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.
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 [1–6]. 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] := ρ0ρ1 − ρ1ρ0 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 [8–12]. The entanglement of bipartite states might be partially broadcast via local operations [13–15]. 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 [18–21]. 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 [23–49]. 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,k〈j∣ρ∣k〉∣j〉〈k∣ on ${ \mathcal H }$ is called a real state [23], if
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 ${\mathscr{R}}({ \mathcal H })$.
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,k〈k∣O∣j〉∣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 [50–52], 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}.
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
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
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
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
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
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
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.,
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
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
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]
$\begin{eqnarray}{M}_{{\rm{g}}}(\rho ):=1-\mathop{\max }\limits_{\sigma \in {\mathscr{R}}({ \mathcal H })}F{\left(\rho ,\sigma \right)}^{2},\end{eqnarray}$
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
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
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
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:
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
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
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
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
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, μj ≠ μs for any j ≠ s. Since [A, B] = 0, we have
where ${{\bf{0}}}_{{d}_{0}}$ is the d0 × d0 zero matrix, ${d}_{0}=d-{\sum }_{k=1}^{r}2{d}_{k}$, λk > 0, λk ≠ λj for any k ≠ j, and Ak is a 2dk × 2dk skew-symmetric matrix defined as
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
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
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(ρa ⨂ ρb, σa ⨂ σb) = F(ρa, σa)F(ρb, σb) [55], we have
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].
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
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 ${\sigma }^{{ab}}\in D({{ \mathcal H }}^{a}\otimes {{ \mathcal H }}^{b})$,
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.
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