Entanglement in Bipartite Quantum Systems with Fast Local Unitary Control

Entanglement is one of the distinguishing features of quantum mechanics, and it lies at the core of emerging quantum technologies. Generating sufficient entanglement is a prerequisite for pure state quantum computation [28], it is at the root of quantum cryptography [5, 11] and it can act as a resource for increasing the sensitivity of quantum sensors [7]. At the same time many basic questions about entanglement remain unanswered, and our theoretical understanding of multipartite and mixed state entanglement is limited [4]. One setting in which entanglement is well-understood is the case of a bipartite quantum system in a pure state, where the singular value decomposition (a.k.a. the Schmidt decomposition) can be employed. Similar tools can also be applied to indistinguishable subsystems [15] (bosons and fermions), where the singular value decomposition is to be replaced with the Autonne–Takagi and Hua factorization respectively.

In this paper we apply quantum control theory [8, 6] to closed bipartite quantum systems subject to fast local unitary control. Factoring out the fast controllable degrees of freedom leads to a reduced control system defined on the singular values of the state. Such reduced control systems have been addressed in [1, Ch. 22] under simplifying assumptions. In particular, there, the reduced state space is assumed to be a smooth manifold without singularities. In our case singularities occur where two singular values coincide or one singular value vanishes, and in fact they constitute the main technical difficulty. In [20] this assumption is removed such that we can apply the results therein to the present setting.

Similar ideas have been applied to open Markovian quantum systems with fast unitary control in [25, 31, 23] and by the author et al. in [22]. There, the reduced control system describes the evolution of the eigenvalues of the density matrix (a positive semidefinite matrix of unit trace) representing a mixed quantum state. In quantum thermodynamics, a natural simplification of this system has been explored in [9, 29, 24, 30].

Finally, in an upcoming paper [21] we will apply the methods derived here combined with optimal control theory to derive explicit control solutions for low dimensional systems.

We start by defining the bilinear control system describing a closed bipartite quantum system with fast local unitary control in Section 2.1. We consider the case of distinguishable subsystems, as well as that of bosonic and fermionic subsystems. In Section 2.2 we show that these systems are related to certain matrix diagonalizations (which themselves are related to symmetric Lie algebras) and we derive corresponding reduced control systems in Section 2.3. In Section 2.4 we show how some controllability assumptions can be weakened if one neglects the global phase of the quantum state. Then we turn to some applications. In Section 3.1 we use the reduced control system to prove that the full bilinear control system is always controllable and stabilizable. In Section 3.2 we derive some quantum speed limits for the evolution of the singular values. The connection between the present quantum setting and symmetric Lie algebras is drawn in Appendix A, and this allows us to apply the results from [20].

Let two finite dimensional Hilbert spaces $\mathbb{C}^{d_{1}}$ and $\mathbb{C}^{d_{2}}$ of dimensions $d_{1},d_{2}\geq 2$ representing the subsystems be given. The total Hilbert space of the bipartite system is then $\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}$.¹¹1Abstractly the symbol $\otimes$ denotes the tensor product, but since we always work with concrete vectors and matrices we interpret $\otimes$ as the Kronecker product. This identifies the vector spaces $\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}$ and $\mathbb{C}^{d_{1}d_{2}}$ and similarly for matrices. We denote by $\mathfrak{u}(d)$ the unitary Lie algebra consisting of skew-Hermitian matrices in $d$ dimensions. Our goal is to study the full bilinear control system [16, 10] defined by the following controlled Schrödinger equation²²2Throughout the paper we set $\hbar=1$ and thus write the (uncontrolled) Schrödinger equation as $\ket{\dot{\psi}(t)}=-\mathrm{i}\mkern 1.0muH_{0}\ket{\psi(t)}$. on $\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}$:

where $H_{0}\in\mathrm{i}\mkern 1.0mu\mathfrak{u}(d_{1})\otimes\mathrm{i}\mkern 1.0mu\mathfrak{u}(d_{2})\cong\mathrm{i}\mkern 1.0mu\mathfrak{u}(d_{1}d_{2})$ is the drift Hamiltonian (or coupling Hamiltonian), $E_{i}\in\mathrm{i}\mkern 1.0mu\mathfrak{u}(d_{1})$ and $F_{j}\in\mathrm{i}\mkern 1.0mu\mathfrak{u}(d_{2})$ are the control Hamiltonians, and $u_{i}$ and $v_{j}$ are the corresponding control functions. We make the following key assumptions:

1. (I)

   The control functions $u_{i}$ and $v_{j}$ are locally integrable, in particular they may be unbounded.

2. (II)

   The control Hamiltonians generate the full local unitary Lie algebra:

   $$\langle\mathrm{i}\mkern 1.0muE_{i}\otimes\mathds{1},\,\mathds{1}\otimes\mathrm{i}\mkern 1.0muF_{j}:\,i=1,\ldots,m_{1},\,j=1,\ldots,m_{2}\rangle_{\mathsf{\mathsf{Lie}}}=\mathfrak{u}_{\mathrm{loc}}(d_{1},d_{2}),$$

where $\mathfrak{u}_{\mathrm{loc}}(d_{1},d_{2}):=(\mathfrak{u}(d_{1})\otimes\mathds{1})+(\mathds{1}\otimes\,\mathfrak{u}(d_{2}))$ is the Lie algebra of the Lie group $\operatorname{U}_{\mathrm{loc}}(d_{1},d_{2}):=\operatorname{U}(d_{1})\otimes\operatorname{U}(d_{2})$ of local unitary transformations. Put simply, we have fast control over $\operatorname{U}_{\mathrm{loc}}(d_{1},d_{2})$.

This covers the case of two distinguishable subsystems. However, we also wish to treat systems composed of two indistinguishable subsystems. In this case both subsystems have the same dimension $d:=d_{1}=d_{2}$. In the bosonic case, the state $\ket{\psi}$ is unchanged by swapping the two subsystems, i.e., $U_{\mathrm{swap}}\ket{\psi}=\ket{\psi}$, where $U_{\mathrm{swap}}\ket{\psi_{1}}\otimes\ket{\psi_{2}}=\ket{\psi_{2}}\otimes\ket{\psi_{1}}$. These “symmetric” states lie in the space $\mathrm{Sym}^{2}(\mathbb{C}^{d})$. In the fermionic case, swapping yields a phase factor of $-1$, i.e., $U_{\mathrm{swap}}\ket{\psi}=-\ket{\psi}$. Such “skew-symmetric” states are contained in the space $\bigwedge^{2}(\mathbb{C}^{d})$. In both cases the set of local unitaries applicable to the system is restricted to symmetric local unitaries $\operatorname{U}_{\mathrm{loc}}^{s}(d):=\{V\otimes V:V\in\operatorname{U}(d)\}$. The corresponding Lie algebra is $\mathfrak{u}_{\mathrm{loc}}^{s}(d):=\{\mathrm{i}\mkern 1.0muE\otimes\mathds{1}+\mathds{1}\otimes\mathrm{i}\mkern 1.0muE:\mathrm{i}\mkern 1.0muE\in\mathfrak{u}(d)\}$ and is isomorphic to $\mathfrak{u}(d)$.³³3Note however that the map $\operatorname{U}(d)\to\operatorname{U}_{\mathrm{loc}}^{s}(d)$ given by $V\mapsto V\otimes V$ is a double cover with kernel $\{\mathds{1},-\mathds{1}\}$. The set of all coupling Hamiltonians applicable to such systems is the set $\mathfrak{u}^{s}(d^{2})=\{\mathrm{i}\mkern 1.0muH\in\mathfrak{u}(d^{2}):U_{\mathrm{swap}}HU_{\mathrm{swap}}^{*}=H\}$. Hence, in the case of two indistinguishable subsystems the bilinear control system takes the form:

where $H_{0}\in\mathrm{i}\mkern 1.0mu\mathfrak{u}^{s}(d^{2})$ and $E_{1},\dots,E_{m}\in\mathrm{i}\mkern 1.0mu\mathfrak{u}(d)$. Assumption I remains unchanged, but Assumption II is slightly modified to state:

1. (III)

   The local control Hamiltonians $\mathrm{i}\mkern 1.0muE_{i}\otimes\mathds{1}_{2}+\mathds{1}_{1}\otimes\mathrm{i}\mkern 1.0muE_{i}$ for $i=1,\ldots,m$ generate the full Lie algebra $\mathfrak{u}_{\mathrm{loc}}(d)$.

Note that the only difference between the bosonic and fermionic case is that the initial state of the control system (B’) lies in $\mathrm{Sym}^{2}(\mathbb{C}^{d})$ and $\bigwedge^{2}(\mathbb{C}^{d})$ respectively.

In order to derive and understand the reduced control system (which we will introduce in the next section) obtained by factoring out the local unitary action, we must first understand the mathematical structure of this action. In the case of distinguishable subsystems, the local unitary action corresponds the complex singular value decomposition, an thus the only invariants of a state under this action are its singular values, which therefore are the natural choice for the reduced state. The bosonic and fermionic cases correspond to less well-known matrix decompositions called the Autonne–Takagi and Hua factorization respectively. Importantly all of these matrix decompositions also correspond to certain symmetric Lie algebras, and this is the key to applying the results on reduced control systems from [20] to the full bilinear control systems (B) and (B’).

Let $\{\ket{i}_{1}\}_{i=1}^{d_{1}}$ and $\{\ket{j}_{2}\}_{j=1}^{d_{2}}$ denote the standard orthonormal bases⁴⁴4In the following we will usually omit the index denoting the subsystem, since it will be clear form the order. of $\mathbb{C}^{d_{1}}$ and $\mathbb{C}^{d_{2}}$ respectively, and let $\ket{\psi}\in\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}$ be a state vector. The components $(\psi_{ij})_{i,j=1}^{d_{1},d_{2}}$ of $\ket{\psi}$ are uniquely given by

Hence every bipartite state can be represented by a matrix.⁵⁵5Our convention is consistent with [4, Sec. 9.2] and [15]. In other contexts one often defines the vectorization operation $\mathrm{vec}(\cdot)$ which turns a matrix into a vector by stacking its columns and satisfies $\mathrm{vec}(AXB)=(B^{\top}\otimes A)\mathrm{vec}(X)$. Our convention is slightly different in that, identifying $\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}\cong\mathbb{C}^{d_{1}d_{2}}$ via the Kronecker product, we may write $\ket{\psi}=\mathrm{vec}(\psi^{\top})$. More precisely, we have used the canonical isomorphism

where $(\cdot)^{\prime}$ denotes the dual space. We will use $\psi\in\mathbb{C}^{d_{1},d_{2}}$ to denote the matrix corresponding to $\ket{\psi}$ under this isomorphism and vice versa. For distinguishable subsystems, the matrix representing $\psi$ is an arbitrary complex matrix in $\mathbb{C}^{d_{1},d_{2}}$ (the constraint induced by the normalization of the state $\ket{\psi}$ will be discussed in Remark 2.3 below). For indistinguishable subsystems, it holds that $d:=d_{1}=d_{2}$, and the matrix $\psi\in\mathfrak{sym}(\mathbb{C},d)$ is symmetric in the bosonic case and $\psi\in\mathfrak{asym}(\mathbb{C},d)$ is skew-symmetric in the fermionic case.

Let $V\otimes W\in\operatorname{U}_{\mathrm{loc}}(d_{1},d_{2})$ be a local unitary, and set $\ket{\phi}=V\otimes W\ket{\psi}$. The coordinates of $V$ are then defined by $V=\sum_{k,i=1}^{d_{1}}V_{ki}\ket{k}\!\bra{i}$, and similarly $W=\sum_{l,j=1}^{d_{2}}W_{lj}\ket{l}\!\bra{j}$. Then $\phi_{kl}=\sum_{i,j=1}^{d_{1},d_{2}}V_{ki}W_{lj}\psi_{ij}$. In matrix form this can be rewritten as $\phi=V\psi W^{\top}$. Another way to state this is $V\otimes W\ket{\psi}=\ket{V\psi W^{\top}}$. Note that in the case of indistinguishable subsystems we have $V=W$. This suggests a connection to certain matrix diagonalizations, namely:

• •

  The complex singular value decomposition in the distinguishable subsystems case (often referred to as the Schmidt decomposition in the context of quantum mechanics). It states that for any complex matrix $\psi\in\mathbb{C}^{d_{1},d_{2}}$ there exist unitary matrices $V\in\operatorname{U}(d_{1})$ and $W\in\operatorname{U}(d_{2})$ such that $V\psi W^{*}$ is real and diagonal. The correspondence is established by

  $$\ket{\phi}=V\otimes\overline{W}\ket{\psi}\iff\phi=V\psi W^{*}.$$

• •

  The Autonne–Takagi factorization in the bosonic case. It states that for any complex symmetric matrix $\psi\in\mathfrak{sym}(\mathbb{C},d)$, there is a unitary $V\in\operatorname{U}(d)$ such that $V\psi V^{\top}$ is real and diagonal. The correspondence is then given by

  $$\ket{\phi}=V\otimes V\ket{\psi}\iff\phi=V\psi V^{\top}.$$

• •

  The Hua factorization in the fermionic case. It states that for any complex skew-symmetric matrix $\psi\in\mathfrak{asym}(\mathbb{C},d)$, there is a unitary $V\in\operatorname{U}(d)$ such that $V\psi V^{\top}$ is real and quasi-diagonal in the following sense: if $d$ is even, then $V\psi V^{\top}$ is block diagonal with blocks of size $2\times 2$, if $d$ is odd, there is an additional block of size $1\times 1$ in the lower right corner. Note that the quasi-diagonal matrix is still skew-symmetric, and so the diagonal is zero. The correspondence to local unitary state transformations is as in the bosonic case.

Note that the Autonne–Takagi factorization and the Hua factorization are special cases of singular value decompositions, and hence the resulting (quasi/̄)diagonal matrix will have the singular values on its (quasi/̄)diagonal. In the first case we denote by $\Sigma\subset\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}$ the subspace of “real diagonal states” corresponding to the set of real diagonal matrices $\mathfrak{diag}(d_{1},d_{2},\mathbb{R})$ under the isomorphism (1). Clearly $\Sigma$ has dimension ${d_{\min}}:=\min(d_{1},d_{2})$. In the second case we write $\Sigma\subset\mathrm{Sym}^{2}(\mathbb{C}^{d})$ for the $d$-dimensional subspace corresponding to the real diagonal matrices $\mathfrak{diag}(d,\mathbb{R})$. Similarly, in the third case we write $\Xi\subset\bigwedge^{2}(\mathbb{C}^{d})$ for the $\lfloor d/2\rfloor$-dimensional subspace of states corresponding to the real (skew-symmetric) quasi-diagonal matrices $\mathfrak{qdiag}(d,\mathbb{R})$. We will use the following maps to send the singular values to their corresponding (quasi/̄)diagonal state:

A convenient shorthand notation is $\ket{\sigma}=\operatorname{diag}(\sigma)$ resp. $\ket{\xi}=\operatorname{qdiag}(\xi)$. We will always use the standard Euclidean inner product on $\mathbb{R}^{n}$, and on $\mathbb{C}^{n}$ we will use the real part $\operatorname{Re}(\braket{\cdot}{\cdot})$ of the standard inner product. Then, due to the inclusion of the factor $1/{\sqrt{2}}$ it holds that the maps above are $\mathbb{R}$-linear isometric isomorphisms. Furthermore we denote⁶⁶6The symbol ${\mathrel{\ooalign{$\hss\shortdownarrow$\hss\cr\hss\raisebox{2.0pt}{\scalebox{0.6}{$-$}}\hss}}}$ is a combination of $\shortdownarrow$ and $+$. by $\Sigma_{\mathrel{\ooalign{$\hss\shortdownarrow$\hss\cr\hss\raisebox{2.0pt}{\scalebox{0.6}{$-$}}\hss}}}\subset\Sigma$ the cone of states $\operatorname{diag}(\sigma)$ where the diagonal elements $(\sigma_{i})_{i=1}^{d_{\min}}$ are non-negative and arranged in non-increasing order, and analogously we write $\Xi_{\mathrel{\ooalign{$\hss\shortdownarrow$\hss\cr\hss\raisebox{2.0pt}{\scalebox{0.6}{$-$}}\hss}}}\subset\Xi$ for the quasi-diagonal states $\operatorname{qdiag}(\xi)$ where the $(\xi_{i})_{i=1}^{\lfloor d/2\rfloor}$ are non-negative and arranged in non-increasing order. The set $\Sigma_{\mathrel{\ooalign{$\hss\shortdownarrow$\hss\cr\hss\raisebox{2.0pt}{\scalebox{0.6}{$-$}}\hss}}}$ resp. $\Xi_{\mathrel{\ooalign{$\hss\shortdownarrow$\hss\cr\hss\raisebox{2.0pt}{\scalebox{0.6}{$-$}}\hss}}}$ is called the Weyl chamber.

Conversely to the diagonal embeddings we also define the following orthogonal projections:

More precisely these are the orthogonal projections on $\Sigma$ and $\Xi$ followed by $\operatorname{diag}^{-1}$ and $\operatorname{qdiag}^{-1}$ respectively.

Due to Assumptions I and II (resp. III), we can move arbitrarily quickly within the local unitary orbits of the system if we ignore the drift term [10, Prop. 2.7]. With the drift this is still approximately true. In the previous section we have shown that using local unitary transformations, we can always obtain a state of (quasi/̄)diagonal form which is completely determined by the singular values of the state. In particular, within the bilinear control systems (B) and (B’) two states are effectively equivalent if and only if they have the same singular values (up to order and sign). This strongly suggests that there should exist a “reduced” control system, defined on the singular values — or rather the Schmidt sphere (cf. Remark 2.3). This is indeed the case. The reduced control system is defined in greater generality in [20, Sec. 2.1] using symmetric Lie algebras, which unify many well-known matrix diagonalizations, such as the ones encountered in the previous section. No knowledge of symmetric Lie algebras is presupposed here, but the connections are expounded in Appendix A.

Let us briefly motivate the definition of the reduced control system. Let $\ket{\psi}$ be a solution to the full control system (B). Assume that the corresponding matrix $\psi$ can be diagonalized in a differentiable way as $\psi(t)=V(t)\tilde{\sigma}(t)W^{\top}(t)$, and that it is regular⁷⁷7We say that $\psi$ is regular if its singular values are distinct and non-zero.. Here $\tilde{\sigma}(t)$ is the diagonal matrix with diagonal elements $\sigma(t)$. Then by differentiating (cf. [19, Lem. 2.3]) we obtain that $\dot{\sigma}=-H_{V\otimes W}\sigma$ (and analogously $\dot{\sigma}=-H_{V\otimes V}^{s}\sigma$ or $\dot{\xi}=-H_{V\otimes V}^{a}\xi$ in the bosonic and fermionic cases) where

We call $H_{V\otimes W}$, $H_{V\otimes V}^{s}$ and $H_{V\otimes V}^{a}$ the induced vector fields. The collection of induced vector fields is denoted

Note that these are linear vector fields on $\mathbb{R}^{{d_{\min}}}$, $\mathbb{R}^{d}$ and $\mathbb{R}^{\lfloor d/2\rfloor}$ respectively, and hence they can be represented as matrices in the respective standard basis. We will later see that these are indeed skew-symmetric matrices and that the corresponding dynamics preserve the Schmidt sphere. The following proposition gives the explicit expressions.

We see immediately that if the drift Hamiltonian is local, then the induced vector fields vanish:

Finally we can define the reduced control systems:

As mentioned previously, and shown in Lemma 2.11 below, the induced vector fields preserve the Schmidt sphere, and hence we may define the reduced control systems directly on the Schmidt sphere.

The main result of [20] is the equivalence of the full bilinear control system (B) resp. (B’) and the reduced control system (R), resp. (${\sf R}^{s}$) or (${\sf R}^{a}$), proven in Theorems 3.8 and 3.14 therein. In our case this specializes to the following result.

First we need to define the quotient maps $\mathrm{sing}^{\mathrel{\ooalign{$\hss\shortdownarrow$\hss\cr\hss\raisebox{2.0pt}{\scalebox{0.6}{$-$}}\hss}}}:\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}\to\mathbb{R}^{{d_{\min}}}$ and $\mathrm{qsing}^{\mathrel{\ooalign{$\hss\shortdownarrow$\hss\cr\hss\raisebox{2.0pt}{\scalebox{0.6}{$-$}}\hss}}}:\bigwedge^{2}(\mathbb{C}^{d})\to\mathbb{R}^{\lfloor d/2\rfloor}$. Given a (possibly not normalized) vector $\ket{\psi}\in\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}$ (or $\mathrm{Sym}^{2}(\mathbb{C}^{d})$), the map $\mathrm{sing}^{\mathrel{\ooalign{$\hss\shortdownarrow$\hss\cr\hss\raisebox{2.0pt}{\scalebox{0.6}{$-$}}\hss}}}$ yields the singular values of the corresponding matrix $\psi\in\mathbb{C}^{d_{1},d_{2}}$, chosen non-negative and arranged in non-increasing order. Similarly, for $\ket{\xi}\in\bigwedge^{2}(\mathbb{C}^{d})$, the map $\mathrm{qsing}^{\mathrel{\ooalign{$\hss\shortdownarrow$\hss\cr\hss\raisebox{2.0pt}{\scalebox{0.6}{$-$}}\hss}}}$ yields the singular values of the skew-symmetric matrix $\xi$, except that we keep only one singular value of each pair and multiply it by $\sqrt{2}$ to keep the normalization. Note that when restricting the domain of $\mathrm{sing}^{\mathrel{\ooalign{$\hss\shortdownarrow$\hss\cr\hss\raisebox{2.0pt}{\scalebox{0.6}{$-$}}\hss}}}$ and $\mathrm{qsing}^{\mathrel{\ooalign{$\hss\shortdownarrow$\hss\cr\hss\raisebox{2.0pt}{\scalebox{0.6}{$-$}}\hss}}}$ to (normalized) quantum states, the image will lie in the respective Schmidt sphere, and even in the Weyl chamber $S_{\mathrel{\ooalign{$\hss\shortdownarrow$\hss\cr\hss\raisebox{2.0pt}{\scalebox{0.6}{$-$}}\hss}}}^{{d_{\min}}-1}$ resp. $S_{\mathrel{\ooalign{$\hss\shortdownarrow$\hss\cr\hss\raisebox{2.0pt}{\scalebox{0.6}{$-$}}\hss}}}^{d-1}$ and $S_{\mathrel{\ooalign{$\hss\shortdownarrow$\hss\cr\hss\raisebox{2.0pt}{\scalebox{0.6}{$-$}}\hss}}}^{\lfloor d/2\rfloor-1}$.

Recall that here and throughout the paper we use Assumptions I and II (resp. III), unless stated otherwise.

The Equivalence Theorem 2.10 shows that the full bilinear control system (B) resp. (B’) and the reduced control system (R), resp. (${\sf R}^{s}$) or (${\sf R}^{a}$), contain essentially the same information. Hence for every control theoretic notion, such as controllability and stabilizability, there is a specialized equivalence result, see [20, Sec. 4] for an overview. As a first consequence we obtain:

Let us also recall the equivalence of reachable sets here, which is arguably the most useful consequence. First we give the definitions of reachable sets in the reduced control system (R). The definitions for other control systems are entirely analogous. The reachable set of $\sigma_{0}$ at time $T$ is defined as

for any $T\geq 0$. By $\mathsf{reach}_{\ref{eq:reduced}}(\sigma_{0}):=\bigcup_{T\geq 0}\mathsf{reach}_{\ref{eq:reduced}}(\sigma_{0},T)$ we denote the all time reachable set of $\sigma_{0}$, and by $\mathsf{reach}_{\ref{eq:reduced}}(\sigma_{0},[0,T]):=\bigcup_{t\in[0,T]}\mathsf{reach}_{\ref{eq:reduced}}(\sigma_{0},t)$ we denote the reachable set of $\sigma_{0}$ up to time $T$.

The following result is an immediate consequence of the Equivalence Theorem 2.10 and [20, Prop. 4.3].

The Equivalence Theorem 2.10 guarantees the existence of an approximate lift, but its proof also provides a way to find corresponding control functions. Under some additional assumptions we can give an explicit formula for the controls of an exact lift, see [20, Prop. 3.10]. In particular this requires the solution to be smooth and regular, and the controls of the bilinear system (B) resp. (B’) to linearly span the corresponding Lie algebra. Before stating the result we define some notation.

The group of local unitary operations $\operatorname{U}_{\mathrm{loc}}(d_{1},d_{2})$ and its symmetric counterpart $\operatorname{U}_{\mathrm{loc}}^{s}(d)$ act on the state spaces $\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}$, $\mathrm{Sym}^{2}(\mathbb{C}^{d})$ and $\bigwedge^{2}(\mathbb{C}^{d})$ respectively. The corresponding infinitesimal action of the Lie algebras $\mathfrak{u}_{\mathrm{loc}}(d_{1},d_{2})$ and $\mathfrak{u}_{\mathrm{loc}}^{s}(d)$ can be determined as in Section 2.2 using the formula $(\mathrm{i}\mkern 1.0muE\otimes\mathds{1}+\mathds{1}\otimes\mathrm{i}\mkern 1.0muF)\ket{\psi}=\ket{\mathrm{i}\mkern 1.0mu(E\psi+\psi\overline{F})}$. In the special case where the state is regular and diagonal this function (more precisely its negative) gets a special name:

where $\tilde{\sigma}\in\mathfrak{diag}(d_{1},d_{2},\mathbb{R})$ and $\tilde{\xi}\in\mathfrak{qdiag}(d,\mathbb{R})$ denote the (quasi-)diagonal matrices corresponding to $\sigma$ and $\xi$. Although these maps are not bijective, by restricting the domain to the orthocomplement of the kernel and the codomain to the image, inverse maps can be defined. Indeed, this is nothing but the Moore–Penrose pseudoinverse. Explicit expressions are given in Lemmas A.7, A.10 and A.13.

To use these inverse maps we have to understand the images of the maps $\operatorname{ad}_{\sigma}^{d}$, $\operatorname{ad}_{\sigma}^{s}$, and $\operatorname{ad}_{\xi}^{a}$. It turns out that, for regular $\sigma$ resp. $\xi$, these images are exactly given by the orthocomplement of the diagonal subspaces $\Sigma,\Xi$. We denote the orthogonal projection on $\mathbb{C}^{d_{1}}\otimes\mathbb{C}^{d_{2}}$ with kernel $\Sigma$ by $\Pi_{\Sigma}^{\perp}$, and use the same notation on $\mathrm{Sym}^{2}(\mathbb{C}^{d})$. In the matrix picture, this map simple removes the real part of the diagonal elements of $\psi$. Similarly, on $\bigwedge^{2}(\mathbb{C}^{d})$, the orthogonal projection with kernel $\Xi$ is denoted $\Pi_{\Xi}^{\perp}$ and it removes the real part of the quasi-diagonal of $\psi$.

With these definitions [20, Prop. 3.10] can be specialized as follows: