Welcome to visit Communications in Theoretical Physics,
Quantum Physics and Quantum Information

Can imaginarity be broadcast via real operations?

  • Linshuai Zhang 1, 2 ,
  • Nan Li , 1, 2
Expand
  • 1Academy of Mathematics and Systems Science, University of Chinese Academy of Sciences, Beijing 100190, China
  • 2School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

Received date: 2024-07-02

  Revised date: 2024-08-03

  Accepted date: 2024-08-12

  Online published: 2024-09-20

Copyright

© 2024 Institute of Theoretical Physics CAS, Chinese Physical Society and IOP Publishing. All rights, including for text and data mining, AI training, and similar technologies, are reserved.

Abstract

Imaginarity has proven to be a valuable resource in various quantum information processing tasks. A natural question arises: can the imaginarity of quantum states be broadcast via real operations? In this work, we present explicit structures for nonreal states whose imaginarity can be broadcast and cloned. That is, for a nonreal state, its imaginarity can be cloned if and only if it is a direct sum of several maximally imaginary states under orthogonal transformation, and its imaginarity can be broadcast if and only if it is a direct sum of a real state and some nonreal qubit states which are mixtures of two orthogonal maximally imaginary states under orthogonal transformation. In particular, we show that for a nonreal pure state, its imaginarity cannot be broadcast unless it is a maximally imaginary state. Furthermore, we derive a trade-off relation on the imaginarity broadcasting of pure states in terms of the measure of irreversibility of quantum states concerning real operations and the geometric measure of imaginarity. In addition, we demonstrate that any faithful measure of imaginarity is not superadditive.

Cite this article

Linshuai Zhang , Nan Li . Can imaginarity be broadcast via real operations?[J]. Communications in Theoretical Physics, 2024 , 76(11) : 115104 . DOI: 10.1088/1572-9494/ad6de5

1. Introduction

One of the fundamental distinctions between quantum information and classical information is that quantum information cannot be broadcast [16]. Recall that a set of quantum states {ρj: j = 0, 1} on a Hilbert space ${ \mathcal H }$ can be broadcast [4], if there exists a quantum operation ${ \mathcal E }:D({ \mathcal H })\to D({{ \mathcal H }}^{a}\otimes {{ \mathcal H }}^{b})$ such that
$\begin{eqnarray}{\mathrm{tr}}_{a}{ \mathcal E }({\rho }_{j})={\rho }_{j},\quad {\mathrm{tr}}_{b}{ \mathcal E }({\rho }_{j})={\rho }_{j},\quad j=0,1.\end{eqnarray}$
Here, ${{ \mathcal H }}^{a}={{ \mathcal H }}^{b}={ \mathcal H }$ and $D({ \mathcal H })$ denotes the set of quantum states on a Hilbert space ${ \mathcal H }$. If the quantum operation ${ \mathcal E }$ further satisfies that
$\begin{eqnarray}{ \mathcal E }({\rho }_{j})={\rho }_{j}\otimes {\rho }_{j},\quad j=0,1,\end{eqnarray}$
then {ρj: j = 0, 1} can be cloned [4]. The no-broadcasting theorem tells us that the set of quantum states {ρj: j = 0, 1} can be broadcast if and only if the commutator [ρ0, ρ1] := ρ0ρ1ρ1ρ0 vanishes [4, 6]. The no-cloning theorem says that {ρj: j = 0, 1} can be cloned if and only if ρ0 = ρ1 or ρ0ρ1 = 0 [4].
The development of quantum information theory has expanded the study of cloning and broadcasting from quantum states to the information encoded within them, including total correlations, quantum correlations, entanglement, quantum steering, quantum Fisher information, coherence, asymmetry, and magic. For instance, it has been demonstrated that the total correlations of a bipartite state can be broadcast if and only if the state is classical-classical [7]. The quantum correlations of a bipartite state can be broadcast if and only if the state is classical-quantum [812]. The entanglement of bipartite states might be partially broadcast via local operations [1315]. The quantum steering of a bipartite state can be perfectly cloned if and only if the state is of zero discord [16]. The quantum Fisher information of quantum states cannot be cloned, while it might be broadcast even when the states do not commute with each other [17]. The coherence and asymmetry of quantum states cannot be broadcast [1821]. Broadcasting of any magic state via stabilizer operations is impossible [22].
In recent years, the resource theory of imaginarity has been established and imaginarity has proven to be an important resource in quantum information tasks both theoretically and experimentally [2349]. For example, Wu et al. have shown the necessity of imaginarity for distinguishing quantum states both theoretically and experimentally [24, 25]. They have proven that certain real bipartite states can be completely distinguished through local operations and classical communication, but if these operations are limited to real local measurements, it becomes impossible to distinguish these states with nonzero probability. Furthermore, they show that imaginarity can serve as a resource in optical experiments: under certain commonly assumed conditions in the experiments, the number of optical components required to perform real operations can be reduced by half compared to the number needed for general quantum operations.
Now a natural question arises: can the imaginarity of quantum states be broadcast via real operations? The main purpose of this work is to address this question. After introducing the explicit definitions of imaginarity broadcasting and cloning, we provide a complete description of quantum states whose imaginarity can be broadcast and cloned. Furthermore, to study imaginarity broadcasting from a quantitative perspective, we establish a trade-off relation on imaginarity broadcasting for pure states.
The remainder of this work is organized as follows. In section 2, we briefly review the resource theory of imaginarity. In section 3, we introduce the definitions of imaginarity cloning and broadcasting, and provide a complete characterization for nonreal states whose imaginarity can be broadcast and cloned. In addition, we prove that any faithful measure of imaginarity is not superadditive. In section 4, we establish a trade-off relation on imaginarity broadcasting for pure states. Finally, we conclude with a discussion in section 5. We present detailed proofs of our results in the appendices.

2. Preliminaries

In this section, we briefly review real states (free states), real operations (free operations), and measures of imaginarity in the resource theory of imaginarity.
Consider a quantum system described by a d-dimensional Hilbert space ${ \mathcal H }$ with a fixed orthonormal basis {∣j⟩: j = 0, 1, ⋯ ,d − 1}. A state ρ = ∑j,kjρk⟩∣j⟩⟨k∣ on ${ \mathcal H }$ is called a real state [23], if
$\begin{eqnarray}\langle j| \rho | k\rangle \in {\mathbb{R}},\quad j,k=0,1,\cdots ,d-1,\end{eqnarray}$
where ${\mathbb{R}}$ denotes the set of real numbers. In this case, $\overline{\rho }=\rho $ with $\overline{\rho }:={\sum }_{j,k}\overline{\langle j| \rho | k\rangle }| j\rangle \langle k| $. Otherwise, the state ρ is a nonreal state and has the resource of imaginarity. We will denote the set of real states by ${\mathscr{R}}({ \mathcal H })$.
Recall that any state ρ such that
$\begin{eqnarray}\rho =O| +{\rm{i}}\rangle \langle +{\rm{i}}| {O}^{{\rm{T}}}\end{eqnarray}$
is called a maximally imaginary state in Ref. [23]. Here, $| +{\rm{i}}\rangle =(| 0\rangle +{\rm{i}}| 1\rangle )/\sqrt{2}$ with ${\rm{i}}=\sqrt{-1}$ the imaginary unit, and O is an orthogonal operator on ${ \mathcal H }$ with respect to the basis {∣j⟩: j = 0, 1,…,d − 1}, i.e., $\overline{O}=O$ and OOT = 1, where 1 is the identity operator on ${ \mathcal H }$, and OT := ∑j,kkOj⟩∣j⟩⟨k∣ denotes the transpose of O with respect to the basis {∣j⟩: j = 0, 1,…,d − 1}.
A quantum operation ${ \mathcal E }:D({{ \mathcal H }}^{a})\to D({{ \mathcal H }}^{b})$ is called a real operation [23, 37], if it has an operator-sum representation
$\begin{eqnarray}{ \mathcal E }(\cdot )=\displaystyle \sum _{n}{K}_{n}(\cdot ){K}_{n}^{\dagger },\end{eqnarray}$
where {Kn} are real Kraus operators, i.e., ${\sum }_{n}{K}_{n}^{\dagger }{K}_{n}={{\bf{1}}}^{a}$ and $\langle {j}^{b}| {K}_{n}| {k}^{a}\rangle \in {\mathbb{R}}$ for any j, k, n. Here, ${{ \mathcal H }}^{\alpha }$ is a dα-dimensional Hilbert space with a fixed orthonormal basis {∣jα⟩: j = 0, 1, ⋯ ,dα − 1} for α = a, b, and 1a is the identity operator on ${{ \mathcal H }}^{a}$.
A functional $M:D({ \mathcal H })\to [0,\infty )$ is called a measure of imaginarity [23], if it satisfies the following conditions:
(M1) Non-negativity. M(ρ) ≥ 0 for any state $\rho \in D({ \mathcal H })$, and the equality holds if $\rho \in {\mathscr{R}}({ \mathcal H })$.
(M2) Monotonicity. $M({ \mathcal E }(\rho ))\leqslant M(\rho )$ for any state $\rho \in D({ \mathcal H })$ and any real operation ${ \mathcal E }$.
Furthermore, an imaginarity measure M is called faithful [23], if M(ρ) = 0 ensures that $\rho \in {\mathscr{R}}({ \mathcal H })$.
In addition, similar to the resource theory of coherence [5052], one may require an imaginarity measure to meet the following conditions [26]:
(M3) Strong monotonicity. For any state $\rho \in D({ \mathcal H })$ and any real operation ${ \mathcal E }$ with real Kraus operators {Kn}, ∑npnM(ρn) ≤ M(ρ), where pn = $\mathrm{tr}({K}_{n}\rho {K}_{n}^{\dagger })$, ${\rho }_{n}={K}_{n}\rho {K}_{n}^{\dagger }/{p}_{n}$.
(M4) Convexity. M(∑jpjρj) ≤ ∑jpjM(ρj) for any probability distribution {pj} and any states {ρj}.
(M5) Additivity for direct-sum states.
$\begin{eqnarray}M(p{\rho }^{a}\oplus (1-p){\rho }^{b})={pM}({\rho }^{a})+(1-p)M({\rho }^{b}),\end{eqnarray}$
for any p ∈ [0, 1] and any states ${\rho }^{a}\in D({{ \mathcal H }}^{a})$, ${\rho }^{b}\in D({{ \mathcal H }}^{b})$. $p{\rho }^{a}\oplus (1-p){\rho }^{b}\in D({{ \mathcal H }}^{a}\oplus {{ \mathcal H }}^{b})$ is of the direct-sum form with respect to the given bases {∣ja⟩: j = 0, 1, ⋯ ,da − 1} on ${{ \mathcal H }}^{a}$ and {∣kb⟩: k = 0, 1, ⋯ ,db − 1} on ${{ \mathcal H }}^{b}$.
It has been shown that (M3) and (M4) are equivalent to (M2) and (M5) [26].
So far, many imaginarity measures have been proposed, such as the robustness of imaginarity [23, 25], the trace norm of imaginarity [23], the geometric measure of imaginarity [25, 37], the relative entropy of imaginarity [26], the weight of imaginarity [26], the imaginarity measure based on the fidelity for Gaussian states [38], the Tsallis relative entropy of imaginarity [39], the l1 norm of imaginarity [41], and the α-z-Rényi relative entropy of imaginarity [43].

3. Imaginarity cloning and broadcasting

In this section, after introducing the definitions of imaginary cloning and broadcasting, we provide a complete characterization for nonreal states whose imaginarity can be cloned and broadcast. Finally, we prove that any faithful imaginarity measure is not superadditive.

Given a basis $\{| j\rangle :j=0,1,\cdots ,d-1\}$ for a d-dimensional Hilbert space ${ \mathcal H }$, we say that the imaginarity of a nonreal state ρ on ${ \mathcal H }$ can be broadcast if there exists a real operation ${ \mathcal E }:D({ \mathcal H })\to D({{ \mathcal H }}^{a}\otimes {{ \mathcal H }}^{b})$ such that

$\begin{eqnarray}{\mathrm{tr}}_{b}{ \mathcal E }(\rho )=\rho ,\quad {\mathrm{tr}}_{a}{ \mathcal E }(\rho )=\rho .\end{eqnarray}$
Here, we put ${{ \mathcal H }}^{a}={{ \mathcal H }}^{b}={ \mathcal H }$ and let $\{| j\rangle :j=0,1,\cdots ,d-1\}$ denote both the basis for ${{ \mathcal H }}^{\alpha }$, $\alpha =a,b$. Furthermore, if ${ \mathcal E }(\rho )$ not only satisfies equation (7) but also is uncorrelated, i.e.,
$\begin{eqnarray}{ \mathcal E }(\rho )=\rho \otimes \rho ,\end{eqnarray}$
we say that the imaginarity of ρ can be cloned.

Obviously, if the imaginarity of ρ can be cloned, then its imaginarity can be broadcast, but the converse is not true.
In fact, any maximally imaginary state defined by equation (4) can always be cloned because it can be transformed into any quantum state on arbitrary dimensional Hilbert space by real operations [23]. Here, we provide an explicit structure for the real operation that can clone the maximally imaginary state ∣ + i⟩. Indeed, consider the quantum operation $\tilde{{ \mathcal E }}:D({ \mathcal H })\to D({{ \mathcal H }}^{a}\otimes {{ \mathcal H }}^{b})$ defined by
$\begin{eqnarray}\tilde{{ \mathcal E }}(\rho ):= O(\rho \otimes | 0\rangle \langle 0| ){O}^{{\rm{T}}}\end{eqnarray}$
for any state $\rho \in D({ \mathcal H })$. Here ${{ \mathcal H }}^{a}={{ \mathcal H }}^{b}={ \mathcal H }\,=\mathrm{span}\{| 0\rangle ,| 1\rangle \}$ and O is an operator on ${{ \mathcal H }}^{a}\otimes {{ \mathcal H }}^{b}$ defined by
$\begin{eqnarray}\begin{array}{rcl}O & := & | +{\rm{i}}\rangle \langle +{\rm{i}}| \otimes | +{\rm{i}}\rangle \langle 0| +| -{\rm{i}}\rangle \langle -{\rm{i}}| \otimes | -{\rm{i}}\rangle \langle 0| \\ & & +\,| -{\rm{i}}\rangle \langle +{\rm{i}}| \otimes | +{\rm{i}}\rangle \langle 1| +| +{\rm{i}}\rangle \langle -{\rm{i}}| \otimes | -{\rm{i}}\rangle \langle 1| ,\end{array}\end{eqnarray}$
with $| \pm {\rm{i}}\rangle =(| 0\rangle \pm {\rm{i}}| 1\rangle )/\sqrt{2}$. It is easy to verify that O is an orthogonal operator. Thus, $\tilde{{ \mathcal E }}$ is a real operation and satisfies
$\begin{eqnarray}\tilde{{ \mathcal E }}(| +{\rm{i}}\rangle \langle +{\rm{i}}| )=| +{\rm{i}}\rangle \langle +{\rm{i}}| \otimes | +{\rm{i}}\rangle \langle +{\rm{i}}| ,\end{eqnarray}$
which implies that the imaginarity of the maximally imaginary state ∣ + i⟩ can be cloned via the real operation $\tilde{{ \mathcal E }}$. Hence, any maximally imaginary state defined by equation (4) can be cloned via a real operation.
Now, it is natural to ask: besides the set of maximally imaginary states, can the imaginarity of other nonreal states be cloned via real operations? The following theorem provides a surprising result: apart from maximally imaginary states and the direct-sum states of several maximally imaginary states under an orthogonal transformation, the imaginarity of other nonreal states cannot be cloned.

Given a basis $\{| j\rangle :j=0,1,\cdots ,d-1\}$ for a d-dimensional Hilbert space ${ \mathcal H }$, the following facts about a nonreal quantum state $\rho \in D({ \mathcal H })$ are equivalent:

(1) The imaginarity of ρ can be cloned via a real operation.
(2) $\{\rho ,\bar{\rho }\}$ can be cloned via a quantum operation.
(3) $\rho \bar{\rho }=0$.
(4) There exists an orthogonal operator O such that
$\begin{eqnarray}O\rho {O}^{{\rm{T}}}=\underset{k=1}{\overset{r}{\bigoplus }}{\alpha }_{k}| +{\rm{i}}\rangle \langle +{\rm{i}}{| }_{k}.\end{eqnarray}$
Here, αk > 0, ${\sum }_{k=1}^{r}{\alpha }_{k}=1$ with r ≤ (d − 1)/2, and ∣ ± i⟩⟨ ± i∣k denote the density operators of $| \pm {\rm{i}}{\rangle }_{k}:=(| 2k-2\rangle \,\pm {\rm{i}}| 2k-1\rangle )/\sqrt{2}$ for k = 1, ⋯ , r.
See appendix A for the proof.
Let ${{ \mathcal H }}_{k}:=\mathrm{span}\{| 2k-2\rangle ,| 2k-1\rangle \}$ for k = 1, ⋯ ,r, and ${{ \mathcal H }}_{0}:=\mathrm{span}\{| j\rangle :j=2r,\cdots ,d-1\}$. Then ${{ \mathcal H }}_{k}=\mathrm{span}\{| +{\rm{i}}{\rangle }_{k},| -{\rm{i}}{\rangle }_{k}\}$ and ${ \mathcal H }={\bigoplus }_{k=0}^{r}{{ \mathcal H }}_{k}$. We remark that due to the isomorphism between the internal and external direct sums, we do not distinguish between the sum and the direct sum, and thus consider ∣ ± i⟩k as a state on both ${{ \mathcal H }}_{k}$ and ${ \mathcal H }$.
According to theorem 1, if we are limited to the set of qubit states or pure states, then maximally imaginary states are the only nonreal states whose imaginarity can be cloned.
The following theorem completely characterizes nonreal states whose imaginarity can be broadcast. That is, the imaginarity of a nonreal state can be broadcast if and only if it can be transformed by an orthogonal operator into a direct sum of some real state and several nonreal qubit states with their eigenvectors being maximally imaginary states. Note that we follow the notations in theorem 1.

Given a basis $\{| j\rangle :j=0,1,\cdots ,d-1\}$ for a d-dimensional Hilbert space ${ \mathcal H }$, the following facts about a nonreal quantum state $\rho \in D({ \mathcal H })$ are equivalent:

(1) The imaginarity of ρ can be broadcast via a real operation.
(2) $\{\rho ,\bar{\rho }\}$ can be broadcast via a quantum operation.
(3) $[\rho ,\bar{\rho }]=0$.
(4) There exists an orthogonal operator O such that
$\begin{eqnarray}O\rho {O}^{{\rm{T}}}=\underset{k=1}{\overset{r}{\bigoplus }}\left({\alpha }_{k}| +{\rm{i}}\rangle \langle +{\rm{i}}{| }_{k}+{\beta }_{k}| -{\rm{i}}\rangle \langle -{\rm{i}}{| }_{k}\right)\bigoplus {p}_{0}{\rho }_{0}.\end{eqnarray}$
Here αk, βk, p0 ≥ 0, αk + βk > 0, ${\sum }_{k=1}^{r}({\alpha }_{k}+{\beta }_{k})+{p}_{0}\,=1$, and ρ0 is a real state on ${{ \mathcal H }}_{0}$.
See appendix B for the proof.
According to theorem 2, if we are limited to the set of qubit states, the only nonreal states whose imaginarity can be broadcast are the states such that
$\begin{eqnarray}\begin{array}{rcl}O\rho {O}^{{\rm{T}}} & = & p| +{\rm{i}}\rangle \langle +{\rm{i}}| +(1-p)| -{\rm{i}}\rangle \langle -{\rm{i}}| \\ & = & \displaystyle \frac{1+(2p-1){\sigma }_{y}}{2},\end{array}\end{eqnarray}$
for 0 ≤ p ≤ 1, p ≠ 1/2, and some orthorgonal operator O. Here, σy is the second Pauli operator. When limited to the set of pure states, maximally imaginary states are the only nonreal states whose imaginarity can be broadcast.
Finally, we emphasize that if there is a faithful imaginarity measure M satisfying the superadditivity, i.e.,
$\begin{eqnarray}M({\rho }^{{ab}})\geqslant M({\rho }^{a})+M({\rho }^{b})\end{eqnarray}$
for any state ${\rho }^{{ab}}\in D({{ \mathcal H }}^{a}\otimes {{ \mathcal H }}^{b})$ with ${\rho }^{a}:={\mathrm{tr}}_{b}{\rho }^{{ab}}\in D({{ \mathcal H }}^{a})$ and ${\rho }^{b}:={\mathrm{tr}}_{a}{\rho }^{{ab}}\in D({{ \mathcal H }}^{b})$, then by contradiction it can be easily derived that imaginarity broadcasting of any nonreal state is impossible. Suppose that there exists a nonreal state $\rho \in D({ \mathcal H })$ whose imaginarity can be broadcast. From definition 1, we know that there exists a real operation ${ \mathcal E }:D({ \mathcal H })\to D({{ \mathcal H }}^{a}\otimes {{ \mathcal H }}^{b})$ such that ${\tilde{\rho }}^{a}={\tilde{\rho }}^{b}=\rho $ with ${\tilde{\rho }}^{{ab}}:={ \mathcal E }(\rho )$. Then by the monotonicity and the superadditivity of M, we have
$\begin{eqnarray}M(\rho )\geqslant M({ \mathcal E }(\rho ))\geqslant M({\tilde{\rho }}^{a})+M({\tilde{\rho }}^{b})=2M(\rho ),\end{eqnarray}$
which implies that M(ρ) = 0. By the faithfulness of M, we get that $\rho \in {\mathscr{R}}({ \mathcal H })$, which contradicts the assumption.
From the above discussion, we conclude that imaginarity broadcasting of any nonreal state is impossible if there exists a faithful imaginarity measure satisfying the superadditivity. However, according to theorems 1 and 2, the maximally imaginary state can be broadcast and cloned, which implies the following theorem.

Any faithful imaginarity measure is not superadditive.

4. Trade-off relation

In this section, we study the imaginarity broadcasting of nonreal states from a quantitative perspective. Consider a quantum state ρ on a d-dimensional Hilbert space ${ \mathcal H }$ with a fixed orthonormal basis {∣j⟩: j = 0, 1, ⋯ ,d − 1} and a real operation ${ \mathcal E }:D({ \mathcal H })\to D({{ \mathcal H }}^{a}\otimes {{ \mathcal H }}^{b})$ with ${{ \mathcal H }}^{a}={{ \mathcal H }}^{b}={ \mathcal H }$ as before. We will establish a trade-off relation for pure states between the degree of irreversibility $\mathrm{irrev}(\rho ,{\tilde{\rho }}^{a})$ in the state transformation $\rho \to {\tilde{\rho }}^{a}:= {\mathrm{tr}}_{b}{ \mathcal E }(\rho )$ and the amount of imaginarity contained in the state ${\tilde{\rho }}^{b}:= {\mathrm{tr}}_{a}{ \mathcal E }(\rho )$ quantified by the geometric measure of imaginarity ${M}_{{\rm{g}}}({\tilde{\rho }}^{b})$.
Firstly, we introduce a measure of the irreversibility of the state transformation $\rho \to {\tilde{\rho }}^{a}:={\mathrm{tr}}_{b}{ \mathcal E }(\rho )$ as
$\begin{eqnarray}\mathrm{irrev}(\rho ,{\tilde{\rho }}^{a}):=1-F({\tilde{\rho }}^{a}\to \rho ),\end{eqnarray}$
where $F{({\tilde{\rho }}^{a}\to \rho ):={\max }_{{{ \mathcal E }}^{a}}F(\rho ,{{ \mathcal E }}^{a}({\tilde{\rho }}^{a}))}^{2}$ is the maximal transformation fidelity introduced in Ref. [25] with the maximization taken over all real operations ${{ \mathcal E }}^{a}:D({{ \mathcal H }}^{a})\to D({ \mathcal H })$, and $F(\rho ,\sigma ):=\mathrm{tr}\sqrt{\sqrt{\rho }\sigma \sqrt{\rho }}$ is the fidelity between states ρ and σ [53]. From the definition and the properties of the fidelity, we can easily obtain that $0\leqslant \mathrm{irrev}(\rho ,{\tilde{\rho }}^{a})\leqslant 1$, and $\mathrm{irrev}(\rho ,{\tilde{\rho }}^{a})=0$ if and only if there exists a real operation ${{ \mathcal E }}^{a}$ such that ${{ \mathcal E }}^{a}({\tilde{\rho }}^{a})=\rho $. In this sense, this measure captures the extent to which the original state ρ cannot be perfectly recovered from the transformed state ${\tilde{\rho }}^{a}$ via real operations, reflecting the inherent irreversibility of the state transformation $\rho \to {\tilde{\rho }}^{a}$.
Next, we choose the geometric measure of imaginarity to quantify the amount of imaginarity contained in a quantum state. Recall that the geometric measure of imaginarity of a state ρ is defined as [25, 37]
$\begin{eqnarray}{M}_{{\rm{g}}}(\rho ):=1-\mathop{\max }\limits_{\sigma \in {\mathscr{R}}({ \mathcal H })}F{\left(\rho ,\sigma \right)}^{2},\end{eqnarray}$
and it is shown that
$\begin{eqnarray}{M}_{{\rm{g}}}(\rho )=\displaystyle \frac{1-F(\rho ,\bar{\rho })}{2}.\end{eqnarray}$
To establish our trade-off relation, we prove some useful conclusions concerning the geometric measure of imaginarity in the following lemma.

${M}_{{\rm{g}}}$ satisfies the following properties.

(i) For any quantum state ρ on ${ \mathcal H }$,
$\begin{eqnarray}{M}_{{\rm{g}}}(\rho )\leqslant \displaystyle \frac{1}{2},\end{eqnarray}$
and the equality holds if and only if $\rho \bar{\rho }=0$.
(ii) For any quantum states ρa on ${{ \mathcal H }}^{a}$ and ρb on ${{ \mathcal H }}^{b}$,
$\begin{eqnarray}{M}_{{\rm{g}}}({\rho }^{a}\otimes {\rho }^{b})={M}_{{\rm{g}}}({\rho }^{a})+{M}_{{\rm{g}}}({\rho }^{b})-2{M}_{{\rm{g}}}({\rho }^{a}){M}_{{\rm{g}}}({\rho }^{b}).\end{eqnarray}$
(iii) For any quantum states ρ and σ on ${ \mathcal H }$,
$\begin{eqnarray}| {M}_{{\rm{g}}}(\rho )-{M}_{{\rm{g}}}(\sigma )| \leqslant 2\sqrt{1-F(\rho ,\sigma )}.\end{eqnarray}$
See appendix C for the proof.
Note that combining theorem 1 and property (i) of lemma 1 implies that the quantum states that maximize the geometric measure of imaginarity are precisely those nonreal states whose imaginarity can be cloned via real operations.
With these preparations, we present our trade-off relation in the following theorem.

Let $\rho =| \psi \rangle \langle \psi | $ be a pure state on ${ \mathcal H }$, ${ \mathcal E }:D({ \mathcal H })\to D({{ \mathcal H }}^{a}\otimes {{ \mathcal H }}^{b})$ be a real operation, ${\tilde{\rho }}^{a}:={\mathrm{tr}}_{b}{ \mathcal E }(\rho )$, and ${\tilde{\rho }}^{b}:={\mathrm{tr}}_{a}{ \mathcal E }(\rho )$, then

$\begin{eqnarray}{M}_{{\rm{g}}}({\tilde{\rho }}^{b})\left(1-2{M}_{{\rm{g}}}(\rho )\right)\leqslant 2\sqrt{\mathrm{irrev}(\rho ,{\tilde{\rho }}^{a})}.\end{eqnarray}$

See appendix D for the proof.
It should be emphasized that from the trade-off relation, it is easy to get that for pure states, maximally imaginary states are the only nonreal states whose imaginarity can be cloned. In fact, we can deduce a stronger conclusion: for a nonreal pure state ρ = ∣ψ⟩⟨ψ∣ on ${ \mathcal H }$ which is not the maximally imaginary state, there does not exist a real operation ${ \mathcal E }:D({ \mathcal H })\to D({{ \mathcal H }}^{a}\otimes {{ \mathcal H }}^{b})$ such that
$\begin{eqnarray}{\tilde{\rho }}^{a}=\rho ,\quad {\tilde{\rho }}^{b}\rlap{/}{\in }{\mathscr{R}}({{ \mathcal H }}^{b}).\end{eqnarray}$
Indeed, if there exists a real operation ${ \mathcal E }:D({ \mathcal H })\to D({{ \mathcal H }}^{a}\otimes {{ \mathcal H }}^{b})$ satisfying equation (24), then ρ can be perfectly recovered from ${\tilde{\rho }}^{a}$ via the identity channel ${ \mathcal I }$ on ${ \mathcal H }$ and thus $\mathrm{irrev}(\rho ,{\tilde{\rho }}^{a})=0$. On the other hand, since ρ is not the maximally imaginary state, and thus Mg(ρ) ≠ 1/2, from theorem 4 it follows that ${M}_{{\rm{g}}}({\tilde{\rho }}^{b})=0$. Hence, using the faithfulness of Mg, we get that ${\tilde{\rho }}^{b}$ must be a real state which contradicts the second equality in equation (24).
In the following, we provide an example to illustrate the inequality (23) in theorem 4. Consider a qubit system described by ${ \mathcal H }$ with the given basis {∣0⟩, ∣1⟩}. Let ρθ = ∣ψθ⟩⟨ψθ∣ with
$\begin{eqnarray*}| {\psi }_{\theta }\rangle =\cos \displaystyle \frac{\theta }{2}| 0\rangle +\mathrm{isin}\displaystyle \frac{\theta }{2}| 1\rangle \end{eqnarray*}$
and θ ∈ [0, π]. Performing the real operation $\tilde{{ \mathcal E }}:D({ \mathcal H })\to D({ \mathcal H }\otimes { \mathcal H })$ defined by equation (9) on ρθ, we get that $\tilde{{ \mathcal E }}({\rho }_{\theta })=| {\tilde{\psi }}_{\theta }\rangle \langle {\tilde{\psi }}_{\theta }| $, where
$\begin{eqnarray*}| {\tilde{\psi }}_{\theta }\rangle =\displaystyle \frac{\cos \tfrac{\theta }{2}+\sin \tfrac{\theta }{2}}{\sqrt{2}}| +{\rm{i}}\rangle | +{\rm{i}}\rangle +\displaystyle \frac{\cos \tfrac{\theta }{2}-\sin \tfrac{\theta }{2}}{\sqrt{2}}| -{\rm{i}}\rangle | -{\rm{i}}\rangle .\end{eqnarray*}$
Direct calculations show that
$\begin{eqnarray*}{\tilde{\rho }}_{\theta }^{a}={\tilde{\rho }}_{\theta }^{b}=\displaystyle \frac{1+\sin \theta }{2}| +{\rm{i}}\rangle \langle +{\rm{i}}| +\displaystyle \frac{1-\sin \theta }{2}| -{\rm{i}}\rangle \langle -{\rm{i}}| .\end{eqnarray*}$
From equation (19), we have
$\begin{eqnarray}{M}_{{\rm{g}}}({\rho }_{\theta })={M}_{{\rm{g}}}({\tilde{\rho }}_{\theta }^{a})={M}_{{\rm{g}}}({\tilde{\rho }}_{\theta }^{b})=\displaystyle \frac{1-| \cos \theta | }{2}.\end{eqnarray}$
To calculate $\mathrm{irrev}({\rho }_{\theta },{\tilde{\rho }}_{\theta }^{a})=1-F({\tilde{\rho }}_{\theta }^{a}\to {\rho }_{\theta })$, applying theorem 7 in Ref. [37], we know that the numerical result of $F({\tilde{\rho }}_{\theta }^{a}\to {\rho }_{\theta })$ can be obtained from the following semidefinite program:
$\begin{eqnarray}\begin{array}{rcl}F({\tilde{\rho }}_{\theta }^{a}\to {\rho }_{\theta }) & = & \max \ \mathrm{tr}({{\rm{\Sigma }}}_{{\rm{\Lambda }}}({\left({\tilde{\rho }}_{\theta }^{a}\right)}^{{\rm{T}}}\otimes {\rho }_{\theta }))\\ & & {\rm{s}}.{\rm{t}}.\,{{\rm{\Sigma }}}_{{\rm{\Lambda }}}\geqslant 0,\\ & & \,{\mathrm{tr}}_{b}{{\rm{\Sigma }}}_{{\rm{\Lambda }}}\leqslant {\bf{1}},\\ & & \,{{\rm{\Sigma }}}_{{\rm{\Lambda }}}^{{\rm{T}}}={{\rm{\Sigma }}}_{{\rm{\Lambda }},}\\ & & \,\mathrm{tr}({{\rm{\Sigma }}}_{{\rm{\Lambda }}}({\left({\tilde{\rho }}_{\theta }^{a}\right)}^{{\rm{T}}}\otimes {\bf{1}}))=1,\end{array}\end{eqnarray}$
Based on the above analysis, we plot the graphs of ${M}_{{\rm{g}}}({\tilde{\rho }}_{\theta }^{b})\left(1-2{M}_{{\rm{g}}}({\rho }_{\theta })\right)$ and $2\sqrt{\mathrm{irrev}({\rho }_{\theta },{\tilde{\rho }}_{\theta }^{a})}$ as functions of θ in figure 1, from which we can see that $2\sqrt{\mathrm{irrev}({\rho }_{\theta },{\tilde{\rho }}_{\theta }^{a})}$ is lower bound by ${M}_{{\rm{g}}}({\tilde{\rho }}_{\theta }^{b})\left(1-2{M}_{{\rm{g}}}({\rho }_{\theta })\right)$. In particular, when θ = 0, π/2, and π, the bound is tight. For both the cases of θ = 0 (∣ψθ⟩ = ∣0⟩) and θ = π (∣ψθ⟩ = ∣1⟩), ∣ψθ⟩ are real, which implies that ${\tilde{\rho }}_{\theta }^{b}$ are real, i.e., ${M}_{{\rm{g}}}({\tilde{\rho }}_{\theta }^{b})=0$. On the other hand, for the real state ∣ψθ⟩, $\mathrm{irrev}({\rho }_{\theta },{\tilde{\rho }}_{\theta }^{a})=0$. Thus, both sides of the inequality (23) are zero. For the case of θ = π/2, ∣ψθ⟩ = ∣ + i⟩ is a maximally imaginary state, thus Mg(ρθ) = 1/2. On the other hand, since it can be broadcast via a real operation, we have that $\mathrm{irrev}({\rho }_{\theta },{\tilde{\rho }}_{\theta }^{a})=0$. So, in this case, both sides of the inequality (23) are also zero.
Figure 1. Graphs of ${M}_{{\rm{g}}}({\tilde{\rho }}_{\theta }^{b})\left(1-2{M}_{{\rm{g}}}({\rho }_{\theta })\right)$ (blue dashed line) and $\mathrm{irrev}({\rho }_{\theta },{\tilde{\rho }}_{\theta }^{a})$ (red solid line) versus θ.

5. Discussion

In this work, we have introduced the definitions of imaginarity cloning and broadcasting, providing a complete characterization of nonreal states whose imaginarity can be cloned and broadcast. We have shown that the imaginarity of a nonreal state can be cloned if and only if it is a direct sum of maximally imaginary states under an orthogonal transformation, and that its imaginarity can be broadcast if and only if it is a direct sum of some real state and several nonreal qubit states whose eigenvectors are composed by maximally imaginary states under an orthogonal transformation. We have demonstrated that any faithful imaginarity measure is not superadditive. Additionally, we have introduced the definition of the irreversibility in the state transformation and established several properties of the geometric measure of imaginarity. In particular, a quantum state ρ maximizes the geometric measure of imaginarity if and only if its imaginarity can be cloned. Finally, we have established a trade-off relation based on the irreversibility of the state transformation and the geometric measure of imaginarity.
An interesting future direction for research would be to explore the weaker form of imaginarity cloning and broadcasting. For example, there are two naturally weaker forms of imaginarity cloning and broadcasting:
(i) We say that the imaginarity resource of a nonreal state ρ on ${ \mathcal H }$ can be weakly broadcast, if there exists a real operation ${ \mathcal E }:D({ \mathcal H })\to D({{ \mathcal H }}^{a}\otimes {{ \mathcal H }}^{b})$ such that
$\begin{eqnarray}M({\mathrm{tr}}_{a}{ \mathcal E }(\rho ))=M({\mathrm{tr}}_{b}{ \mathcal E }(\rho ))=M(\rho ),\end{eqnarray}$
where M is a faithful imaginarity measure. Furthermore, we say that the imaginarity of ρ can be weakly cloned if ${ \mathcal E }(\rho )$ not only satisfies equation (27) but also is a product state.
(ii) We say that the imaginarity of a nonreal state ρ on ${ \mathcal H }$ can be weakly broadcast, if there exists a real operation ${ \mathcal E }:D({ \mathcal H })\to D({{ \mathcal H }}^{a}\otimes {{ \mathcal H }}^{b})$ such that
$\begin{eqnarray}{\mathrm{tr}}_{b}{ \mathcal E }(\rho )=\rho ,\quad {\mathrm{tr}}_{a}{ \mathcal E }(\rho )\notin {\mathscr{R}}({{ \mathcal H }}^{b}).\end{eqnarray}$
Furthermore, we say that the imaginarity of ρ can be weakly cloned if ${ \mathcal E }(\rho )$ not only satisfies equation (28) but also is a product state.
Another future direction is to extend the trade-off relationship we have established from pure states to mixed states. It is worth further investigating whether a similar trade-off relation can be established for mixed states.

Appendix A. Proof of theorem 1

(1) ⇒ (2). Suppose that the imaginarity of ρ can be cloned. Then there exists a real operation ${ \mathcal E }:D({ \mathcal H })\to D({{ \mathcal H }}^{a}\otimes {{ \mathcal H }}^{b})$ such that
$\begin{eqnarray}{ \mathcal E }(\rho )=\rho \otimes \rho .\end{eqnarray}$
Combining equation (A1) and the fact that ${ \mathcal E }$ is a real operation, we have
$\begin{eqnarray}{ \mathcal E }(\bar{\rho })=\bar{\rho }\otimes \bar{\rho }.\end{eqnarray}$
Hence, $\{\rho ,\bar{\rho }\}$ can be cloned via the real operation ${ \mathcal E }$.
(2) ⇒ (1). Suppose that $\{\rho ,\bar{\rho }\}$ can be cloned. Then there exists a quantum operation ${ \mathcal E }:D({ \mathcal H })\to D({{ \mathcal H }}^{a}\otimes {{ \mathcal H }}^{b})$ such that
$\begin{eqnarray}{ \mathcal E }(\rho )=\rho \otimes \rho ,\quad { \mathcal E }(\bar{\rho })=\bar{\rho }\otimes \bar{\rho }.\end{eqnarray}$
Consider the quantum operation $\tilde{{ \mathcal E }}:D({ \mathcal H })\to D({{ \mathcal H }}^{a}\otimes {{ \mathcal H }}^{b})$ defined by
$\begin{eqnarray}\tilde{{ \mathcal E }}(\sigma ):=\displaystyle \frac{1}{2}{ \mathcal E }(\sigma )+\displaystyle \frac{1}{2}\overline{{ \mathcal E }(\bar{\sigma })}\end{eqnarray}$
for any state $\sigma \in D({ \mathcal H })$. Let {Kn} be a set of Kraus operators for ${ \mathcal E }$, i.e., ${ \mathcal E }(\cdot )={\sum }_{n}{K}_{n}(\cdot ){K}_{n}^{\dagger }$, ${\sum }_{n}{K}_{n}^{\dagger }{K}_{n}={\bf{1}}$. Then $\tilde{{ \mathcal E }}$ defined by equation (A4) can be expressed as
$\begin{eqnarray}\tilde{{ \mathcal E }}(\sigma )=\displaystyle \frac{1}{2}\displaystyle \sum _{n}{K}_{n}\sigma {K}_{n}^{\dagger }+\displaystyle \frac{1}{2}\displaystyle \sum _{n}{\overline{K}}_{n}\sigma {\overline{K}}_{n}^{\dagger }\end{eqnarray}$
for any state $\sigma \in D({ \mathcal H })$. By lemma 1 in Ref. [37], we know that $\tilde{{ \mathcal E }}$ has real Kraus operators given by $\{({K}_{n}+{\overline{K}}_{n})/2,{\rm{i}}({K}_{n}-{\overline{K}}_{n})/2\}$, which implies that $\tilde{{ \mathcal E }}$ is a real operation. Furthermore, combining equations (A3) and (A4), one can easily check that $\tilde{{ \mathcal E }}$ satisfies
$\begin{eqnarray}\tilde{{ \mathcal E }}(\rho )=\rho \otimes \rho .\end{eqnarray}$
Hence, the imaginarity of ρ can be cloned via the real operation $\tilde{{ \mathcal E }}$.
(2) ⇔ (3). The equivalence between (2) and (3) can be directly obtained from the no-cloning theorem [4].
(3) ⇒ (4). Suppose that a nonreal state satisfies that
$\begin{eqnarray}\rho \bar{\rho }=0.\end{eqnarray}$
Let $\mathrm{Re}(\rho ):=(\rho +\bar{\rho })/2$ and $\mathrm{Im}(\rho )\,:=\,-{\rm{i}}(\rho -\bar{\rho })/2$. Then
$\begin{eqnarray}\rho =\mathrm{Re}(\rho )+\mathrm{iIm}(\rho ),\end{eqnarray}$
and thus
$\begin{eqnarray}\rho \bar{\rho }=\mathrm{Re}{\left(\rho \right)}^{2}+\mathrm{Im}{\left(\rho \right)}^{2}+{\rm{i}}\left(\mathrm{Im}(\rho )\mathrm{Re}(\rho )-\mathrm{Re}(\rho )\mathrm{Im}(\rho )\right).\end{eqnarray}$
From equations (A7) and (A9), we have
$\begin{eqnarray}\mathrm{Re}{\left(\rho \right)}^{2}={\left(\mathrm{iIm}(\rho )\right)}^{2}.\end{eqnarray}$
Furthermore, by corollary 2.2.11 in Ref. [54], there exists an orthogonal operator O such that
$\begin{eqnarray}\begin{array}{rcl}O(\mathrm{iIm}(\rho )){O}^{{\rm{T}}} & = & \underset{k=1}{\overset{r}{\bigoplus }}{\lambda }_{k}\left(\begin{array}{cc}0 & -{\rm{i}}\\ {\rm{i}} & 0\end{array}\right)\bigoplus {{\bf{0}}}_{d-2r}\\ & = & \displaystyle \sum _{k=1}^{r}{\lambda }_{k}\left(| +{\rm{i}}\rangle \langle +{\rm{i}}{| }_{k}-| -{\rm{i}}\rangle \langle -{\rm{i}}{| }_{k}\right),\end{array}\end{eqnarray}$
where λk > 0 and 0d−2r denotes the zero operator on ${{ \mathcal H }}_{0}$. Note that here we do not distinguish the operator and its matrix representation in the given basis. Therefore, equations (A10) and (A11) imply that
$\begin{eqnarray}O\mathrm{Re}(\rho ){O}^{{\rm{T}}}=\displaystyle \sum _{k=1}^{r}{\lambda }_{k}(| +{\rm{i}}\rangle \langle +{\rm{i}}{| }_{k}+| -{\rm{i}}\rangle \langle -{\rm{i}}{| }_{k}).\end{eqnarray}$
Combining equations (A11) and (A12) implies that
$\begin{eqnarray}\begin{array}{rcl}O\rho {O}^{{\rm{T}}} & = & O\mathrm{Re}(\rho ){O}^{{\rm{T}}}+O(\mathrm{iIm}(\rho )){O}^{{\rm{T}}}\\ & = & \displaystyle \sum _{k=1}^{r}2{\lambda }_{k}| +{\rm{i}}\rangle \langle +{\rm{i}}{| }_{k}.\end{array}\end{eqnarray}$
(4) ⇒ (3). Suppose that there exists an orthogonal operator O such that
$\begin{eqnarray}O\rho {O}^{{\rm{T}}}=\displaystyle \sum _{k=1}^{r}{\alpha }_{k}| +{\rm{i}}\rangle \langle +{\rm{i}}{| }_{k}.\end{eqnarray}$
By direct calculation, we have
$\begin{eqnarray}\begin{array}{rcl}O\rho \bar{\rho }{O}^{{\rm{T}}} & = & (O\rho {O}^{{\rm{T}}}){\left(O\bar{\rho }{O}^{{\rm{T}}}\right)}^{{\rm{T}}}\\ & = & \left(\displaystyle \sum _{k=1}^{r}{\alpha }_{k}| +{\rm{i}}\rangle \langle +{\rm{i}}{| }_{k}\right)\left(\displaystyle \sum _{k^{\prime} =1}^{r}{\alpha }_{k^{\prime} }| -{\rm{i}}\rangle \langle -{\rm{i}}{| }_{k^{\prime} }\right)\\ & = & 0,\end{array}\end{eqnarray}$
which implies that $\rho \bar{\rho }=0$. This completes the proof of theorem 1.

Appendix B. Proof of theorem 2

Before proving theorem 2, we need to prove the following lemma that will play a crucial role in the proof of theorem 2.

Let A be a $2d\times 2d$ skew-symmetric matrix defined as

$\begin{eqnarray}A=\left(\begin{array}{ccccccc}0 & -1 & & & & & \\ 1 & 0 & & & & & \\ & & 0 & -1 & & & \\ & & 1 & 0 & & & \\ & & & & \ddots & & \\ & & & & & 0 & -1\\ & & & & & 1 & 0\end{array}\right),\end{eqnarray}$
and B be a $2d\times 2d$ real symmetric matrix. If $[A,B]=0$ and ${\rm{i}}\lambda A+{pB}$ is a positive matrix for $\lambda \gt 0$, $p\geqslant 0$, then there exists an orthogonal matrix O such that
$\begin{eqnarray}\begin{array}{l}O({\rm{i}}\lambda A+{pB}){O}^{{\rm{T}}}\\ =\,{0}_{{m}_{0}}\underset{j=1}{\overset{r}{\bigoplus }}\left(p{\mu }_{j}{{\bf{1}}}_{{m}_{j}-2{r}_{j}}\underset{s=1}{\overset{{r}_{j}}{\bigoplus }}\left(\begin{array}{cc}p{\mu }_{j} & -{\rm{i}}\lambda {\lambda }_{j,s}\\ {\rm{i}}\lambda {\lambda }_{j,s} & p{\mu }_{j}\end{array}\right)\right),\end{array}\end{eqnarray}$
where ${m}_{0}=2d-{\sum }_{j=1}^{r}{m}_{j}$, $r\in {{\mathbb{Z}}}_{+}$, ${m}_{j}\in {{\mathbb{Z}}}_{+}$, ${r}_{j}\in {{\mathbb{Z}}}_{+}$, ${\mu }_{j}\gt 0$, ${\mu }_{j}\ne {\mu }_{k}$ for $j\ne k$, ${\sum }_{j=1}^{r}{\mu }_{j}{m}_{j}=1$, and ${\lambda }_{j,s}\gt 0$.

Below, we first demonstrate the proof of lemma 2.
By corollary 2.5.11 in Ref. [54], for real symmetric matrix B, there exists a 2d × 2d orthogonal matrix O such that
$\begin{eqnarray}{{OBO}}^{{\rm{T}}}={{\bf{0}}}_{{m}_{0}}\underset{j=1}{\overset{r}{\bigoplus }}{\mu }_{j}{1}_{{m}_{j}},\end{eqnarray}$
where ${{\bf{0}}}_{{m}_{0}}$ is the m0 × m0 zero matrix with ${m}_{0}:=2d-{\sum }_{j=1}^{r}{m}_{j}$, ${{\bf{1}}}_{{m}_{j}}$ is the mj × mj identity matrix, and μj > 0, μjμs for any js. Since [A, B] = 0, we have
$\begin{eqnarray}[{{OAO}}^{{\rm{T}}},{{OBO}}^{{\rm{T}}}]=0.\end{eqnarray}$
From equations (B3) and (B4), and the fact that OAOT is a skew-symmetric matrix, we know that
$\begin{eqnarray}{{OAO}}^{T}={C}_{0}\underset{j=1}{\overset{r}{\bigoplus }}{C}_{j},\end{eqnarray}$
where C0 is an m0 × m0 skew-symmetric matrix, and Cj is an mj × mj skew-symmetric matrix. Therefore, by combining equations (B3) and (B5), we have
$\begin{eqnarray}O({\rm{i}}\lambda A+{pB}){O}^{{\rm{T}}}={\rm{i}}\lambda {C}_{0}\mathop{\mathop{\bigoplus }\limits^{r}}\limits_{j=1}({\rm{i}}\lambda {C}_{j}+p{\mu }_{j}{{\bf{1}}}_{{m}_{j}}).\end{eqnarray}$
Then, by combining equation (B6) and the fact that O(iλA + pB)OT is positive for λ > 0, we get that C0 = 0. Hence,
$\begin{eqnarray}O({\rm{i}}\lambda A+{pB}){O}^{{\rm{T}}}={{\bf{0}}}_{{m}_{0}}\mathop{\mathop{\bigoplus }\limits^{r}}\limits_{j=1}({\rm{i}}\lambda {C}_{j}+p{\mu }_{j}{{\bf{1}}}_{{m}_{j}}).\end{eqnarray}$
For any skew-symmetric matrix Cj, by corollary 2.5.11 in Ref. [54], there exists an mj × mj orthogonal matrix Oj such that
$\begin{eqnarray}{O}_{j}{C}_{j}{O}_{j}^{{\rm{T}}}={{\bf{0}}}_{{m}_{j}-2{r}_{{m}_{j}}}\underset{s=1}{\overset{{r}_{{m}_{j}}}{\bigoplus }}{\lambda }_{j,s}\left(\begin{array}{cc}0 & -1\\ 1 & 0\end{array}\right),\end{eqnarray}$
where λj,s > 0. Let
$\begin{eqnarray}\mathop{O}\limits^{\sim }={{\bf{1}}}_{{m}_{0}}\underset{j=1}{\overset{r}{\bigoplus }}{O}_{j}.\end{eqnarray}$
Then, equations (B7)-(B9) imply that
$\begin{eqnarray}\begin{array}{c}\rm{\unicode{x000A0}}\rm{\unicode{x000A0}}\tilde{O}O({\rm{i}}\lambda A+pB){O}^{{\rm{T}}}{\tilde{O}}^{{\rm{T}}}\\ =\,{{\bf{0}}}_{{m}_{0}}\mathop{\mathop{\bigoplus }\limits^{r}}\limits_{j=1}\left(p{\mu }_{j}{{\bf{1}}}_{{m}_{j}-2{r}_{{m}_{j}}}\mathop{\mathop{\bigoplus }\limits^{{r}_{{m}_{j}}}}\limits_{s=1}\left(\begin{array}{cc}p{\mu }_{j} & -{\rm{i}}\lambda {\lambda }_{j,s}\\ {\rm{i}}\lambda {\lambda }_{j,s} & p{\mu }_{j}\end{array}\right)\right).\end{array}\end{eqnarray}$
This completes the proof of lemma 2.
Now we proceed to prove theorem 2.
(1) ⇔ (2). The equivalence between (1) and (2) can be derived using a similar method as the proof of theorem 1.
(2) ⇔ (3). The equivalence between (2) and (3) can be directly obtained from the no-broadcasting theorem [4, 6].
(3) ⇒ (4). Let ρ be a nonreal state such that
$\begin{eqnarray}[\rho ,\bar{\rho }]=0,\end{eqnarray}$
which implies that
$\begin{eqnarray}[\mathrm{Re}(\rho ),\mathrm{Im}(\rho )]=0.\end{eqnarray}$
By corollary 2.5.11 in Ref. [54], there exists a d × d orthogonal operator O such that
$\begin{eqnarray}O{\rm{Im}}(\rho ){O}^{{\rm{T}}}={{\bf{0}}}_{{d}_{0}}\mathop{\mathop{\bigoplus }\limits^{r}}\limits_{k=1}{\lambda }_{k}{A}_{k},\end{eqnarray}$
where ${{\bf{0}}}_{{d}_{0}}$ is the d0 × d0 zero matrix, ${d}_{0}=d-{\sum }_{k=1}^{r}2{d}_{k}$, λk > 0, λkλj for any kj, and Ak is a 2dk × 2dk skew-symmetric matrix defined as
$\begin{eqnarray}{A}_{k}=\left(\begin{array}{ccccccc}0 & -1 & & & & & \\ 1 & 0 & & & & & \\ & & 0 & -1 & & & \\ & & 1 & 0 & & & \\ & & & & \ddots & & \\ & & & & & 0 & -1\\ & & & & & 1 & 0\end{array}\right).\end{eqnarray}$
From equation (B12), we have
$\begin{eqnarray}[O\mathrm{Re}(\rho ){O}^{{\rm{T}}},O\mathrm{Im}(\rho ){O}^{{\rm{T}}}]=0.\end{eqnarray}$
Therefore, by combining equations (B13) and (B15), we have
$\begin{eqnarray}O\mathrm{Re}(\rho ){O}^{{\rm{T}}}={p}_{0}{B}_{0}\underset{k=1}{\overset{r}{\bigoplus }}{p}_{k}{B}_{k},\end{eqnarray}$
where p0 ≥ 0, B0 is a d0 × d0 real symmetric matrix, pk > 0, and Bk is a 2dk × 2dk real symmetric matrix satisfying
$\begin{eqnarray}[{A}_{k},{B}_{k}]=0.\end{eqnarray}$
Hence, from equations (B13) and (B16), it follows that
$\begin{eqnarray}\begin{array}{rcl}O\rho {O}^{{\rm{T}}} & = & O\mathrm{Re}(\rho ){O}^{{\rm{T}}}+{\rm{i}}O\mathrm{Im}(\rho ){O}^{{\rm{T}}}\\ & = & {p}_{0}{B}_{0}\underset{k=1}{\overset{r}{\bigoplus }}({\rm{i}}{\lambda }_{k}{A}_{k}+{p}_{k}{B}_{k}).\end{array}\end{eqnarray}$
By combining equations (B17) and (B18) and lemma 2, we can obtain that for each k, there exists an orthogonal operator Ok such that
$\begin{eqnarray}\begin{array}{c}{O}_{k}({\rm{i}}{\lambda }_{k}{A}_{k}+{p}_{k}{B}_{k}){O}_{k}^{{\rm{T}}}={{\bf{0}}}_{{m}_{k,0}}\,\\ \,\mathop{\mathop{\bigoplus }\limits^{{r}_{k}}}\limits_{j=1}\left({p}_{k}{\mu }_{k,j}{{\bf{1}}}_{{m}_{k,j}-2{r}_{k,j}}\mathop{\mathop{\bigoplus }\limits^{{r}_{k,j}}}\limits_{s=1}\left(\begin{array}{cc}{p}_{k}{\mu }_{k,j} & -{\rm{i}}{\lambda }_{k}{\lambda }_{k,j,s}\\ {\rm{i}}{\lambda }_{k}{\lambda }_{k,j,s} & p{\mu }_{k,j}\end{array}\right)\right),\end{array}\end{eqnarray}$
where ${m}_{k,0}=2{d}_{k}-{\sum }_{j=1}^{{r}_{k}}{m}_{k,j}$, ${r}_{k}\in {{\mathbb{Z}}}_{+}$, ${m}_{k,j}\in {{\mathbb{Z}}}_{+}$, μk,j > 0, ${\mu }_{k,j}\ne {\mu }_{k,{j}^{{\prime} }}$ for $j\ne {j}^{{\prime} }$, ${\sum }_{j=1}^{{r}_{k}}{\mu }_{k,j}{m}_{k,j}=1$, and λk,j,s > 0. Let $\tilde{O}={{\bf{1}}}_{d-{\sum }_{k=}^{r}2{d}_{k}}{\bigoplus }_{k=1}^{r}{O}_{k}$. By combining equations (B18) and (B19), we have
$\begin{eqnarray}\begin{array}{c}\tilde{O}O\rho {O}^{{\rm{T}}}{\tilde{O}}^{{\rm{T}}}={p}_{0}{B}_{0}\,\\ \,\mathop{\mathop{\bigoplus }\limits^{r}}\limits_{k=1}\left({{\bf{0}}}_{{m}_{k,0}}\mathop{\mathop{\bigoplus }\limits^{{r}_{k}}}\limits_{j=1}\left({p}_{k}{\mu }_{k,j}{{\bf{1}}}_{{m}_{k,j}-2{r}_{k,j}}\mathop{\mathop{\bigoplus }\limits^{{r}_{k,j}}}\limits_{s=1}\left(\begin{array}{cc}{p}_{k}{\mu }_{k,j} & -{\rm{i}}{\lambda }_{k}{\lambda }_{k,j,s}\\ {\rm{i}}{\lambda }_{k}{\lambda }_{k,j,s} & p{\mu }_{k,j}\end{array}\right)\right)\right).\end{array}\end{eqnarray}$
Furthermore, there exists a d × d permutation matrix O1 such that
$\begin{eqnarray}\begin{array}{c}{O}_{1}\tilde{O}O\rho {O}^{{\rm{T}}}{\tilde{O}}^{{\rm{T}}}{O}_{1}^{{\rm{T}}}\\ \rm{\unicode{x000A0}}\rm{\unicode{x000A0}}=\,\left({p}_{0}{B}_{0}\mathop{\mathop{\bigoplus }\limits^{r}}\limits_{k=1}\left({{\bf{0}}}_{{m}_{k,0}}\mathop{\mathop{\bigoplus }\limits^{{r}_{k}}}\limits_{j=1}{p}_{k}{\mu }_{k,j}{{\bf{1}}}_{{m}_{k,j}-2{r}_{k,j}}\right)\right)\\ \mathop{\mathop{\bigoplus }\limits^{r}}\limits_{k=1}\mathop{\mathop{\bigoplus }\limits^{{r}_{k}}}\limits_{j=1}\mathop{\mathop{\bigoplus }\limits^{{r}_{k,j}}}\limits_{s=1}\left(\begin{array}{cc}{p}_{k}{\mu }_{k,j} & -{\rm{i}}{\lambda }_{k}{\lambda }_{k,j,s}\\ {\rm{i}}{\lambda }_{k}{\lambda }_{k,j,s} & {p}_{k}{\mu }_{k,j}\end{array}\right).\end{array}\end{eqnarray}$
Let
$\begin{eqnarray*}\begin{array}{c}{q}_{0}\,=\,{p}_{0}+\displaystyle \sum _{k=1}^{r}\displaystyle \sum _{j=1}^{{r}_{k}}{p}_{k}{\mu }_{k,j}({m}_{k,j}-2{r}_{k,j}),\,\\ {\sigma }_{0}\,=\,\displaystyle \frac{1}{{q}_{0}}\left({p}_{0}{B}_{0}\mathop{\mathop{\bigoplus }\limits^{r}}\limits_{k=1}\left({{\bf{0}}}_{{m}_{k,0}}\mathop{\mathop{\bigoplus }\limits^{{r}_{k}}}\limits_{j=1}{p}_{k}{\mu }_{k,j}{{\bf{1}}}_{{m}_{k,j}-2{r}_{k,j}}\right)\right),\\ {\sigma }_{k,j,s}\,=\,\displaystyle \frac{1}{2{p}_{k}{\mu }_{k,j}}\left(\begin{array}{cc}{p}_{k}{\mu }_{k,j} & -{\rm{i}}{\lambda }_{k}{\lambda }_{k,j,s}\\ {\rm{i}}{\lambda }_{k}{\lambda }_{k,j,s} & {p}_{k}{\mu }_{k,j}\end{array}\right).\,\end{array}\end{eqnarray*}$
It is obvious that σ0 is a real state, σk,j,s is a nonreal qubit state whose eigenvectors are composed by two orthogonal maximally imaginary states, and
$\begin{eqnarray}{O}_{1}\tilde{O}O\rho {O}^{{\rm{T}}}{\tilde{O}}^{{\rm{T}}}{O}_{1}^{{\rm{T}}}={q}_{0}{\sigma }_{0}\underset{k=1}{\overset{r}{\bigoplus }}\underset{j=1}{\overset{{r}_{k}}{\bigoplus }}\underset{s=1}{\overset{{r}_{k,j}}{\bigoplus }}2{p}_{k}{\mu }_{{k}_{j}}{\sigma }_{k,j,s}.\end{eqnarray}$
Hence, there exists an orthogonal operator O2 such that
$\begin{eqnarray}\begin{array}{l}{O}_{2}{O}_{1}\tilde{O}O\rho {O}^{{\rm{T}}}{\tilde{O}}^{{\rm{T}}}{O}_{1}^{{\rm{T}}}{O}_{2}^{{\rm{T}}}\\ =\displaystyle \sum _{l=1}^{m}\left({\alpha }_{l}| +{\rm{i}}\rangle \langle +{\rm{i}}{| }_{l}+{\beta }_{l}| -{\rm{i}}\rangle \langle -{\rm{i}}{| }_{l}\right)\oplus {q}_{0}{\sigma }_{0},\end{array}\end{eqnarray}$
where $m={\sum }_{k=1}^{r}{\sum }_{j=1}^{{r}_{k}}{r}_{k,j}$, αl, βl ≥ 0, αl + βl > 0, ${\sum }_{l=1}^{m}({\alpha }_{l}+{\beta }_{l})+{q}_{0}\,=\,1$.
(4) ⇒ (3). By direct calculations, we have
$\begin{eqnarray}\begin{array}{rcl}\rho \bar{\rho } & = & \bar{\rho }\rho \\ & = & {O}^{{\rm{T}}}\left(\underset{k=1}{\overset{r}{\bigoplus }}{\alpha }_{k}{\beta }_{k}\left(| +{\rm{i}}\rangle \langle +{\rm{i}}{| }_{k}+| -{\rm{i}}\rangle \langle -{\rm{i}}{| }_{k}\right)\bigoplus {p}_{0}^{2}{\rho }_{0}^{2}\right)O.\end{array}\end{eqnarray}$
This completes the proof of theorem 2.

Appendix C. Proof of lemma 1

For item (i), using the fact that 0 ≤ F(ρ, σ) ≤ 1 for any states ρ and σ, we can deduce that ${M}_{{\rm{g}}}(\rho )=(1-F(\rho ,\bar{\rho }))/2\leqslant 1/2$, and the equality holds if and only if $F(\rho ,\bar{\rho })=0$. Combining the fact that F(ρ, σ) = 0 if and only if ρσ = 0 [55], we can easily obtain that Mg(ρ) = 1/2 if and only if $\rho \bar{\rho }=0$.
For item (ii), using the fact that F(ρaρb, σaσb) = F(ρa, σa)F(ρb, σb) [55], we have
$\begin{eqnarray}\begin{array}{rcl}{M}_{{\rm{g}}}({\rho }^{a}\otimes {\rho }^{b}) & = & \displaystyle \frac{1}{2}\left(1-F({\rho }^{a}\otimes {\rho }^{b},\overline{{\rho }^{a}}\otimes \overline{{\rho }^{b}})\right)\\ & = & \displaystyle \frac{1}{2}\left(1-F({\rho }^{a},\overline{{\rho }^{a}})F({\rho }^{b},\overline{{\rho }^{b}})\right)\\ & = & \displaystyle \frac{1}{2}\left(1-(1-2{M}_{{\rm{g}}}({\rho }^{a}))(1-2{M}_{{\rm{g}}}({\rho }^{b}))\right)\\ & = & {M}_{{\rm{g}}}({\rho }^{a})+{M}_{{\rm{g}}}({\rho }^{b})-2{M}_{{\rm{g}}}({\rho }^{a}){M}_{{\rm{g}}}({\rho }^{b}).\end{array}\end{eqnarray}$
For item (iii), assume that Mg(ρ) ≥ Mg(σ). Then, we have
$\begin{eqnarray}\begin{array}{l}\ \ \ | {M}_{{\rm{g}}}(\rho )-{M}_{{\rm{g}}}(\sigma )| \\ =\,\displaystyle \frac{1}{2}\left(F(\sigma ,\bar{\sigma })-F(\rho ,\bar{\rho })\right)\\ =\,\displaystyle \frac{1}{4}\left({D}_{{\rm{B}}}{\left(\rho ,\bar{\rho }\right)}^{2}-{D}_{{\rm{B}}}{\left(\sigma ,\bar{\sigma }\right)}^{2}\right)\\ \leqslant \,\displaystyle \frac{1}{4}\left({D}_{{\rm{B}}}(\rho ,\bar{\rho })+{D}_{{\rm{B}}}(\sigma ,\bar{\sigma })\right)\\ \times \,\left({D}_{{\rm{B}}}(\rho ,\sigma )+{D}_{{\rm{B}}}(\sigma ,\bar{\sigma })+{D}_{{\rm{B}}}(\bar{\sigma },\bar{\rho })-{D}_{{\rm{B}}}(\sigma ,\bar{\sigma })\right)\\ \leqslant \,\sqrt{2}{D}_{{\rm{B}}}(\rho ,\sigma )\\ =\,2\sqrt{1-F(\rho ,\sigma )},\end{array}\end{eqnarray}$
where the first inequality follows from the triangle inequality of the Bures distance ${D}_{{\rm{B}}}(\rho ,\sigma ):= \sqrt{2\left(1-F(\rho ,\sigma )\right)}$ [56], and the second inequality follows from the fact that${D}_{{\rm{B}}}(\rho ,\sigma )={D}_{{\rm{B}}}(\bar{\sigma },\bar{\rho })\leqslant \sqrt{2}$.

Appendix D. Proof of theorem 4

For the proof of theorem 4, we use a method similar to that in Ref. [20].
Let Λa be the optimal real operation such that
$\begin{eqnarray}\mathrm{irrev}(\rho ,{\tilde{\rho }}^{a})=1-F{\left(\rho ,{{\rm{\Lambda }}}^{a}({\tilde{\rho }}^{a})\right)}^{2}.\end{eqnarray}$
Consider the recovery state σab on ${{ \mathcal H }}^{a}\otimes {{ \mathcal H }}^{b}$ relative to the recovery operation Λa, which is defined as
$\begin{eqnarray}{\sigma }^{{ab}}\,:=\,({{\rm{\Lambda }}}^{a}\otimes {{ \mathcal I }}^{b})\circ { \mathcal E }(\rho ),\end{eqnarray}$
where ${{ \mathcal I }}^{b}$ is the identity channel on ${{ \mathcal H }}^{b}$ and ◦ denotes the composition of quantum operations. From the definition of σab, it is easy to check that
$\begin{eqnarray}{\sigma }^{a}:= {\mathrm{tr}}_{b}{\sigma }^{{ab}}={{\rm{\Lambda }}}^{a}({\tilde{\rho }}^{a}).\end{eqnarray}$
Therefore, we have
$\begin{eqnarray}\begin{array}{rcl}\mathrm{irrev}(\rho ,{\tilde{\rho }}^{a}) & = & 1-F{\left(\rho ,{{\rm{\Lambda }}}^{a}({\tilde{\rho }}^{a})\right)}^{2}\\ & \geqslant & 1-F(\rho \otimes {\tilde{\rho }}^{b},{\sigma }^{{ab}})\\ & \geqslant & \displaystyle \frac{1}{4}| {M}_{{\rm{g}}}(\rho \otimes {\tilde{\rho }}^{b})-{M}_{{\rm{g}}}({\sigma }^{{ab}}){| }^{2},\end{array}\end{eqnarray}$
where the first equality follows from equation (D1), the second inequality follows from lemma 1, and the first inequality follows from equation (D3) and the following lemma:
Lemma 3 (Marvian–Spekkens [20]). For any pure state ${\rho }^{a}=| \psi \rangle \langle \psi | \in D({{ \mathcal H }}^{a})$, any state ${\rho }^{b}\in D({{ \mathcal H }}^{b})$, and any state ${\sigma }^{{ab}}\in D({{ \mathcal H }}^{a}\otimes {{ \mathcal H }}^{b})$,
$\begin{eqnarray}F({\rho }^{a}\otimes {\rho }^{b},{\sigma }^{{ab}})\geqslant F{\left({\rho }^{a},{\sigma }^{a}\right)}^{2}.\end{eqnarray}$
From the monotonicity of Mg and the fact that $({{\rm{\Lambda }}}^{a}\otimes {{ \mathcal I }}^{b})\circ { \mathcal E }$ is a real operation, we have
$\begin{eqnarray}{M}_{{\rm{g}}}(\rho )\geqslant {M}_{{\rm{g}}}({{\rm{\Lambda }}}^{a}\otimes {{ \mathcal I }}^{b})\circ { \mathcal E }(\rho ))={M}_{{\rm{g}}}({\sigma }^{{ab}}).\end{eqnarray}$
Furthermore, applying lemma 1, we have
$\begin{eqnarray}{M}_{{\rm{g}}}(\rho \otimes {\tilde{\rho }}^{b})={M}_{{\rm{g}}}(\rho )+{M}_{{\rm{g}}}({\tilde{\rho }}^{b})-2{M}_{{\rm{g}}}(\rho ){M}_{{\rm{g}}}({\tilde{\rho }}^{b}).\end{eqnarray}$
Therefore, by combining equations (D6) and (D7), we have
$\begin{eqnarray}{M}_{{\rm{g}}}(\rho \otimes {\tilde{\rho }}^{b})-{M}_{{\rm{g}}}({\sigma }^{{ab}})\geqslant {M}_{{\rm{g}}}({\tilde{\rho }}^{b})(1-2{M}_{{\rm{g}}}(\rho ))\geqslant 0.\end{eqnarray}$
From equations (D4) and (D8), we conclude that
$\begin{eqnarray}{M}_{{\rm{g}}}({\tilde{\rho }}^{b})(1-2{M}_{{\rm{g}}}(\rho ))\leqslant 2\sqrt{\mathrm{irrev}(\rho ,{\tilde{\rho }}^{a})}.\end{eqnarray}$
This completes the proof of theorem 4.

This work was supported by the National Key R&D Program of China under Grant No. 2020YFA0712700, the National Natural Science Foundation of China under Grant No.12341103 and the Youth Innovation Promotion Association of CAS under Grant No. 2020002.

1
Wootters W K, Zurek W H 1982 A single quantum cannot be cloned Nature 299 802

DOI

2
Dieks D 1982 Communication by EPR devices Phy. Lett. A 92 271

DOI

3
Yuen H P 1986 Amplification of quantum states and noiseless photon amplifiers Phys. Lett. A 113 405

DOI

4
Barnum H, Caves C M, Fuchs C A, Jozsa R, Schumacher B 1996 Noncommuting mixed states cannot be broadcast Phys. Rev. Lett. 76 2818

DOI

5
Scarani V, Iblisdir S, Gisin N, Acín A 2005 Quantum cloning Rev. Mod. Phys. 77 1225

DOI

6
Barnum H, Barrett J, Leifer M, Wilce A 2007 Generalized no-broadcasting theorem Phys. Rev. Lett. 99 240501

DOI

7
Piani M, Horodecki P, Horodecki R 2008 No-local-broadcasting theorem for multipartite quantum correlations Phys. Rev. Lett. 100 090502

DOI

8
Luo S, Li N, Cao X 2009 Relation between “no broadcasting” for noncommuting states and “no local broadcasting” for quantum correlations Phys. Rev. A 79 054305

DOI

9
Luo S, Sun W 2010 Decomposition of bipartite states with applications to quantum no-broadcasting theorems Phys. Rev. A 82 012338

DOI

10
Luo S 2010 On quantum no-broadcasting Lett. Math. Phys. 92 143

DOI

11
Chatterjee S, Sazim S, Chakrabarty I 2016 Broadcasting of quantum correlations: Possibilities and impossibilities Phys. Rev. A 93 042309

DOI

12
Mundra R, Patel D, Chakrabarty I, Ganguly N, Chatterjee S 2019 Broadcasting of quantum correlations in qubit-qudit systems Phys. Rev. A 100 042319

DOI

13
Buzek V, Vedral V, Plenio M B, Knight P L, Hillery M 1997 Broadcasting of entanglement via local copying Phys. Rev. A 55 3327

DOI

14
Koashi M, Imoto N 1998 No-cloning theorem of entangled states Phys. Rev. Lett. 81 4264

DOI

15
Adhikari S, Majumdar A S, Nayak N 2008 Broadcasting of continuous-variable entanglement Phys. Rev. A 77 042301

DOI

16
Zhu D, Shang W-M, Zhang F-L, Chen J-L 2022 Quantum cloning of steering Chin. Phys. Lett. 39 070302

DOI

17
Lu X M, Sun Z, Wang X, Luo S, Oh C H 2013 Broadcasting quantum fisher information Phys. Rev. A 87 050302

DOI

18
Sharma U K, Chakrabarty I, Shukla M K 2017 Broadcasting quantum coherence via cloning Phys. Rev. A 96 052319

DOI

19
Lostaglio M, Müller M P 2019 Coherence and asymmetry cannot be broadcast Phys. Rev. Lett. 123 020403

DOI

20
Marvian I, Spekkens R W 2019 No-broadcasting theorem for quantum asymmetry and coherence and a trade-off relation for approximate broadcasting Phys. Rev. Lett. 123 020404

DOI

21
Yang C, Guo Z, Zhang C, Cao H 2022 Broadcasting coherence via incoherent operations Linear Mult. Alg 21 70

DOI

22
Zhang Z, Feng L, Luo S 2024 No-broadcasting of magic states Phys. Rev. A 110 012462

DOI

23
Hickey A, Gour G 2018 Quantifying the imaginarity of quantum mechanics J. Phys. A 51 414009

DOI

24
Wu K-D, Kondra T V, Rana S, Scandolo C M, Xiang G-Y, Li C-F, Guo G-C, Streltsov A 2021 Operational resource theory of imaginarity Phys. Rev. Lett. 126 090401

DOI

25
Wu K-D, Kondra T V, Rana S, Scandolo C M, Xiang G-Y, Li C-F, Guo G-C, Streltsov A 2021 Resource theory of imaginarity: Quantification and state conversion Phys. Rev. A 103 032401

DOI

26
Xue S, Guo J, Li P, Ye M, Li Y 2021 Quantification of resource theory of imaginarity Quantum Inf. Process. 20 1

DOI

27
Renou M-O, Trillo D, Weilenmann M, Le T P, Tavakoli A, Gisin N, Acín A, Navascués M 2021 Quantum theory based on real numbers can be experimentally falsified Nature 600 625

DOI

28
Zhu H 2021 Hiding and masking quantum information in complex and real quantum mechanics Phys. Rev. Res. 3 033176

DOI

29
Zhang R-Q, Hou Z, Li Z, Zhu H, Xiang G-Y, Li C-F, Guo G-C 2021 Experimental masking of real quantum states Phys. Rev. Appl. 16 024052

DOI

30
Prasannan N, De S, Barkhofen S, Brecht B, Silberhorn C, Sperling J 2021 Experimental entanglement characterization of two-rebit states Phys. Rev. A 103 L040402

DOI

31
Li Z-D 2022 Testing real quantum theory in an optical quantum network Phys. Rev. Lett. 128 040402

DOI

32
Li N, Luo S, Sun Y 2022 Brukner-zeilinger invariant information in the presence of conjugate symmetry Phys. Rev. A 106 032404

DOI

33
Chen M-C 2022 Ruling out real-valued standard formalism of quantum theory Phys. Rev. Lett. 128 040403

DOI

34
Miyazaki J, Matsumoto K 2022 Imaginarity-free quantum multiparameter estimation Quantum 6 665

DOI

35
Wu D 2022 Experimental refutation of real-valued quantum mechanics under strict locality conditions Phys. Rev. Lett. 129 140401

DOI

36
Bednorz A, Batle J 2022 Optimal discrimination between real and complex quantum theories Phys. Rev. A 106 042207

DOI

37
Kondra T V, Datta C, Streltsov A 2023 Real quantum operations and state transformations New J. Phys. 25 093043

DOI

38
Xu J 2023 Imaginarity of gaussian states Phys. Rev. A 108 062203

DOI

39
Xu J 2023 Quantifying the imaginarity of quantum states via Tsallis relative entropy arXiv:2311.12547

40
Li H-b, Hua M, Zheng Q, Zhi Q-j, Ping Y 2023 Relationship between robustness of imaginarity and quantum coherence Eur. Phys. J. D 77 28

DOI

41
Chen Q, Gao T, Yan F 2023 Measures of imaginarity and quantum state order Sci. Chin. Phys. Mech. Astron. 66 280312

DOI

42
Zhang J, Luo Y, Li Y 2023 Imaginaring and deimaginaring power of quantum channels and the trade-off between imaginarity and entanglement Quantum Inf. Process. 22 405

DOI

43
Chen X, Lei Q 2024 Imaginarity measure induced by relative entropy arXiv:2404.00637

44
Yao J, Chen H, Mao Y-L, Li Z-D, Fan J 2024 Proposals for ruling out real quantum theories in an entanglement-swapping quantum network with causally independent sources Phys. Rev. A 109 012211

DOI

45
Xu J 2024 Coherence and imaginarity of quantum states arXiv:2404.06210

46
Fernandes C, Wagner R, Novo L, Galvão E F 2024 Unitary-invariant witnesses of quantum imaginarity arXiv:2403.15066

47
Chen X, Lei Q 2024 Imaginarity of quantum channels: Refinement and alternative arXiv:2405.06222

48
Chen B, Huang X, Fei S-M 2024 On complementarity and distribution of imaginarity in finite dimensions Res. Phys. 60 107671

DOI

49
Wu K-D, Kondra T V, Scandolo C M, Rana S, Xiang G-Y, Li C-F, Guo G-C, Streltsov A 2024 Resource theory of imaginarity in distributed scenarios Commun. Phys. 7 171

DOI

50
Baumgratz T, Cramer M, Plenio M B 2014 Quantifying coherence Phys. Rev. Lett. 113 140401

DOI

51
Girolami D 2014 Observable measure of quantum coherence in finite dimensional systems Phys. Rev. Lett. 113 170401

DOI

52
Yu X-D, Zhang D-J, Xu G F, Tong D M 2016 Alternative framework for quantifying coherence Phys. Rev. A 94 060302

DOI

53
Nielsen M A, Chuang I L 2000 Quantum Computation and Quantum Information Cambridge University Press

54
Horn R A, Johnson C R 2012 Matrix Analysis 2nd edn. Cambridge Cambridge University Press

55
Luo S, Zhang Q 2004 Informational distance on quantum-state space Phys. Rev. A 69 032106

DOI

56
Bures D 1969 An extension of Kakutani's theorem on infinite product measures to the tensor product of semifinite w*-algebras Trans. Am. Math. Soc. 135 199

DOI

Outlines

/