \rangle . \end{aligned}$$

  (3.1)

• (2) For $X\in B(H_A\otimes H_B)$ and $\mathcal {L}\in B(B(H_A), B(H_B))$, we have

  $$\begin{aligned} \langle \mathcal {T}_{\overline{\pi _A}\otimes \pi _B}X, \mathcal {L}\rangle =\langle X, \mathcal {T}_{\pi _A,\pi _B}\mathcal {L}\rangle . \end{aligned}$$

  (3.2)

Proof

Both two identities follow from Proposition 2.2. Indeed, we have

$$\begin{aligned} \langle \mathcal {T}_{\pi _A,\pi _B}\Phi , \Psi \rangle&= {\text {Tr}}((C_{(\mathcal {T}_{\pi _A,\pi _B}\Phi )})^*C_{\Psi })= {\text {Tr}}((\mathcal {T}_{\overline{\pi _A}\otimes \pi _B}C_\Phi )^*C_{\Psi })\\&={\text {Tr}}(C_\Phi ^* \mathcal {T}_{\overline{\pi _A}\otimes \pi _B}C_{\Psi }) = {\text {Tr}}(C_{\Phi }^*C_{(\mathcal {T}_{\pi _A,\pi _B}\Psi )})=\langle \Phi , \mathcal {T}_{\pi _A,\pi _B}\Psi \rangle \end{aligned}$$

which implies (3.1). Also, we can repeat a similar argument to prove (3.2). $\square$

Recall that positivity, complete positivity, and EB property are preserved under the twirling operation $\mathcal {L}\mapsto \mathcal {T}_{\pi _A,\pi _B}\mathcal {L}$ [26, Proposition 2.1]. More generally, we have $\mathcal {T}_{\pi _A,\pi _B}(\mathcal {K}) \subseteq \mathcal {K}$ for any mapping cone $\mathcal {K}$ by the following Proposition 3.2.

Proposition 3.2

For a closed convex cone $\mathcal {K}\in B^h(B(H_A),B(H_B))$, the following are equivalent:

• (1) $\mathcal {T}_{\pi _A,\pi _B}\mathcal {K}\subset \mathcal {K}$,

• (2) $\mathcal {T}_{\pi _A,\pi _B}(\mathcal {K}^{\circ })\subset \mathcal {K}^{\circ }$,

• (3) $\mathcal {T}_{\pi _B,\pi _A}(\mathcal {K}^{*})\subset \mathcal {K}^{*}$,

• (4) $\mathcal {T}_{\overline{\pi _A}\otimes \pi _B}C_{\mathcal {K}}\subset C_{\mathcal {K}}$.

In this case, we have $\mathcal {T}_{\overline{\pi _A}\otimes \pi _B}C_{\mathcal {K}}={\textrm{Inv}}(\overline{\pi _A}\otimes \pi _B)\cap C_{\mathcal {K}}$ and $\mathcal {T}_{\pi _A,\pi _B}\mathcal {K}=\textrm{Cov}(\pi _A,\pi _B)\cap \mathcal {K}$. Moreover, the above conditions hold if $\mathcal{C}\mathcal{P}_{BB}\circ \mathcal {K}\circ \mathcal{C}\mathcal{P}_{AA}\subset \mathcal {K}$.

Proof

The equivalence $(1)\Leftrightarrow (3) \Leftrightarrow (4)$ is a direct consequence from Proposition 2.2. For $(1)\Rightarrow (2)$, it is enough to see that

$$\begin{aligned} \langle \mathcal {T}_{\pi _A,\pi _B}\Phi ,\Psi \rangle =\langle \Phi ,\mathcal {T}_{\pi _A,\pi _B}\Psi \rangle \ge 0 \end{aligned}$$

for any $\Phi \in \mathcal {K}^{\circ }$ and $\Psi \in \mathcal {K}$. The other direction $(2)\Rightarrow (1)$ follows from $(1)\Rightarrow (2)$ since $\mathcal {K}^{\circ \circ }=\mathcal {K}$.

The second conclusion follows from the properties $\mathcal {T}_{\overline{\pi _A}\otimes \pi _B}\circ \mathcal {T}_{\overline{\pi _A}\otimes \pi _B}=\mathcal {T}_{\overline{\pi _A}\otimes \pi _B}$ and $\mathcal {T}_{\pi _A,\pi _B}\circ \mathcal {T}_{\pi _A,\pi _B}=\mathcal {T}_{\pi _A,\pi _B}$. For the last assertion, it is enough to note that the twirling operations preserve positivity of linear maps, and that $\mathcal {T}_{\pi _A,\pi _B}\Phi$ is approximated by convex combinations of $\textrm{Ad}_{\pi _B(x)^*}\circ \Phi \circ \textrm{Ad}_{\pi _A(x)}\in \mathcal {K}$ for $x\in G$. $\square$

Recall that $\mathcal {K}$ and $C_{\mathcal {K}^{\circ }}$ determine each other via the generalized Horodecki criterion (Proposition 2.1). One of our main results in this section is to establish an analogous result for $\textrm{Cov}(\pi _A,\pi _B)\cap \mathcal {K}$ and ${\textrm{Inv}}(\overline{\pi _A}\otimes \pi _B)\cap C_{\mathcal {K}^{\circ }}$ as follows.

Theorem 3.3

Let $\mathcal {K}\subset B^h(B(H_A),B(H_B))$ be a closed convex cone satisfying $\mathcal{C}\mathcal{P}_{BB}\circ \mathcal {K}\circ \mathcal{C}\mathcal{P}_{AA}\subset \mathcal {K}$.

• (1) The following are equivalent for $\mathcal {L}\in \textrm{Cov}(\pi _A,\pi _B)$:

  • (a) $\mathcal {L}\in \textrm{Cov}(\pi _A,\pi _B)\cap \mathcal {K}$,

  • (b) $({\text {id}}_A\otimes \mathcal {L}^*)(X)\in \textbf{P}_{AA}$ for every $X\in {\textrm{Inv}}(\overline{\pi _A}\otimes \pi _B)\cap C_{\mathcal {K}^{\circ }}$,

  • (c) $\langle X, \mathcal {L}\rangle \ge 0$ for every $X\in {\textrm{Inv}}(\overline{\pi _A}\otimes \pi _B)\cap C_{\mathcal {K}^{\circ }}$.

• (2) The following are equivalent for $X\in \textrm{Inv}(\overline{\pi _A}\otimes \pi _B)$:

  • (a) $X\in {\textrm{Inv}}(\overline{\pi _A}\otimes \pi _B)\cap C_{\mathcal {K}}$,

  • (b) $({\text {id}}_A\otimes \mathcal {L}^*)(X)\in \textbf{P}_{AA}$ for every $\mathcal {L}\in \textrm{Cov}(\pi _A,\pi _B)\cap \mathcal {K}^{\circ }$,

  • (c) $\langle X,\mathcal {L}\rangle \ge 0$ for every $\mathcal {L}\in \textrm{Cov}(\pi _A,\pi _B)\cap \mathcal {K}^{\circ }$.

Proof

Let us prove only the first part. The other one is analogous. Note that $(a)\Rightarrow (b)$ follows from Proposition 2.1 and $(b)\Rightarrow (c)$ is clear from the relation (2.6), so it suffices to prove the direction $(c)\Rightarrow (a)$. Since $\mathcal {L}\in \textrm{Cov}(\pi _A, \pi _B)$, we have

$$\begin{aligned} \langle X, \mathcal {L}\rangle = \langle X, \mathcal {T}_{\pi _A,\pi _B}\mathcal {L}\rangle = \langle \mathcal {T}_{\overline{\pi _A}\otimes \pi _B}X, \mathcal {L}\rangle \end{aligned}$$

for all $X\in C_{\mathcal {K}^{\circ }}$ by Lemma 3.1. Now $\mathcal {T}_{\overline{\pi _A}\otimes \pi _B}X\in \textrm{Inv}(\overline{\pi _A}\otimes \pi _B)\cap C_{\mathcal {K}^{\circ }}$ by Proposition 3.2, so the assumption (c) implies that $\langle X,\mathcal {L}\rangle \ge 0$ for all $X\in C_{\mathcal {K}^{\circ }}$. Therefore, Proposition 2.1 implies $\mathcal {L} \in \mathcal {K}$ again. $\square$

Note that $\textrm{Cov}(\pi _A,\pi _B)\cap \mathcal {K}$ plays as witnesses for ${\textrm{Inv}}(\overline{\pi _A}\otimes \pi _B)\cap C_{\mathcal {K}^{\circ }}$ when $\mathcal{C}\mathcal{P}_{BB}\circ \mathcal {K}\circ \mathcal{C}\mathcal{P}_{AA}\subset \mathcal {K}$. We can further prove that much fewer witnesses from $\textrm{Cov}(\pi _A,\pi _B)\cap \mathcal {K}$ are enough if $\mathcal {K}$ is a nonzero mapping cone, i.e., $\mathcal {K}\subset \mathcal {POS}_{AB}$. Let us start with a compact convex subset

$$\begin{aligned} \textrm{Cov}_1(\pi _A,\pi _B)\cap \mathcal {K}:=\left\{ \Phi \in {\textrm{Cov}}(\pi _A,\pi _B)\cap \mathcal {K}:{\text {Tr}}C_{\Phi }=1\right\} . \end{aligned}$$

Remark 3.4

[26, Lemma 3.6] implies that $(d_B/d_A)\cdot \textrm{Cov}_1(\pi _A,\pi _B)$ is the set of all unital $(\pi _A,\pi _B)$-covariant maps if $\pi _A$ is irreducible, and that $\textrm{Cov}_1(\pi _A,\pi _B)$ is the set of all trace-preserving $(\pi _A,\pi _B)$-covariant maps if $\pi _B$ is irreducible. Note that the notation $\textrm{Cov}_1(\pi _A,\pi _B)$ is used differently in [26].

Now we can extend Theorem 3.9 (6) of [26] to a general context of mapping cones. More precisely, Theorem 3.3 implies that only the extreme points of $\textrm{Cov}_1(\pi _A,\pi _B)\cap \mathcal {K}^{\circ }$ are enough as witnesses for $\textrm{Inv}(\overline{\pi _A}\otimes \pi _B)\cap C_\mathcal {K}$ since every compact convex set in a finite-dimensional space can be written as the convex hull of its extreme points.

Corollary 3.5

If $\mathcal {K}\subset \mathcal {POS}_{AB}$ is a nonzero mapping cone, then the following are equivalent for $X\in \textrm{Inv}(\overline{\pi _A}\otimes \pi _B)$:

• (1) $X\in \textrm{Inv}(\overline{\pi _A}\otimes \pi _B)\cap C_\mathcal {K}$,

• (2) $({\text {id}}_A\otimes \mathcal {L}^*)(X)\in \textbf{P}_{AA}$ for every $\mathcal {L}\in \textrm{Ext}(\textrm{Cov}_1(\pi _A,\pi _B)\cap \mathcal {K}^{\circ })$,

• (3) $\langle X,\mathcal {L}\rangle \ge 0$ for every $\mathcal {L}\in \textrm{Ext}(\textrm{Cov}_1(\pi _A,\pi _B)\cap \mathcal {K}^{\circ })$.

Recall that the main purpose of Theorem 3.9 (6) in [26] was to use $\textrm{Ext}(\textrm{Cov}_1(\pi _A,\pi _B)\cap \mathcal {POS})$ to study separability of $\rho \in \textrm{Inv}(\overline{\pi _A}\otimes \pi _B)\cap \textbf{D}$. A merit of our general approach is that Corollary 3.5 with the case $\mathcal {K}=\mathcal{S}\mathcal{P}_k$ provides a new systematic way to compute the Schmidt numbers of $\rho \in \textrm{Inv}(\overline{\pi _A}\otimes \pi _B)\cap \textbf{D}$ using $\textrm{Ext}(\textrm{Cov}_1(\pi _A,\pi _B)\cap \mathcal {POS}_k)$ as a family of witnesses.

Theorem 3.6

Let $\rho \in \textrm{Inv}(\overline{\pi _A}\otimes \pi _B)$. Then the following are equivalent:

• (1) $\textrm{SN}(\rho )\le k$,

• (2) $({\text {id}}_A\otimes \mathcal {L})(\rho )\in \textbf{P}_{AA}$ for every $\mathcal {L}\in \textrm{Ext}(\textrm{Cov}_1(\pi _B,\pi _A)\cap \mathcal {POS}_k)$,

• (3) $\langle \Omega _A|({\text {id}}_A\otimes \mathcal {L})(\rho )|\Omega _A\rangle \ge 0$ for every $\mathcal {L}\in \textrm{Ext}(\textrm{Cov}_1(\pi _B,\pi _A)\cap \mathcal {POS}_k)$.

The above Theorem 3.6 will be applied for concrete applications in Sect. 4.

4 Quantum objects with orthogonal group symmetry

There are very few examples whose k-positivity or Schmidt numbers have been fully characterized. Even in the following cases

$$\begin{aligned}&\mathcal {L}^{(d)}_{a,b}(Z):= (1-a-b)\frac{{\text {Tr}}(Z)}{d}I_d+aZ+bZ^{\top }, \end{aligned}$$

(4.1)

$$\begin{aligned}&\rho _{a,b}^{(d)}:=\frac{1-a-b}{d^2}I_d\otimes I_d+a|\Omega _d\rangle \langle \Omega _d|+\frac{b}{d}{F_d}, \end{aligned}$$

(4.2)

their k-positivity and Schmidt numbers have not been fully characterized. Here, $\displaystyle |\Omega _d\rangle :=\frac{1}{\sqrt{d}}\sum \nolimits _{j=1}^d|jj\rangle$ is the maximally entangled state and $\displaystyle F_d:=\sum \nolimits _{i,j=1}^d |ij\rangle \langle ji|$ is the flip matrix. Let us write $\mathcal {L}_{a,b}:=\mathcal {L}^{(d)}_{a,b}$ and $\rho _{a,b}:=\rho ^{(d)}_{a,b}$ for simplicity. On the problem of k-positivity, the answers for some special cases $\mathcal {L}_{a,0}$, $\mathcal {L}_{0,b}$, and $\mathcal {L}_{a,1-a}$, were obtained from [35], and some other cases were considered in [36]. On the other hand, Schmidt numbers of the isotropic states $\rho _{a,0}$ [16] were computed in [34], and Schmidt numbers of the Werner states $\rho _{0,b}$ [38] were also known (see [23, Theorem 1.7.4] for example). Despite the partial answers to the cases of single parameters, the problems of the general cases $\mathcal {L}_{a,b}$ and $\rho _{a,b}$ remain open. To our best knowledge, our result is the first example of computations of the Schmidt numbers for non-trivial two-dimensional families of quantum states in arbitrarily high-dimensional settings.

A crucial observation is that the above quantum objects $\mathcal {L}_{a,b}$ and $\rho _{a,b}$ are linked via the standard orthogonal group symmetries. Let G be the orthogonal group $\mathcal {O}(d)$, and let $\pi _A(O)=\pi _B(O)=O$ be the fundamental representation of $\mathcal {O}(d)$. In this case, we denote by $\textrm{Cov}(O,O)=\textrm{Cov}(\pi _A,\pi _B)$ and $\textrm{Inv}(O\otimes O)=\textrm{Inv}(\overline{\pi _A}\otimes \pi _B)$ for simplicity. Then we have $\rho _{a,b}=C_{\mathcal {L}_{a,b}}$ and $\textrm{Cov}_1(O,O)=\left\{ \mathcal {L}_{a,b}:a,b\in \mathbb {C}\right\}$, as noted in [15, 37]. Moreover, $\mathcal {L}_{a,b}$ is Hermitian-preserving if and only if $a,b\in \mathbb R$.

In this section, we aim to establish a complete characterization of k-positivity of $\mathcal {L}_{a,b}$ and Schmidt number of $\rho _{a,b}$ in terms of the parameters a and b. Then k-block positivity of $\rho _{a,b}$ and k-superpositivity of $\mathcal {L}_{a,b}$ are immediate through the Choi-Jamiołkowski map.

Our strategy consists of two steps. The first step is to employ some methodologies from [35] to study k-positivity of $\mathcal {L}_{a,b}\in \textrm{Cov}_1(O,O)$ in a direct way (Theorem 4.1), and the second step is to apply Theorem 3.6 to compute the Schmidt numbers of all $\rho _{a,b}\in \textrm{Inv}(O\otimes O)\cap \textbf{D}$ accurately (Theorems 4.4 and 4.9).

4.1 k-positivity of orthogonally covariant maps

Note that positivity and complete positivity of $\mathcal {L}_{p,q}$ were completely characterized recently in a more general setting, namely the hyperoctahedrally covariant maps [26, Section 4]. This section is devoted to characterizing k-positivity of all $\mathcal {L}_{p,q}\in \textrm{Cov}_1(O,O)$, which generalizes the results from [35]. Indeed, for the following convex subsets

$$\begin{aligned} P_k:=\left\{ (p,q)\in \mathbb R^2:\mathcal {L}_{p,q}\in \mathcal {POS}_k\right\} ,~1\le k\le d, \end{aligned}$$

(4.3)

we prove $P_1\supsetneq P_2\supsetneq \cdots \supsetneq P_d$ with explicit geometric and algebraic descriptions. First of all, the geometric structures of the convex subsets $P_k$ can be categorized into four distinct cases.

• (1) The region $P_1$ is trapezoid-shaped with vertices (1, 0), $\left( 0,-\frac{1}{d-1}\right)$, $\left( -\frac{1}{d-1},0\right)$, and (0, 1).

• (2) If $1<k\le \frac{d}{2}$, the region $P_k$ is quadrilateral-shaped with vertices (1, 0), $\left( 0,-\frac{1}{d-1}\right)$, $\left( -\frac{1}{kd-1},0\right)$, and $\left( -\frac{1}{kd+k-1}, \frac{k}{kd+k-1}\right)$.

• (3) If $\frac{d}{2}<k<d$, the region $P_k$ is bounded by a piecewise-linear curve joining $\left( -\frac{2}{d^2+d-2},\frac{d}{d^2+d-2}\right) , (1,0), \left( 0,-\frac{1}{d-1}\right)$, and $\left( -\frac{1}{kd-1},0\right)$ in that order, and by a conic (i.e., a quadratic curve) passing through $\left( -\frac{1}{kd-1},0\right)$ and $\left( -\frac{2}{d^2+d-2},\frac{d}{d^2+d-2}\right)$.

• (4) Lastly, the region $P_d$ is a triangle with vertices $(1,0), \left( 0,-\frac{1}{d-1}\right)$, and $\left( -\frac{2}{d^2+d-2}, \frac{d}{d^2+d-2}\right)$.

A visualization of the above characterizations for special cases $d=3$ and $d=4$ are given in the following Fig. 2.

Fig. 2

The regions of k-positive maps $\mathcal {L}_{p,q}^{(d)}$ for $d=3,4$

We now present explicit algebraic descriptions of the regions $P_k$ (equivalently, $\mathcal {POS}_k$) in the following theorem.

Theorem 4.1

Let $\mathcal {L}_{p,q}$ be a linear map of the form (4.1). Then

• (1) $\mathcal {L}_{p,q}\in \mathcal {POS}$ if and only if ${\left\{ \begin{array}{ll} p-(d-1)q\le 1, \\ q-(d-1)p\le 1,\\ frac{1}{d-1}\le p+q\le 1.\end{array}\right. }$

• (2) $\mathcal {L}_{p,q}\in \mathcal {POS}_k$ ($1<k\le \frac{d}{2}$) if and only if ${\left\{ \begin{array}{ll} p-(d-1)q\le 1, \\ p+(d+1)q\le 1, \\ (kd-1)p+(d-1)q\ge -1,\\ q-(kd-1)p\le 1.\end{array}\right. }$

• (3) $\mathcal {L}_{p,q}\in \mathcal {POS}_k$ ($\frac{d}{2}<k<d$) if and only if ${\left\{ \begin{array}{ll} p-(d-1)q\le 1, \\ p+(d+1)q\le 1, \\ (kd-1)p+(d-1)q\ge -1,\\ f_k(p,q)\le 0,\end{array}\right. }$ where $f_k(x,y)$ is a quadratic polynomial explicitly given by

  $$\begin{aligned} f_k(x,y)&=(kd-1)x^2-(d^3-kd^2-kd-d+2)xy+(d-1)y^2 \nonumber \\&\quad -(kd-2)x-(d-2)y-1. \end{aligned}$$

  (4.4)

• (4) $\mathcal {L}_{p,q}\in \mathcal{C}\mathcal{P}$ if and only if ${\left\{ \begin{array}{ll} p-(d-1)q\le 1, \\ p+(d+1)q\le 1, \\ (d+1)p+q\ge -\frac{1}{d-1}.\end{array}\right. }$

As mentioned already, (1) and (4) of Theorem 4.1, i.e., positivity and complete positivity of $\mathcal {L}_{p,q}$, were fully characterized in a recent paper [26, Section 4] for more general models. The authors studied linear maps of the form

$$\begin{aligned} \psi _{a,b,c}(Z):=a\frac{{\text {Tr}}(Z)}{d}I_d+bZ+cZ^{\top }+(1-a-b-c)\textrm{diag}(Z), \end{aligned}$$

(4.5)

and exhibited a full characterization of positivity and complete positivity. Thus, our main focus is to prove (2) and (3) of Theorem 4.1. Let us recall a criterion of k-positivity proposed in [35, Lemma 1].

Proposition 4.2

Let $1\le k\le d$. Then a linear map $\mathcal {L}:M_d\rightarrow M_d$ is k-positive if and only if the bipartite matrix

$$\begin{aligned} C_k^v(\mathcal {L}):=\sum _{i,j=1}^k |i\rangle \langle j|\otimes \mathcal {L}(|v_i\rangle \langle v_j|)\in M_k\otimes M_d \end{aligned}$$

(4.6)

is positive semidefinite for any choice of an orthonormal subset $\left\{ v_1,\ldots , v_k\right\}$ of $\mathbb {C}^d$.

The following lemma plays a crucial role in applying the above Proposition 4.2 to prove (2) and (3) of Theorem 4.1.

Lemma 4.3

For $1\le k\le d$, we have

$$\begin{aligned} \min \sum _{j,j'=1}^k|\langle {v_j}|\overline{v_{j'}}\rangle |^2=\max (2k-d,0), \end{aligned}$$

(4.7)

where $\overline{w}$ is the complex conjugation of $w\in \mathbb {C}^d$ and the minimum is taken over all orthonormal subsets $\left\{ v_1,\ldots , v_k\right\} \subset \mathbb {C}^d$.

Proof

If $2k\le d$, then we can take $|v_j\rangle =\frac{1}{\sqrt{2}}(|2j-1\rangle +i|2j\rangle )$ ($j=1,\ldots , k$) and check $\langle {v_j}|\overline{v_{j'}}\rangle =0$ for every $j,j'$, so the equality (4.7) holds. From now, let us focus on the case $2k>d$. First, we consider an arbitrary orthonormal subset $\left\{ v_1,\ldots , v_k\right\} \subset \mathbb {C}^d$ and orthogonal projections $\Pi _v=\sum _{j=1}^k |v_j\rangle \langle v_j|$ and $\Pi _{\overline{v}}=\sum _{j=1}^k |\overline{v_j}\rangle \langle \overline{v_j}|$. Then we can observe that

$$\begin{aligned} \sum _{j,j'} |\langle {v_j}|\overline{v_{j'}}\rangle |^2=\text {Tr}(\Pi _v \Pi _{\overline{v}})&\ge \text {dim}(\text {Ran}(\Pi _{v})\cap \text {Ran}(\Pi _{\overline{v}})) \end{aligned}$$

and the right-hand side is equal to

$$\begin{aligned} \text {dim}(\text {Ran}(\Pi _{v}))+\text {dim}(\text {Ran}(\Pi _{\overline{v}})) -\text {dim}(\text {Ran}(\Pi _{v})+\text {Ran}(\Pi _{\overline{v}})). \end{aligned}$$

Thus we have $\sum _{j,j'} |\langle {v_j}|\overline{v_{j'}}\rangle |^2 \ge 2k-d$. On the other hand, let us take a specific orthonormal subset $\left\{ v_1,v_2,\cdots ,v_k\right\} \subseteq \mathbb {C}^d$ where $|v_j\rangle =\frac{1}{\sqrt{d}}\sum _{l=1}^d\omega ^{l(j-1)}|l\rangle$ ($j=1,\ldots , k$) and $\omega =\exp (\frac{2\pi i}{d})$. Then we can check that the desired equality $\sum _{j,j'}|\langle {v_j}|\overline{v_{j'}}\rangle |^2=2k-d$ holds from the relation

$$\begin{aligned} \langle {v_j}|\overline{v_{j'}}\rangle =\frac{1}{d}\sum _{l=1}^d \omega ^{-l(j+j'-2)}={\left\{ \begin{array}{ll} 1 &{} \text {if }j+j'-2\equiv 0 \text {mod} d, \\ 0 &{} \text {otherwise}.\end{array}\right. } \end{aligned}$$

$\square$

Proof of Theorem 4.1 (2) and (3)

For an orthonormal subset $\left\{ v_1,\ldots , v_k\right\}$ in $\mathbb {C}^d$, the associated bipartite matrix in (4.6) is given by

$$\begin{aligned} C_k^v(\mathcal {L}_{p,q})=\frac{1-p-q}{d}I_k\otimes I_d+pk|\Omega _k^v\rangle \langle \Omega _k^v|+qF_k^v, \end{aligned}$$

where ${\left\{ \begin{array}{ll} |\Omega _k^v\rangle =\frac{1}{\sqrt{k}}\sum _{j=1}^k|j\rangle \otimes |v_j\rangle \in \mathbb {C}^k\otimes \mathbb {C}^d,\\ F_k^v=\sum _{i,j=1}^k|i\rangle \langle j|\otimes |\overline{v_j}\rangle \langle \overline{v_i}|\in M_{k}(\mathbb {C})\otimes M_{d}(\mathbb {C}).\end{array}\right. }$ Moreover, we can write $F_k^v=\Pi _{\mathcal {S}}^v-\Pi _{\mathcal {A}}^v$, where $\Pi _{\mathcal {S}}^v$ is the projection onto the (symmetric) space $\textrm{span}\left\{ \frac{|i\rangle |\overline{v_j}\rangle +|j\rangle |\overline{v_i}\rangle }{\sqrt{2}}: 1\le i\le j\le k\right\}$ and $\Pi _{\mathcal {A}}^v$ is the projection onto the (anti-symmetric) space $\textrm{span}\left\{ \frac{|i\rangle |\overline{v_j}\rangle -|j\rangle |\overline{v_i}\rangle }{\sqrt{2}}: 1\le i< j\le k\right\}$. Note that $\textrm{Ran}(\Pi _{\mathcal {S}}^v)\perp \textrm{Ran}(\Pi _{\mathcal {A}}^v)$, and $|\Omega _k^v\rangle \perp \textrm{Ran}(\Pi _{\mathcal {A}}^v)$ since

$$\begin{aligned} \left\langle \Omega _k^v\Bigg |\frac{e_i\otimes \overline{v_j}-e_j\otimes \overline{v_i}}{\sqrt{2}}\right\rangle =\frac{1}{\sqrt{2k}}(\langle v_i|\overline{v_j}\rangle - \langle v_j|\overline{v_i}\rangle )=0, \end{aligned}$$

for all $1\le i<j\le k$. Therefore, after rewriting

$$\begin{aligned} C_k^v(\mathcal {L}_{p,q})=\left( A(I_{kd}-\Pi _{\mathcal {A}}^v)+pk|\Omega _k^v\rangle \langle \Omega _k^v|+q\Pi _{\mathcal {S}}^v\right) \oplus \left( A-q\right) \Pi _{\mathcal {A}}^v \end{aligned}$$

with $A:=\frac{1-p-q}{d}$ for an orthogonal decomposition, the desired condition $C_k^v(\mathcal {L}_{p,q})\in \textbf{P}_{kd}$ is equivalent to $A-q\ge 0$ and

$$\begin{aligned} A(I_{kd}-\Pi _{\mathcal {A}}^v)+pk|\Omega _k^v\rangle \langle \Omega _k^v|+q\Pi _{\mathcal {S}}^v\in \textbf{P}\end{aligned}$$

(4.8)

(the condition $k\ge 2$ ensures that $\Pi _{\mathcal {A}}^v$ is nonzero).

A technical difficulty on (4.8) is that $|\Omega _k^v\rangle \langle \Omega _k^v|$ and $\Pi _{\mathcal {S}}^v$ are not simultaneously diagonalizable since

$$\begin{aligned} |\Omega _k^v\rangle \langle \Omega _k^v| \cdot \Pi _{\mathcal {S}}^v \ne \Pi _{\mathcal {S}}^v \cdot |\Omega _k^v\rangle \langle \Omega _k^v| \end{aligned}$$

in general unless $\left\{ v_i\right\} _{i=1}^k\subset \mathbb R^d$. To overcome this, let us take $|\xi _1\rangle =\Pi _{\mathcal {S}}^v|\Omega _k^v\rangle \in \textrm{Ran}(\Pi _{\mathcal {S}}^v)$ and consider $|\Omega _k^v\rangle =|\xi _1\rangle +|\xi _2\rangle$. Then

$$\begin{aligned} \xi _2\perp (\textrm{Ran}(\Pi _{\mathcal {S}}^v)\cup \textrm{Ran}(\Pi _{\mathcal {A}}^v)). \end{aligned}$$

Moreover, we have the following block matrix decomposition

$$\begin{aligned}&A(I_{kd}-\Pi _{\mathcal {A}}^v)+pk|\Omega _k^v\rangle \langle \Omega _k^v|+q\Pi _{\mathcal {S}}^v \nonumber \\&\quad \cong \small \begin{pmatrix} (A+q)\Pi _{\mathcal {S}}^v+pk|\xi _1\rangle \langle \xi _1| &{} pk|\xi _1\rangle \langle \xi _2| \\ pk|\xi _2\rangle \langle \xi _1| &{} A(I-\Pi _{\mathcal {S}}^v-\Pi _{\mathcal {A}}^v)+pk|\xi _2\rangle \langle \xi _2|\end{pmatrix}. \end{aligned}$$

(4.9)

An important fact to note on (4.9) is that $\textrm{rank}(\Pi _{\mathcal {S}}^v)=\frac{k(k+1)}{2}\ge 2$ and $\textrm{rank}(I-\Pi _{\mathcal {S}}^v-\Pi _{\mathcal {A}}^v)=d^2-k^2\ge 2$ since $1<k<d$. Therefore, the block matrix in (4.9) is positive semidefinite if and only if

$$\begin{aligned} {\left\{ \begin{array}{ll} (a)&{} A+q\ge 0,\\ (b)&{} A\ge 0,\\ (c)&{} A+q+pk\Vert \xi _1\Vert ^2\ge 0,\\ (d)&{} A+pk\Vert \xi _2\Vert ^2\ge 0,\\ (e)&{} \left( A+q+pk\Vert \xi _1\Vert ^2\right) \left( A+pk\Vert \xi _2\Vert ^2\right) \ge (pk)^2\Vert \xi _1\Vert ^2\Vert \xi _2\Vert ^2. \end{array}\right. } \end{aligned}$$

(4.10)

Since $\Vert \xi _1\Vert ^2+\Vert \xi _2\Vert ^2=\Vert \Omega _k^v\Vert ^2=1$, the conditions (d) and (e) can be understood as

$$\begin{aligned} {\left\{ \begin{array}{ll} (d')&{} A+pk-pk\Vert \xi _1\Vert ^2\ge 0,\\ (e')&{} (A+q)(A+pk)-pkq\Vert \xi _1\Vert ^2\ge 0. \end{array}\right. } \end{aligned}$$

(4.11)

Note that the first two conditions (a) and (b) are independent of the choices of an orthonormal subset $\left\{ v_i\right\} _{i=1}^k\subset \mathbb {C}^d$, and that the other inequalities in (c), $(d')$ and $(e')$ are linear in $\Vert \xi _1\Vert ^2$. Since the inequalities in (c), $(d')$ and $(e')$ should hold for all possible choices of $\left\{ v_i\right\} _{i=1}^k\subseteq \mathbb {C}^d$, it suffices to calculate the maximum and minimum values of $\Vert \xi _1\Vert ^2$.

Recall that $|\Omega _k^v\rangle =|\xi _1\rangle \in \textrm{Ran}(\Pi _{\mathcal {S}}^v)$ whenever $\left\{ v_i\right\} _{i=1}^k\subset \mathbb R^d$ (choose $|v_i\rangle =|i\rangle$ for example), so the maximum of $\Vert \xi _1\Vert ^2$ is 1. For the minimum of $\Vert \xi _1\Vert ^2$, the following expression of $\Vert \xi _1\Vert ^2$ in terms of $v_1,v_2,\cdots ,v_k$

$$\begin{aligned} \Vert \xi _1\Vert ^2&=\langle \Omega _k^v|\Pi _{\mathcal {S}}^v|\Omega _k^v\rangle \\ {}&=\langle \Omega _k^v|F_k^v|\Omega _k^v\rangle \quad (\,\because |\Omega _k^v\rangle \perp \textrm{Ran}(\Pi _{\mathcal {A}}^v))\\ {}&=\frac{1}{k}\sum _{j,j'}\langle v_j|\overline{v_{j'}}\rangle \cdot \langle \overline{v_{j'}}|v_j\rangle =\frac{1}{k}\sum _{j,j'}|\langle v_j|\overline{v_{j'}}\rangle |^2. \end{aligned}$$

allows us to apply Lemma 4.3 to conclude that $\min \Vert \xi _1\Vert ^2=\frac{\max (2k-d,0)}{k}$.

To summarize, $\mathcal {L}_{p,q}\in \mathcal {POS}_k$ if and only if the six inequalities (a), (b), (c), $(d')$, $(e')$, and $A-q\ge 0$ hold for $\Vert \xi _1\Vert ^2\in \left\{ 1, m:=\frac{\max (2k-d,0)}{k}\right\}$ and $A=\frac{1-p-q}{d}$. For the cases $1<k\le d/2$, we have $m=0$ and obtain the inequalities in (2). Also, for the cases $d/2<k<d$, we have $m=\frac{2k-d}{k}$ and obtain the inequalities in (3). In particular, the inequality $f_k(p,q)\le 0$ is coming from $(e')$ with $\Vert \xi _1\Vert ^2=\frac{2k-d}{k}$. $\square$

4.2 Schmidt numbers of orthogonally invariant quantum states

Now we are almost ready to compute the Schmidt numbers of all quantum states of the form

$$\begin{aligned} \rho _{a,b}:=\frac{1-a-b}{d^2}I_d\otimes I_d+a|\Omega _d\rangle \langle \Omega _d|+\frac{b}{d}{F_d} \end{aligned}$$

(4.12)

Let us denote by $S_k:=\left\{ (a,b)\in \mathbb R^2:\rho _{a,b}\in \textbf{Sch}_k\right\}$. The main aim of this Section is to prove $S_1\subsetneq S_2\subsetneq \cdots \subsetneq S_d$ with both geometric and algebraic descriptions. Our strategy is to combine Theorem 3.6 and the explicit descriptions of $P_k=\left\{ (p,q)\in \mathbb R^2:\mathcal {L}_{p,q}\in \mathcal {POS}_k\right\}$ (Theorem 4.1).

First of all, Theorem 3.6 implies that we have

$$\begin{aligned}&\textrm{SN}(\rho _{a,b})\le k \nonumber \\&\iff \langle \Omega _d|({\text {id}}_d\otimes \mathcal {L}_{p,q})(\rho _{a,b})|\Omega _d\rangle \ge 0 \text { for all }(p,q)\in \textrm{Ext}(P_k) \nonumber \\ {}&\iff \begin{pmatrix} p&q \end{pmatrix}\begin{pmatrix}d+1 &{} 1 \\ 1 &{} d+1\end{pmatrix}\left( {\begin{array}{c}a\\ b\end{array}}\right) \ge -\frac{1}{d-1} \text { for all }(p,q)\in \textrm{Ext}(P_k) . \end{aligned}$$

(4.13)

Let us denote by

$$\begin{aligned} H_{p,q}=\left\{ (x,y)\in \mathbb {R}^2: px+qy\le 1 \right\} \end{aligned}$$

(4.14)

for all $(p,q)\in \mathbb {R}^2{\setminus }\left\{ (0,0)\right\}$, and by $\alpha :\mathbb {R}^2\rightarrow \mathbb {R}^2$ a linear isomorphism given by

$$\begin{aligned} \alpha : \left( \begin{array}{cc} x\\ y \end{array} \right) \mapsto -(d-1)\left( \begin{array}{cc} d+1&{}1\\ 1&{}d+1 \end{array} \right) \left( \begin{array}{cc} x\\ y \end{array} \right) . \end{aligned}$$

(4.15)

Then we have the following identity:

$$\begin{aligned} S_k=\bigcap _{(p,q)\in \textrm{Ext}(P_k)}H_{\alpha (p,q)}. \end{aligned}$$

(4.16)

This is where a detailed geometric analysis of $P_k$ (Theorem 4.1) manifests its efficacy. Indeed, Theorem 4.1 states that the geometric structures of $P_k$ are categorized into four distinct cases $\left\{ \begin{array}{llll}(1)&{}k=1\\ (2) &{} 1<k\le \frac{d}{2}\\ (3) &{} \frac{d}{2}<k<d\\ (4) &{} k=d \end{array} \right.$. Furthermore, for the three cases (1), (2), (4), the associated regions $P_k$ are compact convex sets with at most four extreme points. Thus, it is enough to use at most four Schmidt number witnesses $\mathcal {L}_{p,q}$ to determine $S_k$ by Theorem 3.6 and (4.16), and the consequence is that $S_k$ is an intersection of at most four closed half-planes for the three cases (1), (2), (4). All our discussions above are summarized into the following theorem.

Theorem 4.4

Let $\rho _{a,b}$ be a bipartite matrix of the form (4.2) and $1\le k\le d$. Then we have

$$\begin{aligned} S_k=\bigcap _{(p,q)\in \textrm{Ext}(P_k)}H_{\alpha (p,q)} \end{aligned}$$

where $\alpha$ and $H_{p,q}$ are from (4.14) and (4.15). Moreover, we have the following algebraic descriptions for the three cases $\left\{ \begin{array}{llll}(1)&{}k=1,\\ (2) &{} 1<k\le \frac{d}{2},\\ (4) &{} k=d. \end{array} \right.$

• (1) $\rho _{a,b}\in \textbf{SEP}$ if and only if ${\left\{ \begin{array}{ll} -\frac{1}{d-1}\le (d+1)a+b\le 1, \\ -\frac{1}{d-1}\le a+(d+1)b\le 1. \end{array}\right. }$

• (2) $\rho _{a,b}\in \textbf{Sch}_k$ ($1<k\le \frac{d}{2}$) if and only if ${\left\{ \begin{array}{ll} -\frac{1}{d-1}\le (d+1)a+b\le \frac{kd-1}{d-1}, \\ a+(d+1)b\le 1, \\ -\frac{d-k+1}{kd+k-1}a+b\ge -\frac{1}{d-1}.\end{array}\right. }$

• (4) $\rho _{a,b}\in \textbf{Sch}_d=\textbf{P}$ if and only if ${\left\{ \begin{array}{ll} a-(d-1)b\le 1, \\ a+(d+1)b\le 1, \\ (d+1)a+b\ge -\frac{1}{d-1}.\end{array}\right. }$

We should remark that the remaining case (3) is quite different from the other cases since there are infinitely many extreme points in $P_k$. In this case, we will utilize some elementary geometric tools from projective geometry to overcome the technical issue. Indeed, we need a quadratic curve to describe $S_k$ for the cases $\frac{d}{2}<k<d$. This excluded case (3) will be discussed with details independently in Sect. 4.2.1.

Although we postpone the proof of the remaining case $\frac{d}{2}<k<d$ to Sect. 4.2.1, let us exhibit a visualized geometric structures of $S_1$, $S_2$, $\cdots$, $S_d$ in the following Fig. 3, particularly for the cases $d=3$ and $d=4$.

As in the case of k-positivity of $\mathcal {L}_{p,q}$, the geometric structures of the convex subsets $S_k$ can be categorized into the following four distinct cases.

• (1) The region $S_1$ is rhombus-shaped with vertices $\left( -\frac{2}{d^2+d-2},\frac{d}{d^2+d-2}\right)$, $\left( \frac{1}{d+2}, \frac{1}{d+2}\right)$, $\left( \frac{d}{d^2+d-2},-\frac{2}{d^2+d-2}\right)$, and $\left( -\frac{1}{d^2+d-2},-\frac{1}{d^2+d-2}\right)$.

• (2) If $1<k\le \frac{d}{2}$, then the region $S_k$ is trapezoid-shaped with vertices $\left( -\frac{2}{d^2+d-2},\frac{d}{d^2+d-2}\right)$, $\left( \frac{kd+k-2}{d^2+d-2}, \frac{d-k}{d^2+d-2}\right)$, $\left( \frac{kd+k-1}{d^2+d-2},-\frac{k+1}{d^2+d-2}\right)$, and $\left( 0,-\frac{1}{d-1}\right)$.

• (3) If $\frac{d}{2}<k<d$, then the region $S_k$ is bounded by a piecewise-linear curve joinig $\left( \frac{d}{3d-2k}, -\frac{2d-2k}{(d-1)(3d-2k)}\right)$, $\left( 0,-\frac{1}{d-1}\right)$, $\left( -\frac{2}{d^2+d-2},\frac{d}{d^2+d-2}\right)$, $\left( \frac{kd+k-2}{d^2+d-2}, \frac{d-k}{d^2+d-2}\right)$ and $\left( \frac{k^2d+k^2+d-3k}{k(d^2+d-2)}, -\frac{(d-k+1)(d-k)}{k(d^2+d-2)}\right)$ in that order, and then joined smoothly by an ellipse from $\left( \frac{k^2d+k^2+d-3k}{k(d^2+d-2)}, -\frac{(d-k+1)(d-k)}{k(d^2+d-2)}\right)$ to $\left( \frac{d}{3d-2k}, -\frac{2d-2k}{(d-1)(3d-2k)}\right)$.

• (4) The region $S_d$ is the same with $P_d=\left\{ (a,b): \mathcal {L}_{a,b}\in \mathcal{C}\mathcal{P}\right\}$, i.e., $S_d$ is a triangle with vertices $(1,0), (0,-\frac{1}{d-1})$, and $(-\frac{2}{d^2+d-2}, \frac{d}{d^2+d-2})$.

4.2.1 Algebraic descriptions of $S_k$ for the cases $\frac{d}{2}<k<d$

Let us focus on explicit algebraic descriptions of

$$\begin{aligned} S_k=\bigcap _{(p,q)\in \textrm{Ext}(P_k)}H_{\alpha (p,q)}=\alpha ^{-1}\bigg (\bigcap _{(p,q)\in \textrm{Ext}(P_k)}H_{p,q}\bigg ) \end{aligned}$$

(4.17)

for the cases $\frac{d}{2}<k<d$. In this case, we have

$$\begin{aligned} \textrm{Ext}(P_k)=\left\{ \left( \frac{-2}{d^2+d-2},\frac{d}{d^2+d-2}\right) , (1,0), \left( 0,\frac{-1}{d-1}\right) , \left( \frac{-1}{kd-1},0\right) \right\} \,\cup \, C_k \end{aligned}$$

Fig. 3

Regions of $\rho ^{(d)}_{a,b}$ of the Schmidt number k for $d=3,4$

by Theorem 4.1 (3). Here, $C_k$ is a conic arc in the second quadrant and is parametrized by a smooth, regular, and strictly convex curve

$$\begin{aligned} \gamma :[0,1]\rightarrow C_k \end{aligned}$$

(4.18)

satisfying $f_k(\gamma (t))\equiv 0$, $\gamma (0)=(-\frac{1}{kd-1},0)$ and $\gamma (1)=(-\frac{2}{d^2+d-2},\frac{d}{d^2+d-2})$. Then (4.17) implies that $\tilde{S}_k:=\alpha (S_k)$ is given by

$$\begin{aligned}&H_{-\frac{2}{d^2+d-2},\frac{d}{d^2+d-2}}\cap H_{1,0}\cap H_{0,-\frac{1}{d-1}}\cap H_{-\frac{1}{kd-1},0}\cap \bigcap _{t\in [0,1]}H_{\gamma (t)}, \end{aligned}$$

(4.19)

and the most technical problem is to demonstrate that $\displaystyle \bigcap \nolimits _{t\in [0,1]}H_{\gamma (t)}$ is a convex set bounded by two lines and one conic arc as in the following Fig. 4.

Here, we need to explain the dual curve and the pole-polar duality from projective geometry [2, 10]. Firstly, we have an explicit formula for the dual curve $\tilde{\gamma }$ of a strictly convex smooth curve $\gamma$. Recall that a plane curve $\gamma : I\rightarrow \mathbb R^2$ defined on an open interval I is called strictly convex if the number of intersection points between $\gamma$ and an arbitrary line is at most 2. If $\gamma$ is smooth, then the strict convexity of $\gamma$ is equivalent to that for every $t\in I$, the image of $\gamma$ is contained in the same half-plane whose boundary is the tangent line $l_t$ at $\gamma (t)$, and $\gamma (t)$ is the unique intersection point of $l_t$ and $\gamma$. For a strictly convex smooth curve $\gamma =(p(t),q(t))$ (with additional conditions in Lemma 4.5), we define its dual curve $\tilde{\gamma }$ by

$$\begin{aligned} \tilde{\gamma }(t):=\left( \frac{q'(t)}{p(t)q'(t)-q(t)p'(t)}, \frac{-p'(t)}{p(t)q'(t)-q(t)p'(t)}\right) . \end{aligned}$$

(4.20)

Fig. 4

Geometric description of the intersection $\displaystyle \bigcap _{t\in [0,1]} H_{\gamma (t)}$

Secondly, the pole-polar duality is a bijective correspondence between $\mathbb {R}^2\setminus \left\{ (0,0)\right\}$ and the set of all lines that are not passing through the origin (0, 0) in $\mathbb {R}^2$. Associated to $(p,q)\in \mathbb {R}^2{\setminus } \left\{ (0,0)\right\}$ is a line

$$\begin{aligned} l=\left\{ (x,y)\in \mathbb {R}^2: px+qy=1\right\} . \end{aligned}$$

In this case, we call l the polar of P and P the pole of l (with respect to the unit circle $C: x^2+y^2=1$), and denote by $l=:\textrm{Polar}(P)$ and $P=:\textrm{Pole}(l)$ respectively. Note that we have

$$\begin{aligned} \tilde{\gamma }(t)=\textrm{Pole}(l_t) \end{aligned}$$

for all $t\in I$, where $l_t$ denotes the line tangent to $\gamma$ at $\gamma (t)$.

The following Lemma 4.5 establishes the connection between the dual curve $\tilde{\gamma }$ and the intersection $\bigcap _{(p,q)\in C_k}H_{p,q}$. This seems a well-known fact, but we provide a proof for readers’ convenience.

Lemma 4.5

Let I be an open interval and $\gamma : I\rightarrow \mathbb R^2{\setminus } \left\{ (0,0)\right\}$ be a smooth, regular, and strictly convex curve. Suppose that for every $t\in I$, the tangent lines $l_t$ at $\gamma (t)$ do not pass through the origin (0, 0), and the origin is in the same (closed) half-plane with $\gamma$ with respect to $l_t$. Then the dual curve $\tilde{\gamma }:I\rightarrow \mathbb R^2{\setminus }\left\{ (0,0)\right\}$ from (4.20) satisfies the following properties.

• (1) $\textrm{Polar}(\gamma (t))$ is tangent to $\tilde{\gamma }$ at $\tilde{\gamma }(t)$ for each $t\in I$.

• (2) $\tilde{\gamma }$ is smooth and strictly convex, and the origin is in the same half-plane with $\widetilde{\gamma }$ with respect to $\textrm{Polar}(\gamma (t))$.

• (3) For any closed interval $[t_0,t_1]\subset I$, the intersection

  $$\begin{aligned} \bigcap _{t\in [t_0,t_1]} H_{\gamma (t)} \end{aligned}$$

  is the largest convex region containing (0, 0), which is bounded by two lines $\textrm{Polar}(\gamma (t_0))$ and $\textrm{Polar}(\gamma (t_1))$ as well as the dual curve $\tilde{\gamma }|_{[t_0,t_1]}$ (as in Fig. 4).

• (4) If $\gamma$ represents a connected part of a conic, then so is $\tilde{\gamma }$.

Proof

Set $\gamma (t)=(p(t),q(t))$. Then the equation of $l_t$ is given by

$$\begin{aligned} q'(t)(x-p(t))-p'(t)(y-q(t))=0. \end{aligned}$$

(4.21)

Note that the given assumptions imply that $p(t)q'(t)-q(t)p'(t)\ne 0$ for all $t\in I$, so by continuity, we may assume $p(t)q'(t)-q(t)p'(t)>0$ for all $t\in I$ without loss of generality. In particular, the dual curve $\widetilde{\gamma }$ from (4.20) is a well-defined smooth curve, and (4.21) implies that $\tilde{\gamma }(t)=\textrm{Pole}(l_t)$ for all $t\in I$. Let us write $\tilde{\gamma }(t)=(\tilde{x}(t),\tilde{y}(t))$ for simplicity and explain why the four conclusions (1)-(4) hold.

(1) It is enough to check that

$$\begin{aligned} p(t)\tilde{x}(t)+q(t)\tilde{y}(t)&=1,\\ p(t)\tilde{x}'(t)+q(t)\tilde{y}'(t)&=0. \end{aligned}$$

Indeed, the first equation comes from the fact that $\tilde{\gamma }(t)=\textrm{Pole}(l_t)$, and the second equation is obtained by differentiating the first equation and the identity $p'(t)\tilde{x}(t)+q'(t)\tilde{y}(t)\equiv 0$ from (4.20).

(2) Note that the strict convexity of $\gamma$ implies that

$$\begin{aligned} q'(t)(p(s)-p(t))-p'(t)(q(s)-q(t))<0, \quad t\ne s\in I, \end{aligned}$$

(4.22)

which is equivalent to

$$\begin{aligned} p(s)\tilde{x}(t)+q(s)\tilde{y}(t)<1, \quad t\ne s\in I. \end{aligned}$$

(4.23)

Thus, both the origin and $\tilde{\gamma }$ are in the same half-plane with respect to $\textrm{Polar}(\gamma (s))$. Moreover, Eq. (4.23) implies that $\tilde{\gamma }$ is strictly convex, and smoothness is immediate from the explicit description (4.20) of $\tilde{\gamma }$.

(3) Let T be the largest convex region containing (0, 0) bounded by two lines $\textrm{Polar}(\gamma (t_0))$, $\textrm{Polar}(\gamma (t_1))$, and the dual curve $\tilde{\gamma }|_{[t_0,t_1]}$. First, it is immediate to see that $T\subseteq \bigcap _{t\in [t_0,t_1]} H_{\gamma (t)}$. Indeed, T should be contained in the same plane with (0, 0) with respect to each tangent line $l_t=\textrm{Pole}(\gamma (t))$ for all $t\in I$, and this implies $T\subseteq H_{\gamma (t)}$ for all $t\in [t_0,t_1]$. On the other side, let us pick an element $(u,v)\notin T$ and let l be the straight line passing through the origin and (u, v). Then l intersects with one of $\textrm{Polar}(\gamma (t_0))$, $\textrm{Polar}(\gamma (t_1))$, and $\tilde{\gamma }|_{[t_0,t_1]}$. For the first two cases, (u, v) and (0, 0) are not on the same half-plane with respect to either $\textrm{Polar}(\gamma (t_0))$ or $\textrm{Polar}(\gamma (t_1))$, so $(u,v)\notin \bigcap _{t\in [t_0,t_1]} H_{\gamma (t)}$. For the remaining case, if we suppose that l contains certain $\tilde{\gamma }(t)$, then (0, 0) and (u, v) are not on the same plane with respect to $H_{\gamma (t)}$. Hence, we can conclude that $T^c\subseteq \left( \bigcap _{t\in [t_0,t_1]} H_{\gamma (t)}\right) ^c$, i.e., $\bigcap _{t\in [t_0,t_1]} H_{\gamma (t)}\subseteq T$.

(4) This is a direct consequence from Plücker’s formula [2, Section 9.1], which states that the degree of the dual curve $\tilde{\gamma }$ is $n(n-1)$ for any non-singular plane algebraic curve $\gamma$ of degree n (in our case, $n=2$). Alternatively, more elementary arguments can be found in [1, 4]. $\square$

From now on, let us focus more on the special case $\gamma : [0,1]\rightarrow C_k$ from (4.18). We may assume that $\gamma$ is extended to the smooth, regular, and strictly convex curve (still denoted by $\gamma$) on an open interval $I\supset [0,1]$ such that $f_k\circ \gamma \equiv 0$. Then Lemma 4.5 (4) implies that there exists a quadratic polynomial $\tilde{f}_k(x,y)$ such that

$$\begin{aligned} \tilde{f}_k\left( (\alpha ^{-1} \circ \tilde{\gamma })(t)\right) \equiv 0. \end{aligned}$$

Here, $\tilde{\gamma }$ is the dual curve of $\gamma$, and $\alpha$ is the linear isomorphism from (4.15). A notable fact is that $\alpha ^{-1}\circ \tilde{\gamma }$ always represents an ellipse. For this conclusion, the following lemma provides more concrete information on the quadratic polynomial $\tilde{f}_k(x,y)$.

Lemma 4.6

The quadratic equation $\tilde{f}_k(x,y)=0$ holds for the following five points $(a_i,b_i)$ $(1\le i\le 5)$

$\left( -\frac{d}{k(d^2+d-2)}, \frac{d^2-kd+d-2k}{k(d^2+d-2)}\right) , \left( \frac{d^2-kd+d-k-1}{d^2+d-2}, -\frac{d-k+1}{d^2+d-2}\right) , \left( \frac{2kd-d^2+2k-d-2}{d^2+d-2}, \frac{2d-2k}{d^2+d-2}\right)$, $\left( \frac{k^2d+k^2+d-3k}{k(d^2+d-2)}, -\frac{(d-k+1)(d-k)}{k(d^2+d-2)}\right) , \left( \frac{d}{3d-2k}, -\frac{2d-2k}{(d-1)(3d-2k)}\right)$,

with the associated tangent lines $l_i$ ($1\le i \le 5$)

$(d+1)x+y=-\frac{1}{d-1}$,   $x+(d+1)y=-\frac{1}{d-1}$,   $x+(d+1)y=1$,

$(d+1)x+y=\frac{kd-1}{d-1}$,    $x-(d-1)y=1$,

respectively. Furthermore, if $\frac{d}{2}<k<d$, the conic determined by the equation $\tilde{f}_k(x,y)=0$ is inscribed in the convex pentagon bounded by the above five tangent lines. In particular, the equation $\tilde{f}_k(x,y)=0$ should represent an ellipse.

Proof

Let us begin with the following expression

$$\begin{aligned} f_k(x,y)=Ax^2+Bxy+Cy^2+Dx+Ey+F \end{aligned}$$

(4.24)

with the coefficients

$$\begin{aligned} {\left\{ \begin{array}{ll}A=kd-1, \quad B=-(d^3-kd^2-kd-d+2), \quad C=d-1, \\ D=-kd+2, \quad E=-d+2, \quad F=-1,\end{array}\right. } \end{aligned}$$

from (4.4). If we write $\gamma (t)=(p(t),q(t))$, then we have

$$\begin{aligned} \tilde{\gamma }(t)=-\left( \frac{2Ap(t)+Bq(t)+D}{Dp(t)+Eq(t)+2F}, \frac{2Cq(t)+Bp(t)+E}{Dp(t)+Eq(t)+2F}\right) \end{aligned}$$

(4.25)

thanks to (4.24) and (4.15).

Recall that $\alpha ^{-1}(\tilde{\gamma }(t))$ are solutions of the equation $\tilde{f}_k(x,y)=0$ for all $t\in I$. Thus, in order to single out five points $(a_i,b_i)$ satisfying $\tilde{f}_k(a_i,b_i)=0$, it is enough to note that the following five points $(p(t_i),q(t_i))$ ($1\le i\le 5$)

$$\begin{aligned} (1,0), (0,1), \left( 0,\frac{-1}{d-1}\right) , \left( \frac{-1}{kd-1},0\right) , \left( \frac{-2}{d^2+d-2},\frac{d}{d^2+d-2}\right) \end{aligned}$$

are solutions to the equation $f_k(x,y)=0$. Then (4.25) provides us with the associated five points $(a_i,b_i)=\alpha ^{-1}(\tilde{\gamma }(t_i))$ listed in the statement. Furthermore, the tangent lines $l_i$ at $(a_i,b_i)$ satisfying $\tilde{f}_k(a_i,b_i)=0$ are given by $\textrm{Polar}(\alpha (p(t_i),q(t_i)))$, by Lemma 4.5 (1). Thus, we can write down what the tangent lines are explicitly, as in the statement. Lastly, it is immediate to check that when $\frac{d}{2}<k<d$, those five tangent lines consist of a convex pentagon, and the corresponding points $(a_i,b_i)$ of tangency are on each of the pentagon’s sides. This observation forces the quadratic equation $\tilde{f}_k(x,y)=0$ to represent an ellipse inscribed in this pentagon. $\square$

Remark 4.7

While the dual quadratic equation $\tilde{f}_k(x,y)=0$ in our consideration always represents an ellipse thanks to Lemma 4.6, the quadratic equation $f_k(x,y)=0$ can represent both an ellipse and a hyperbola. For example, the quadratic equation $f_k(x,y)=0$ for $d=5$ is given by

$$\begin{aligned} (5k-1)p^2-(122-30k)pq+4q^2-(5k-2)p-3q-1=0, \end{aligned}$$

and this represents a hyperbola if $k=3$ and an ellipse if $k=4$.

Finally, we are ready to describe the intersection

$$\begin{aligned} \bigcap _{(p,q)\in C_k}H_{\alpha (p,q)}=\alpha ^{-1}\bigg (\bigcap _{(p,q)\in C_k}H_{p,q}\bigg ) \end{aligned}$$

from (4.19). Recall that $\left( \frac{-1}{kd-1},0\right)$ and $\left( \frac{-2}{d^2+d-2},\frac{d}{d^2+d-2}\right)$ are the two end-points (p, q) of the connected conic arc $C_k$, and their associated points (a, b) satisfying $\tilde{f}_k(a,b)=0$ are given by $\left( \frac{k^2d+k^2+d-3k}{k(d^2+d-2)}, -\frac{(d-k+1)(d-k)}{k(d^2+d-2)}\right)$ and $\left( \frac{d}{3d-2k}, -\frac{2d-2k}{(d-1)(3d-2k)}\right)$. Let us denote by L the line segment between these two points, and let us assume (by changing the sign if necessary) that the inequality $\tilde{f}_k(x,y)\le 0$ represents a filled ellipse.

Corollary 4.8

Let $(a,b)\in \mathbb R^2$. Then $(a,b)\in \displaystyle \bigcap _{(p,q)\in C_k}H_{\alpha (p,q)}$ if and only if (a, b) satisfies $\tilde{f}_k(a,b)\le 0$ or satisfies the following three conditions:

• (1) $(d+1)a+b\le \frac{kd-1}{d-1}$,

• (2) $a-(d-1)b \le 1$,

• (3) $(3d-k+3)a-(kd+k-3)b-\frac{d^2+kd-k-3}{d-1}\le 0$.

Proof

By Lemmas 4.5 (3) and 4.6, the intersection $\displaystyle \bigcap \limits _{(p,q)\in C_k}H_{\alpha (p,q)}$ is the largest convex region containing (0, 0) bounded by the two tangent lines $(d+1)x+y=\frac{kd-1}{d-1}$, $x-(d-1)y=1$ and the dual curve $\tilde{\gamma }|_{[0,1]}$. We refer the readers to Fig. 4 for a visualized understanding. Note that $\tilde{f}_k(x,y)\le 0$ represents a filled ellipse which we denote by E, and E is a subset of the intersection $\displaystyle \bigcap \limits _{(p,q)\in C_k}H_{\alpha (p,q)}$ by Lemma 4.6. Furthermore, $\displaystyle \bigcap _{(p,q)\in C_k}H_{\alpha (p,q)}{\setminus } E$ is a subset of the largest convex region bounded by the two tangent lines $(d+1)x+y=\frac{kd-1}{d-1}$, $x-(d-1)y=1$ and the line segment L between $\left( \frac{k^2d+k^2+d-3k}{k(d^2+d-2)}, -\frac{(d-k+1)(d-k)}{k(d^2+d-2)}\right)$ and $\left( \frac{d}{3d-2k}, -\frac{2d-2k}{(d-1)(3d-2k)}\right)$. Hence, the conclusion follows immediately. $\square$

Now we are ready to complete the proof for the cases $\frac{d}{2}<k<d$.

Theorem 4.9

Let $\rho _{a,b}$ be a bipartite matrix of the form (4.2) and $\frac{d}{2}<k< d$. Then $\rho _{a,b}\in \textbf{Sch}_k$ if and only if $\tilde{f}_k(a,b)\le 0$ or (a, b) satisfies the following inequalities:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\frac{1}{d-1}\le (d+1)a+b\le \frac{kd-1}{d-1},\\ a+(d+1)b\le 1,\\ a-(d-1)b \le 1,\\ (3d-k+3)a-(kd+k-3)b-\frac{d^2+kd-k-3}{d-1}\le 0. \end{array}\right. } \end{aligned}$$

Here, $\tilde{f}_k$ is the quadratic polynomial from Lemma 4.6 such that the inequality $\tilde{f}_k(x,y)\le 0$ represents a filled ellipse.

Proof

Since $\displaystyle \bigcap _{(p,q)\in C_k}H_{p,q}\subseteq H_{-\frac{2}{d^2+d-2},\frac{d}{d^2+d-2}}\cap H_{-\frac{1}{kd-1},0}$, we have

$$\begin{aligned} \tilde{S}_k:=\alpha (S_k)= H_{1,0}\cap H_{0,-\frac{1}{d-1}}\cap \bigcap _{(p,q)\in C_k}H_{p,q} \end{aligned}$$

(4.26)

from (4.19). Thus, we have

$$\begin{aligned} S_k&=\alpha ^{-1}(\tilde{S}_k)=\alpha ^{-1}\left( H_{1,0}\right) \cap \alpha ^{-1} \left( H_{0,-\frac{1}{d-1}}\right) \cap \alpha ^{-1}\left( \bigcap _{(p,q)\in C_k}H_{p,q} \right) \\&=H_{-(d^2-1),-(d-1)}\cap H_{1,d+1} \cap \bigcap _{(p,q)\in C_k}H_{\alpha (p,q)}. \end{aligned}$$

and the conclusion follows immediately from Lemma 4.6 and Corollary 4.8. $\square$

Remark 4.10

It is worth remarking that a small perturbation can produce a drastic increment of the Schmidt number. Recall that $S_d$ is a triangle, and let us parameterize the southern-eastern edge of $S_d$ by $\eta :[0,1]\rightarrow S_d$ such that $\eta (0)=(0,-\frac{1}{d-1})$ and $\eta (1)=(1,0)$. Then $\rho _{\eta (t)}$ is always entangled, and the Schmidt numbers $\mathrm{{SN}}(\rho _{\eta (t)})$ exhibit a monotonically increasing pattern of

$2,\lceil \frac{d}{2}\rceil , \lceil \frac{d}{2}\rceil +1,\lceil \frac{d}{2}\rceil +2,\cdots , d,$

as t increases from 0 to 1. Note that there is a huge gap between 2 and $\lceil \frac{d}{2}\rceil$, which seems entirely new and highly non-trivial. This phenomenon does not appear on the other line segments in the boundary of $S_d$, and some other known cases such as $\rho _{a,0}$ and $\rho _{0,b}$. The only known patterns were $1,2,3,\cdots ,d$ (isotropic states) or 1, 2 (Werner states) to our best knowledge.