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# Distribution and quantification of remotely generated Wigner negativity

## Abstract

Wigner negativity, as a well-known indicator of nonclassicality, plays an essential role in quantum computing and simulation using continuous-variable systems. The conditional preparation of Wigner-negative states through appropriate non-Gaussian operations on an auxiliary mode is common procedure in quantum optics experiments. Motivated by the demand of real-world quantum network, here we investigate the remote creation and distribution of Wigner negativity in the multipartite scenario from a quantitative perspective. By establishing a monogamy relation akin to the generalized Coffman-Kundu-Wootters inequality, we show that the amount of Wigner negativity cannot be freely distributed among different modes. Moreover, for photon subtraction—one of the main experimentally realized non-Gaussian operations—we provide an intuitive method to quantify remotely generated Wigner negativity. Our results pave the way for exploiting Wigner negativity as a valuable resource for numerous quantum information protocols based on non-Gaussian scenario.

## Introduction

Continuous-variable (CV) systems have attained impressive success in quantum information processing1. As an important platform that has been widely studied, Gaussian systems and operations are extensively used in quantum teleportation2, quantum key distribution3, and quantum-enhanced sensing4,5. These protocols come with the advantage of deterministically producing resource states and being analytically tractable due to the Gaussian properties of the states. However, non-Gaussian states and operations have irreplaceable advantages in some CV protocols6, such as entanglement distillation7,8, error correction9, secure quantum communication10, and the verification of Bell nonlocality11. Considerable progress in the controllable generation of multimode non-Gaussian states has been made in recent experiments12,13, which also provide support for the implementation of universal CV quantum computation in the long term14.

For some non-Gaussian states, the Wigner function can reach negative values. This Wigner negativity has been seen as a necessary ingredient in CV quantum computation and simulation to outperform classical devices15,16,17. A common approach to generate Wigner negativity is by means of the action of a conditional operation on initially prepared Gaussian states. In the pursuit of long-distance quantum technologies, it is crucial to develop efficient methods to produce Wigner negativity in a distant node. Recently, it was proven that a necessary requirement for such scheme is the existence of Einstein-Podolsky-Rosen (EPR) steering18,19—a particular type of quantum correlation where local measurements performed on one party can adjust (steer), instantaneously, the state of the other remote party20,21,22. Based on this kind of nonlocal effect, one can remotely produce negativity in the steering mode by applying a set of appropriate operations on the steered mode.

In consideration of the real-world quantum network in the multipartite scenario, it is a worthwhile objective to deeply explore the remote generation and distribution of Wigner negativity over many nodes in an entanglement-based network. As an intermediate type of quantum correlation between entanglement and Bell nonlocality, multipartite quantum steering23 has received extensive attention in recent developments of quantum information theory24,25. It has been successfully implemented in CV optical network26,27,28,29, photonic network30,31,32, and atomic ensembles33. Inspired by the shareability of EPR steering, known as monogamy34,35,36,37,38,39,40, it is interesting to explore how can the remotely generated Wigner negativity be distributed over different modes? Is there any monogamy relations imposing quantitative constraints on that negativity? Does stronger steerability generate more negativity?

Here we present a quantitative investigation of Wigner negativity that is remotely created via multipartite EPR steering, in which non-Gaussian operations performed on one steered node of quantum network produce Wigner negativity in different distant nodes, as shown in Fig. 1. We first investigate to what extent Wigner negativity can be shared by establishing a monogamy relation. This constraints the degree of distributed negativity akin to the Coffman-Kundu-Wootters (CKW) monogamy inequality for steerability36. Basic examples including two typical schemes, i.e. photon subtraction and Fock-state projection, for the remote preparation of Wigner-negative states are given. Then we focus on photon subtraction, one of the commonly used non-Gaussian operations, and find an intuitive measure for the amount of induced Wigner negativity in the steering modes. This allows us to find out the different behaviors of the steerability and the created negativity depending on the squeezing and purities of initial Gaussian states.

## Results

We begin by briefly introducing the theoretical framework of multimode CV quantum optics. The noninteracting quantized electromagnetic field can be treated as a number N of optical modes that behave as quantum harmonic oscillators with different frequencies described by $$\hat{H}=\mathop{\sum }\nolimits_{k = 1}^{N}2{\omega }_{k}({\hat{a}}_{k}^{{\dagger} }{\hat{a}}_{k}+\frac{1}{2})$$. Here, $${\hat{a}}_{k}$$ and $${\hat{a}}_{k}^{{\dagger} }$$ are the annihilation and creation operators of a photon in mode k, satisfying the bosonic commutation relation $$[{\hat{a}}_{k},{\hat{a}}_{{k}^{\prime}}^{{\dagger} }]={\delta }_{k{k}^{\prime}}$$. The corresponding quadrature phase operators for each mode are defined as $${\hat{x}}_{k}={\hat{a}}_{k}+{\hat{a}}_{k}^{{\dagger} }$$ and $${\hat{p}}_{k}=({\hat{a}}_{k}-{\hat{a}}_{k}^{{\dagger} })/i$$. Collecting the quadrature operators for all the modes into a vector $$\hat{{{{\boldsymbol{\xi }}}}}\equiv {({\hat{x}}_{1},{\hat{p}}_{1},...,{\hat{x}}_{N},{\hat{p}}_{N})}^{\top }$$, the covariance matrix (CM) σ is given with elements $${\sigma }_{ij}=\langle {\hat{\xi }}_{i}{\hat{\xi }}_{j}+{\hat{\xi }}_{j}{\hat{\xi }}_{i}\rangle /2-\langle {\hat{\xi }}_{i}\rangle \langle {\hat{\xi }}_{j}\rangle$$. If the system is prepared in a Gaussian state, the properties can be completely determined by its CM. Otherwise, the first and second-order statistical moments are not enough to characterize the non-Gaussian system, and we must resort to a more complete description. Here we choose the Wigner function as a preferred phase space representation for an arbitrary state with density matrix $$\hat{\rho }$$,

$$W({{{\boldsymbol{\xi }}}})={\int}_{{{\mathbb{R}}}^{2N}}\frac{{d}^{2N}{{{\boldsymbol{\alpha }}}}}{{(2\pi )}^{2N}}\exp \left(-i{{{{\boldsymbol{\xi }}}}}^{\top }{{{\boldsymbol{\Omega }}}}{{{\boldsymbol{\alpha }}}}\right)\chi ({{{\boldsymbol{\alpha }}}}),$$
(1)

where $${{{\boldsymbol{\Omega }}}}{ = \bigoplus }_{1}^{N}\left(\begin{array}{cc}0&1\\ -1&0\end{array}\right)$$ is the symplectic form and the Wigner characteristic function $$\chi ({{{\boldsymbol{\alpha }}}})=\,{{\mbox{Tr}}}\,[\hat{\rho }\exp (i{\hat{{{{\boldsymbol{\xi }}}}}}^{\top }{{{\boldsymbol{\Omega }}}}{{{\boldsymbol{\alpha }}}})]$$. A particular attribute of non-Gaussian states is the possibility for this Wigner function to attain negative values that can be quantified as $${{{\mathcal{N}}}}\equiv \int | W({{{\boldsymbol{\xi }}}})| d{{{\boldsymbol{\xi }}}}-1$$41.

In order to effectively generate and distribute Wigner negativity, an indirect scheme was proposed based on EPR steering18. In a two-mode Gaussian system, when there exists steering from Bob to Alice, then an appropriate local Gaussian transformation together with photon subtraction on the steered mode A can remotely generate Wigner negativity in the steering mode B, i.e. $${{{{\mathcal{N}}}}}_{B}\, > \,0$$. The bipartite Gaussian steerability can be quantified by the parameter $${{{{\mathcal{G}}}}}^{B\to A}=\max \{0,\frac{1}{2}\ln \frac{\,{{\mathrm{Det}}}{\sigma }_{B}}{{{\mathrm{Det}}}\,{\sigma }_{AB}}\}$$, where σB and σAB denote the CM for mode B, and the group (AB), respectively42. This formalism was developed for arbitrary conditional operations on an arbitrary number of modes, showing that EPR steering is still necessary to prepare a Wigner-negative state in the steering modes19. The remotely generated Wigner negativity was not quantified, nor are its multimode properties such as the shareability of the negativity among steering modes understood. Especially, since EPR steering is a prerequisite for remote preparation of Wigner negativity, one may intuitively expect that stronger steerability in the initial Gaussian states creates more Wigner negativity. With our quantitative investigation, we show that such a joint increase of steering and Wigner negativity does not always exist.

### Monogamy of remotely generated Wigner negativity

First, we study the multimode character of the remotely generated Wigner negativity by deriving constraints on the distribution of this negativity among various modes in the steering party (B1B2Bn) for a (1 + n)-mode Gaussian state $${\sigma }_{A{B}_{1}{B}_{2}\ldots {B}_{n}}$$. As a fundamental property of EPR steering, the CKW-type monogamy relation reveals that the sum of Gaussian steerability between any two modes cannot exceed their intergroup steerability, i.e., $${{{{\mathcal{G}}}}}^{{B}_{1}{B}_{2}\ldots {B}_{n}\to A}\ge \mathop{\sum }\nolimits_{i = 1}^{n}{{{{\mathcal{G}}}}}^{{B}_{i}\to A}$$, which bounds the achievable key rate of quantum secret sharing36. In analogy with this steering constraint, we establish a monogamy relation for the amount of the remotely generated Wigner negativity:

$${{{{\mathcal{N}}}}}_{{B}_{1}{B}_{2}\ldots {B}_{n}}({{{{\mathcal{L}}}}}_{A| {B}_{1}{B}_{2}\ldots {B}_{n}})\ge \mathop{\sum }\limits_{i=1}^{n}{{{{\mathcal{N}}}}}_{{B}_{i}}({{{{\mathcal{L}}}}}_{A| {B}_{i}}).$$
(2)

Here, $${{{{\mathcal{L}}}}}_{A| {B}_{j}\ldots {B}_{k}}$$ represents the optimal set of local operations on mode A to induce the largest amount of Wigner negativity in the group of modes (BjBk), where the subscript “ABjBk” represents that the choice of $${{{\mathcal{L}}}}$$ depends on the initial Gaussian steering from the group (BjBk) to mode A. Thus generating negativities in different modes requires different optimal operations on the steered mode A. For instance, inducing Wigner negativity in the steering mode Bj, or Bk, or their joint (BjBk), requires different local Gaussian transformations prior to a non-Gaussian operation (e.g. photon subtraction).

### Proof

Without loss of generality, let us focus on a tripartite scenario, in which the steering party B contains two modes B1 and B2. Now we use the fact that $${{{{\mathcal{N}}}}}_{{B}_{1}}({{{{\mathcal{L}}}}}_{A| {B}_{1}})\, > \,0$$ and $${{{{\mathcal{N}}}}}_{{B}_{2}}({{{{\mathcal{L}}}}}_{A| {B}_{2}})\, > \,0$$ cannot be true simultaneously, which is a consequence of another type of Gaussian steering monogamy relation: modes B1 and B2 cannot simultaneously steer mode A under Gaussian measurements34,35. Assuming that mode B1 can steer mode A, negativity can be generated only in the Wigner function of mode B1 under Alice’s local operation $${{{{\mathcal{L}}}}}_{A| {B}_{1}}$$, such that Eq. (2) takes the simpler form $${{{{\mathcal{N}}}}}_{{B}_{1}{B}_{2}}({{{{\mathcal{L}}}}}_{A| {B}_{1}{B}_{2}})\,\geqslant\, {{{{\mathcal{N}}}}}_{{B}_{1}}({{{{\mathcal{L}}}}}_{A| {B}_{1}})$$ (or the analogous expression with swapped B1 ↔ B2).

The local operation $${{{{\mathcal{L}}}}}_{A| {B}_{1}}$$ in the right side is chosen to generate the largest negativity $${{{{\mathcal{N}}}}}_{{B}_{1}}$$ in mode B1, determined by the Gaussian steering of mode A by individual mode B1. Because the Wigner negativity is nonincreasing under partial trace within the group (B1B2)16, it is straightforward as

$$\begin{array}{lll}{{{{\mathcal{N}}}}}_{{B}_{1}}({{{{\mathcal{L}}}}}_{A| {B}_{1}})&=&\int d{{{{\bf{r}}}}}_{{B}_{1}}\left|W[{{{\mathrm{Tr}}}\,}_{{B}_{2}}[{\rho }_{{B}_{1}{B}_{2}}]]({{{{\bf{r}}}}}_{{B}_{1}})\right|-1\\ &=&\int d{{{{\bf{r}}}}}_{{B}_{1}}\left|\int d{{{{\bf{r}}}}}_{{B}_{2}}W[{\rho }_{{B}_{1}{B}_{2}}]({{{{\bf{r}}}}}_{{B}_{1}},{{{{\bf{r}}}}}_{{B}_{2}})\right|-1\\ &\leqslant &\int d{{{{\bf{r}}}}}_{{B}_{1}}\int d{{{{\bf{r}}}}}_{{B}_{2}}\left|W[{\rho }_{{B}_{1}{B}_{2}}]({{{{\bf{r}}}}}_{{B}_{1}},{{{{\bf{r}}}}}_{{B}_{2}})\right|-1\\ &=&{{{{\mathcal{N}}}}}_{{B}_{1}{B}_{2}}({{{{\mathcal{L}}}}}_{A| {B}_{1}}).\end{array}$$

Here, $$W[{\rho }_{{B}_{1}{B}_{2}}]({{{\bf{r}}}})$$ represents the Wigner distribution of the joint state $${\rho }_{{B}_{1}{B}_{2}}$$. Note that the negativity $${{{{\mathcal{N}}}}}_{{B}_{1}{B}_{2}}({{{{\mathcal{L}}}}}_{A| {B}_{1}})$$ in the above inequality is also generated by the same local operation $${{{{\mathcal{L}}}}}_{A| {B}_{1}}$$, which may not be optimal for the group (B1B2). To create the largest negativity $${{{{\mathcal{N}}}}}_{{B}_{1}{B}_{2}}$$, we need to choose optimal local operations $${{{{\mathcal{L}}}}}_{A| {B}_{1}{B}_{2}}$$ based on joint steering of mode A by group (B1B2), leading to $${{{{\mathcal{N}}}}}_{{B}_{1}{B}_{2}}({{{{\mathcal{L}}}}}_{A| {B}_{1}{B}_{2}})\ge {{{{\mathcal{N}}}}}_{{B}_{1}{B}_{2}}({{{{\mathcal{L}}}}}_{A| {B}_{1}})$$. Afterwards it is promptly verified that $${{{{\mathcal{N}}}}}_{{B}_{1}{B}_{2}}({{{{\mathcal{L}}}}}_{A| {B}_{1}{B}_{2}})\ge {{{{\mathcal{N}}}}}_{{B}_{1}{B}_{2}}({{{{\mathcal{L}}}}}_{A| {B}_{1}})\ge {{{{\mathcal{N}}}}}_{{B}_{1}}({{{{\mathcal{L}}}}}_{A| {B}_{1}})$$.

An example is given in Fig. 2. In this linear optical network, a pure three-mode entangled Gaussian state can be generated in the local station by mixing three squeezed inputs, then two of the output modes B1 and B2 are sent to the remote nodes, as illustrated in Fig. 2a. When the first beam splitter is fixed at R1: (1 − R1) = 50:50 and the second beam splitter is adjustable, the steerabilities distributed among three modes in the initial Gaussian state are given in Fig. 2b. Here we choose two typical types of non-Gaussian operations for Alice. One is the single-photon subtraction, which can be effectively realized in experiments12,43. The corresponding Wigner negativities remotely created through photon subtraction on mode A are shown in Fig. 2c. The other conditional preparation of Wigner negativity by local Fock-state projection can be found in panel Fig. 2d, where mode A projects to $$\left|2\right\rangle$$44,45. It is clear that in both cases the two-mode and three-mode Gaussian steering $${{{{\mathcal{G}}}}}^{{B}_{1}\to A}$$, $${{{{\mathcal{G}}}}}^{{B}_{2}\to A}$$ and $${{{{\mathcal{G}}}}}^{{B}_{1}{B}_{2}\to A}$$ are necessary to induce Wigner negativity for individual mode B1, B2, and for the group (B1B2), respectively. However, constrained by the monogamy relation of Gaussian steering34, i.e., modes B1 and B2 cannot steer mode A simultaneously, the negativities generated in the individual mode $${{{{\mathcal{N}}}}}_{{B}_{1}}\, > \,0$$ and $${{{{\mathcal{N}}}}}_{{B}_{2}}\, > \,0$$ can never be satisfied at the same time. This means that when B1 receives a Wigner-negative state, it automatically guarantees that B2 did not acquire Wigner negativity. Meanwhile, the joint Wigner negativities created on the group (B1B2) are significantly higher than the negativity of either individual mode in both cases.

In a more extreme case with R2 = 0.5, no Wigner negativity is created in either mode individually, but their joint Wigner function can be negative, i.e., $${{{{\mathcal{N}}}}}_{{B}_{1}{B}_{2}}({{{{\mathcal{L}}}}}_{A| {B}_{1}{B}_{2}})\, > \,0$$. In Fig. 4 in the following section, we show that in presence of loss such a scenario can be quite common. This reminds of a context of quantum secret sharing where neither B1 nor B2 can acquire Wigner negativity, but only when both cooperate to achieve it. Moreover, such setups where Wigner negativity appears globally but not locally have an appealing interpretation: the operation in mode A created Wigner negativity before beamsplitter R2 and the latter delocalized it over mode B1 and B2. If neither B1 nor B2 locally manifest this Wigner negativity, it must be hidden in a non-trivial correlation between both modes. When the joint state for B1 and B2 is pure—as is the case in Fig. 2d—this is a clear signature of entanglement between the two modes. However, in the more general case where B1 and B2 share a mixed state such conclusion no longer holds. Nevertheless, as shown in Section “Distillation of nonlocal Wigner negativity”, the global negativity of $$W[{\rho }_{{B}_{1}{B}_{2}}]({{{\bf{r}}}})$$ implies that one can always distill Wigner negativity in mode B1 or B2 by performing a suitable measurement on the other mode.

### Quantification of the generated Wigner negativity

EPR steering refers to the ability of one system to adjust the state of another distant system by local measurements. Since remotely generated Wigner negativity is both enabled and constrained by Gaussian steering, one may intuitively expect that stronger steerability induces more Wigner negativity. However, we show that this is not the case by providing an intuitive quantification for some experimentally prominent bipartition (multimode) Gaussian states. We quantify the amount of Wigner negativity in the steering modes, via the purities of initial Gaussian states, and show that purity, rather than steerability, governs the amount of Wigner negativity that can be created. In some cases studied below, less squeezing in the initial Gaussian state (for both pure and mixed cases) produces weaker steerability but remotely creates stronger negativity.

It is well known that any two-mode Gaussian state can be transformed into a standard form46 through local linear unitary Bogoliubov operations (LLUBOs), so that the CM σAB reads

$${\sigma }_{AB,sf}=\left(\begin{array}{cc}{\sigma }_{A}&{\gamma }_{AB}\\ {\gamma }_{AB}^{\top }&{\sigma }_{B}\\ \end{array}\right)=\left(\begin{array}{llll}a&0&{c}_{1}&0\\ 0&a&0&{c}_{2}\\ {c}_{1}&0&b&0\\ 0&{c}_{2}&0&b\\ \end{array}\right)$$
(3)

with a,b ≥ 1 and $$ab-{c}_{1(2)}^{2}\ge 0$$. The two local purities $${\mu }_{A(B)}\equiv 1/\sqrt{\,{{\mbox{Det}}}\,{\sigma }_{A(B)}}=1/a(b)$$ and the global purity $${\mu }_{AB}\equiv 1/\sqrt{\,{{\mbox{Det}}}\,{\sigma }_{AB}}=1/\sqrt{(ab-{c}_{1}^{2})(ab-{c}_{2}^{2})}$$, are invariant under LLUBOs47. Furthermore, because LLUBOs are local and symplectic, the standard form σAB,sf manifests equal quantity of Gaussian steerability possessed in the initial states.

We now restrict our scope to two-mode Gaussian states with c1 = − c2 = c, which include the major experimentally realized CV EPR resources such as the two-mode EPR state with phase-insensitive losses and the two-mode squeezed thermal state. By focusing on the experimentally relevant case where a single-photon subtraction $${{{\mathcal{S}}}}$$ is performed on the steered mode, we derive that the amount of remotely generated Wigner negativity $${{{{\mathcal{N}}}}}_{B}$$ is determined by the purities of initial Gaussian state μA, μB, and μAB:

$${{{{\mathcal{N}}}}}_{B}({{{{\mathcal{S}}}}}_{A| B})=2\left[\frac{{{{\mbox{e}}}}^{\frac{{\mu }_{A}{\mu }_{B}-{\mu }_{AB}{\mu }_{A}}{{\mu }_{AB}-{\mu }_{A}{\mu }_{B}}}({\mu }_{A}{\mu }_{B}-{\mu }_{AB})}{{\mu }_{AB}({\mu }_{A}-1)}-1\right].$$
(4)

Exchanging μA ↔ μB, we can obtain the result for the other direction $${{{{\mathcal{N}}}}}_{A}({{{{\mathcal{S}}}}}_{B| A})$$. The derivation of the above relation is detailed in the section of “Methods”. Gaussian steerability and remotely created Wigner negativity are both determined by the local and global purities, but the dependence is very different. EPR steering provides a necessary bridge to induce Wigner negativity, but it is insufficient to unambiguously quantify the created Wigner negativity.

We explicitly show this point in an example where a two-mode EPR state is distributed over a single lossy channel characterized by ηA (the other channel is assumed ideal), as shown in Fig. 3, which is often used to demonstrate one-way steering26,27,48. The CM of this kind of state is in the standard form (3) with $$a={\eta }_{A}(\cosh 2r-1)+1$$, $$b=\cosh 2r$$, and $${c}_{1}=-{c}_{2}=\sqrt{{\eta }_{A}}\sinh 2r$$, where r is the squeezing parameter. The asymmetric Gaussian steerabilities in two directions are indicated in Fig. 3b, where the Gaussian steerability $${{{{\mathcal{G}}}}}^{A\to B}\, > \,0$$ when ηA > 0.5 while the other direction $${{{{\mathcal{G}}}}}^{B\to A}\, > \,0$$ happens for any ηA > 0. It also shows that higher squeezing level creates stronger steerability (blue lines). By performing a single-photon subtraction on the steered mode B, the Wigner negativity can be created in mode A ($${{{{\mathcal{N}}}}}_{A}({{{{\mathcal{S}}}}}_{B| A})\, > \,0$$) when ηA > 0.5 as well and becomes larger with increasing efficiency ηA (solid lines), as shown in Fig. 3c. Interestingly, for the other direction, by performing a single-photon subtraction on the steered mode A, the generated negativity $${{{{\mathcal{N}}}}}_{B}({{{{\mathcal{S}}}}}_{A| B})$$ does not vary with ηA. This observation can be understood as a consequence of losses in the mode of photon subtraction commute with the subtraction operation itself49. As derived in Eq. (11) in Section “Quantifying the remotely created Wigner negativity”, it can be seen that the Wigner negativity generated in mode B is only determined by the loss in its own channel (i.e. ηB, here it is considered as unity). This highlights the asymmetry of the induced Wigner negativity and suggests a way to remotely generate negativity that is robust to channel loss on the photon-subtraction side. Figure 3c also shows, unlike the Gaussian steerability plotted in Fig. 3b, the lower squeezing level leads to larger negativities (red lines). This can be explained by the purities of initial states given in Fig. 3d. The purities μA, μB, and μAB of the initial states with higher squeezing level are more sensitive to the loss (blue lines). Since purity plays a bigger role than squeezing in generating Wigner negativity50, the case with higher squeezing but lower purities can only lead to weaker Wigner negativities (blue lines).

Besides the examples discussed above, we also analyze the asymmetric Gaussian steerability and the properties of induced Wigner negativity for another important CV EPR resource—two-mode squeezed thermal states51, which are detailed in the section of “Methods”. Moreover, as shown in Section “Necessity of the additional local Gaussian transformation”, we also emphasize that producing EPR resources with CM in a standard form (3) can significantly simplify the procedure for remote generation of Wigner negativity and makes the resulting non-Gaussian state readily available for further applications.

Furthermore, for any three-mode globally pure Gaussian state, i.e., $${\mu }_{A{B}_{1}{B}_{2}}=1$$, the standard form of CM with respect to A − (B1B2) splitting is locally equivalent to

(5)

through Williamson and Bloch-Messiah decompositions47, where $${c}_{1}=\sqrt{{a}^{2}-1}$$. We can see that the system is qualitatively equivalent to a product of a two-mode squeezed vacuum state tensor an uncorrelated vacuum mode, such that the steering property between mode A and modes (B1B2) can be unitarily reduced to the two-mode case. Thus, the generated Wigner negativity of the group (B1B2) is equivalently indicated by

$${{{{\mathcal{N}}}}}_{{B}_{1}{B}_{2}}({{{{\mathcal{S}}}}}_{A| {B}_{1}{B}_{2}})=2\left[{e}^{-\frac{\mu }{1+\mu }}(1+\mu )-1\right],$$
(6)

where $$\mu ={\mu }_{A}={\mu }_{{B}_{1}{B}_{2}}=1/a$$. By the same method, this result (6) is also valid for (1 + n)-mode pure Gaussian states with respect to A − (B1B2Bn) splitting (n > 2).

Finally, as the CM for a more general mixed state contains more parameters, here we numerically show that the amount of remotely generated negativity is still related to the purities of initial Gaussian state. As shown in Fig. 4, we provide an evidence to show this conclusion still holds by a three-mode mixed state. In this scheme, the output modes B1 and B2 are both distributed over lossy channels characterized by $${\eta }_{{B}_{1}}$$ and $${\eta }_{{B}_{2}}$$, respectively, such that the global purity $${\mu }_{A{B}_{1}{B}_{2}}\, < \,1$$ when $${\eta }_{{B}_{1}},{\eta }_{{B}_{2}}\,\ne\, 1$$. For simplicity, we assume that the distance between two remote nodes and the local station is the same, thus $${\eta }_{{B}_{1}}={\eta }_{{B}_{2}}=\eta$$. As shown in Fig. 4b, due to the existence of channel losses (for a fixed value of η = 0.8), the area where neither mode B1 nor mode B2 can individually steer mode A has expanded to 0.242 < R2 < 0.758 compared with the scheme discussed in Fig. 2. The corresponding Wigner negativities remotely created in mode B1 or B2, or the group (B1B2) through a photon subtraction on mode A are given in Fig. 4c. It is clear that both Gaussian steerability and Wigner negativity appear in the same condition. Comparing with Fig. 4c, d, we can find that the amount of remotely created Wigner negativities is still determined by the purities of initial states.

## Discussion

We develop the scheme for remote generation of Wigner negativity through EPR steering to multimode scenario, and show the presence of constraints for distributing Wigner negativity over different modes. So far, multipartite steering has been demonstrated in various Gaussian systems, e.g., linear optical networks26,27, quantum frequency comb29, and Bose–Einstein condensates33. These experimental developments lay a favorable foundation for implementing remote generation of multipartite non-Gaussian states through photon subtraction or other appropriate operations. Furthermore, we present an intuitive and computable quantification of the generated Wigner negativity for bipartition (multimode) system in terms of the local and global purities of initial Gaussian states. Our results deepen the understanding of Wigner negativity as a resource and provide an important framework of non-Gaussian quantum information theory.

Our work also triggers several interesting questions to stimulate further research. For instance, as Gaussian steerability $${{{{\mathcal{G}}}}}^{A\to {B}_{1}}\, > \,0$$ and $${{{{\mathcal{G}}}}}^{A\to {B}_{2}}\, > \,0$$ can happen simultaneously, then by performing a single-photon subtraction on each mode B1, B2, can we achieve more significant increase of the negativity in mode A? In addition, for this direction the Gaussian steerability still follows the CKW-type monogamy constraint, however, this constraint does not hold any more for the generated negativity. We have observed a violation in a pure three-mode state (see Methods), i.e., $${{{{\mathcal{N}}}}}_{A}({{{{\mathcal{L}}}}}_{{B}_{1}{B}_{2}| A})\, < \,{{{{\mathcal{N}}}}}_{A}({{{{\mathcal{L}}}}}_{{B}_{1}| A})+{{{{\mathcal{N}}}}}_{A}({{{{\mathcal{L}}}}}_{{B}_{2}| A})$$. Moreover, after non-Gaussian operations on the steered mode, the resulting system cannot be fully captured by the second-order correlations given in CM. To this day, relatively little is known about the characteristics of non-Gaussian steering52.

## Methods

### Quantifying the remotely created Wigner negativity

It is of particular interest to us is whether stronger steerability in the initial Gaussian states induces more Wigner negativity, as it is enabled and constrained by Gaussian steering. To answer this, we need first quantify the amount of Wigner negativity. In this part, we aim to derive the qualitative measure of Wigner negativity Eq. (4) in the main text by focusing on the experimentally relevant case where a photon subtracted from the steered mode in two-mode Gaussian states c1 = −c2 = c, which include the major experimentally realized CV EPR states.

Let us recall that, any two-mode Gaussian state can be transformed through LLUBOs to the standard form (3). Using the formula derived from ref. 18, we obtain the reduced Wigner function of the steering mode B after the local Gaussian transformation RAB combined with a single-photon subtraction applied on the steered mode A,

$$\begin{array}{rlr}{W}_{B}\left({{{{\boldsymbol{\beta }}}}}_{B}\right)&=\frac{\exp \left\{-\frac{1}{2}\left({{{{\boldsymbol{\beta }}}}}_{B},{\sigma }_{B}^{-1}{{{{\boldsymbol{\beta }}}}}_{B}\right)\right\}}{2\pi \sqrt{{{\mathrm{Det}}}\,{\sigma }_{B}}\left[{{\mathrm{Tr}}}\,\left({R}_{A| B}^{\top }{\sigma }_{A}{R}_{A| B}\right)-2\right]} &\\ & \quad\times\,\left[{{{{\boldsymbol{\beta }}}}}_{B}^{\top }{{\sigma }_{B}^{-1}}^{\top }{\gamma }_{AB}^{\top }{R}_{A| B}{R}_{A| B}^{\top }{\gamma }_{AB}{\sigma }_{B}^{-1}{{{{\boldsymbol{\beta }}}}}_{B}+{{\mathrm{Tr}}}\,({R}_{A| B}^{\top }{V}_{A| B}{R}_{A| B})-2\right]\\ &=\frac{\exp \left\{-\frac{1}{2}\left({{{{\boldsymbol{\beta }}}}}_{B},{\sigma }_{B}^{-1}{{{{\boldsymbol{\beta }}}}}_{B}\right)\right\}}{2\pi \sqrt{{{\mathrm{Det}}}\,{\sigma }_{B}}\left[{{\mathrm{Tr}}}\,\left(\nu {\sigma }_{A}{V}_{A| B}^{-1}\right)-2\right]} \\ & \quad\times\,\left(\nu {{{{\boldsymbol{\beta }}}}}_{B}^{\top }{{\sigma }_{B}^{-1}}^{\top }{\gamma }_{AB}^{\top }{V}_{A| B}^{-1}{\gamma }_{AB}{\sigma }_{B}^{-1}{{{{\boldsymbol{\beta }}}}}_{B}+2\nu -2\right),\end{array}$$
(7)

where $${{{{\boldsymbol{\beta }}}}}_{B}={({x}_{B},{p}_{B})}^{\top }$$ is the coordinate in a multimode phase spaces of subsystem B, $${V}_{A| B}={\sigma }_{A}-{\gamma }_{AB}{\sigma }_{B}^{-1}{\gamma }_{AB}^{\top }$$ is the Schur complement of σB and ν is the corresponding symplectic eigenvalue. The Schur complement VAB can be decomposed through Williamson decomposition via $${V}_{A| B}=\nu {S}_{A| B}^{\top }{S}_{A| B}$$, where SAB is the corresponding symplectic matrix and a local Gaussian transformation $${R}_{A| B}={S}_{A| B}^{-1}$$. When it comes to our particular interest subclass c1 = −c2 = c, it is easy to find out that the Schur complement is a multiple of identity matrix so that there is no need to perform an additional local Gaussian operation. Then we get

$${W}_{B}\left({x}_{B},{p}_{B}\right)=\frac{{{{\mbox{e}}}}^{-\frac{{x}_{B}^{2}+{p}_{B}^{2}}{2b}}\left[2{b}^{2}(a-1)-2b{c}^{2}+{c}^{2}({x}_{B}^{2}+{p}_{B}^{2})\right]}{4\pi {b}^{3}(a-1)},$$
(8)

which is circularly symmetric. It is straightforward to calculate Wigner negativity using integral,

$${{{{\mathcal{N}}}}}_{B}({{{{\mathcal{S}}}}}_{A| B})=\frac{2{c}^{2}{{{\mbox{e}}}}^{\frac{b(a-1)}{{c}^{2}}-1}}{b(a-1)}-2.$$
(9)

By expressing $${{{{\mathcal{N}}}}}_{B}({{{{\mathcal{S}}}}}_{A| B})$$ in terms of purities μAB = 1/(ab − c2), μA = 1/a, μB = 1/b, Eq. (9) becomes

$${{{{\mathcal{N}}}}}_{B}({{{{\mathcal{S}}}}}_{A| B})=2\left[\frac{{{{\mbox{e}}}}^{\frac{{\mu }_{A}{\mu }_{B}-{\mu }_{AB}{\mu }_{A}}{{\mu }_{AB}-{\mu }_{A}{\mu }_{B}}}({\mu }_{A}{\mu }_{B}-{\mu }_{AB})}{{\mu }_{AB}({\mu }_{A}-1)}-1\right].$$
(10)

Figure 3 showed the case of two-mode squeezed vacuum state transmitted with single lossy channel of mode A. To further understand the effect of channel losses, we also take into account the loss in mode B’s channel, characterized by ηB. When both channels are nonideal, the generated negativity in mode B is

$${{{{\mathcal{N}}}}}_{B}=\frac{4{{{\mbox{e}}}}^{(\frac{1}{2{\eta }_{B}}-1){{{\mbox{sech}}}}^{2}r}{\eta }_{B}{\cosh }^{2}r}{1-{\eta }_{B}+{\eta }_{B}\cosh 2r}-2,$$
(11)

which merely depends on ηB but still does not vary with the loss in the channel of the steered mode A.

In the following, we analyze another important kind of experimentally realized CV EPR states–two-mode squeezed thermal states. The CM elements of these states are $$a=({n}_{A}+{n}_{B}+1)\cosh (2r)+({n}_{A}-{n}_{B})$$, $$b=({n}_{A}+{n}_{B}+1)\cosh (2r)-({n}_{A}-{n}_{B})$$, $${c}_{1}=-{c}_{2}=({n}_{A}+{n}_{B}+1)\sinh (2r)$$, where nA, nB are the average number of thermal photons for each subsystem51. We set the thermal noise only on one input mode with nA and leave nB = 0, as illustrated in Fig. 5a. The asymmetric Gaussian steerability in two directions varying with nA is denoted in Fig. 5b, and as a consequence the induced Wigner negativity on the steering mode by applying single-photon subtraction on the steered mode is quantified in Fig. 5c. Note that the effect of thermal noise on the steered mode is more significant than that on the steering mode, which is opposite to the effect of losses on two modes in the main text. As there exists a thermal barrier in the direction $${{{{\mathcal{G}}}}}^{B\to A}$$, and correspondingly, a nonzero $${{{{\mathcal{N}}}}}_{B}({{{{\mathcal{S}}}}}_{A| B})$$ can exist only when nA < 0.682. In the opposite direction, $${{{{\mathcal{G}}}}}^{A\to B}\, > \,0$$ and thus $${{{{\mathcal{N}}}}}_{A}({{{{\mathcal{S}}}}}_{B| A})\, > \,0$$ for arbitrarily large value of thermal noise nA. It is clear that the amount of remotely created Wigner negativity is quantitatively determined by the purities of initial states, as plotted in Fig. 5d.

### Distillation of nonlocal Wigner negativity

In Figs. 2 and 4 we observe cases where the local Wigner functions of B1 and B2 are fully positive, but nevertheless, Wigner negativity arises in the joint state for (B1B2). In this particular situation, one could argue that Wigner negativity is hidden in the nonlocal part of the Wigner function. Nevertheless, it turns out that such hidden negativity can always be unveiled by performing a well-chosen measurement on either B1 or B2. Here we prove this claim.

First of all, we introduce the displaced parity operator

$$\hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{1}},{{{{\bf{r}}}}}_{{B}_{2}}):= {\hat{D}}^{{\dagger} }({{{{\bf{r}}}}}_{{B}_{1}},{{{{\bf{r}}}}}_{{B}_{2}}){(-{\mathbb{1}})}^{\hat{{n}_{{B}_{1}{B}_{2}}}}\hat{D}({{{{\bf{r}}}}}_{{B}_{1}},{{{{\bf{r}}}}}_{{B}_{2}}),$$
(12)

where the displacement operator $$\hat{D}({{{{\bf{r}}}}}_{{B}_{1}},{{{{\bf{r}}}}}_{{B}_{2}})=\hat{D}({{{{\bf{r}}}}}_{{B}_{1}})\otimes \hat{D}({{{{\bf{r}}}}}_{{B}_{2}})$$, with $$\hat{D}({{{{\bf{r}}}}}_{{B}_{k}}):= \exp (i{\hat{{{{\boldsymbol{\xi }}}}}}_{{B}_{k}}^{\top }{{{\boldsymbol{\Omega }}}}{{{{\bf{r}}}}}_{{B}_{k}}/2)$$. Furthermore, $${\hat{n}}_{{B}_{1}{B}_{2}}$$ is the number operator for the joint system (B1B2). Because this number operator can be written as a sum of the local number operators $${\hat{n}}_{{B}_{1}{B}_{2}}={\hat{n}}_{{B}_{1}}+{\hat{n}}_{{B}_{2}}$$, we can now write that

$$\hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{1}},{{{{\bf{r}}}}}_{{B}_{2}})=\hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{1}})\otimes \hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{2}}).$$
(13)

The values of the Wigner function are given by53,54

$$\begin{array}{ll}W[{\rho }_{{B}_{1}{B}_{2}}]({{{{\bf{r}}}}}_{{B}_{1}},{{{{\bf{r}}}}}_{{B}_{2}})=\frac{1}{4{\pi }^{2}}{\langle \hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{1}},{{{{\bf{r}}}}}_{{B}_{2}})\rangle }_{{B}_{1}{B}_{2}}\\ \qquad\qquad\qquad\qquad\,=\frac{1}{4{\pi }^{2}}{\langle \hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{1}})\otimes \hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{2}})\rangle }_{{B}_{1}{B}_{2}},\end{array}$$
(14)

where we introduce the shorthand notation $${\langle \hat{X}\rangle }_{{B}_{1}{B}_{2}}:= {{\mathrm{Tr}}}\,[{\rho }_{{B}_{1}{B}_{2}}\hat{X}]$$.

The identity (14) is now particularly useful to express the correlation between displaced parity measurements on modes B1 and B2 as

$$\begin{array}{ll}{\langle \hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{1}})\otimes \hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{2}})\rangle }_{{B}_{1}{B}_{2}}-{\langle \hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{1}})\rangle }_{{B}_{1}}{\langle \hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{2}})\rangle }_{{B}_{2}}\\ =4{\pi }^{2}\left\{W[{\rho }_{{B}_{1}{B}_{2}}]({{{{\bf{r}}}}}_{{B}_{1}},{{{{\bf{r}}}}}_{{B}_{2}})-W[{\rho }_{{B}_{1}}]({{{{\bf{r}}}}}_{{B}_{1}})W[{\rho }_{{B}_{2}}]({{{{\bf{r}}}}}_{{B}_{2}})\right\}.\end{array}$$
(15)

In the specific case where $${{{{\mathcal{N}}}}}_{{B}_{1}{B}_{2}}\, > \,0$$ and $${{{{\mathcal{N}}}}}_{{B}_{1}}={{{{\mathcal{N}}}}}_{{B}_{2}}=0$$ the above identity implies the existence of values $${{{{\bf{r}}}}}_{{B}_{1}}$$ and $${{{{\bf{r}}}}}_{{B}_{2}}$$ for which $${\langle \hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{1}})\otimes \hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{1}})\rangle }_{{B}_{1}{B}_{2}}-{\langle \hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{1}})\rangle }_{{B}_{1}}{\langle \hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{2}})\rangle }_{{B}_{2}}\, < \,0$$. In other words, there is always a case where the displaced parity measurements are anti-correlated.

A state which has Wigner negativity is always a state where for some phase space coordinates $${{{{\bf{r}}}}}_{{B}_{k}}$$ measurements of the observable $$\hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{k}})$$ provide the outcome −1 with a higher probability than the outcome +1. This observation is important in the light of (15). Let us now fix $${{{{\bf{r}}}}}_{{B}_{1}}$$ and $${{{{\bf{r}}}}}_{{B}_{2}}$$ such that $$W[{\rho }_{{B}_{1}{B}_{2}}]({{{{\bf{r}}}}}_{{B}_{1}},{{{{\bf{r}}}}}_{{B}_{2}})\, < \,0$$. For joint measurements of $$\hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{1}})$$ and $$\hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{2}})$$, we are most likely to obtain opposite parities due to the anti-correlation in (15). More formally phrased, the only way of obtaining this anti-correlation is through

$$\begin{array}{rlr}{{{\rm{Prob}}}}&[\hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{2}})=-1| \hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{1}})=+1]&\\ & > {{{\rm{Prob}}}}[\hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{2}})=+1| \hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{1}})=+1],\\ {{{\rm{Prob}}}}&[\hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{1}})=-1| \hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{2}})=+1]\\ & > {{{\rm{Prob}}}}[\hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{1}})=+1| \hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{2}})=+1].\end{array}$$
(16)

However, this means that when we measure $$\hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{1}})$$ and post-select on measurement outcomes +1, we find that for the state in B2 is given by

$$\begin{array}{lll}W[{\rho }_{{B}_{2}}]({{{{\bf{r}}}}}_{{B}_{2}})\,=\,\frac{1}{2\pi }{\langle \hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{2}})\rangle }_{{B}_{2}}\\ \qquad\qquad\qquad=\,\frac{1}{2\pi }\left({{{\rm{Prob}}}}[\hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{2}})=+1| \hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{1}})=+1]\right.\\ \qquad\qquad\qquad\quad\left.-{{{\rm{Prob}}}}[\hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{2}})=-1| \hat{{{\Pi }}}({{{{\bf{r}}}}}_{{B}_{1}})=+1]\right)\\ \qquad\qquad\qquad <\, 0.\end{array}$$
(17)

In other words, when the global Wigner function in (B1B2) is non-positive, whereas the local Wigner functions in B1 and B2 are positive, conditioning on a positive outcome for a displaced parity measurement on either B1 or B2 allows to prepare a Wigner-negative state in B2 or B1, respectively. Even though this proof shows the existence of some measurement to prepare Wigner negativity, in many cases, one can probably find more convenient measurements to unveil the Wigner negativity.

We note finally that this is a very complementary setting to the one discussed in the remainder of this article. We emphasized that to create and distribute Wigner negativity when the global state is Gaussian, we require quantum steering in this Gaussian state. Yet, when the global state is already Wigner negative, the requirements on the level of the correlations are far less stringent. A good example to keep in mind is that of the state $${\rho }_{{B}_{1}{B}_{2}}=(\left|0,1\right\rangle \left\langle 0,1\right|+\left|1,0\right\rangle \left\langle 1,0\right|)/2$$. This state is globally Wigner negative, but locally leads to positive Wigner functions. Conditioning on a measurement outcome associated with the projector $$\left|0\right\rangle \left\langle 0\right|$$ on either subsystem will project the other subsystem into a state $$\left|1\right\rangle \left\langle 1\right|$$. As such, a simple Gaussian measurement on a global Wigner negative state with only classical correlations can lead to local Wigner negativity. This illustrates that delocalized Wigner negativity is a very strong resource to create local Wigner negativity.

### Necessity of the additional local Gaussian transformation

Here, we prove that the additional local Gaussian transformation RAB required prior to photon subtraction in ref. 18 to optimize the induced Wigner negativity is no longer needed if and only if the CM of two-mode Gaussian states in the standard form σAB,sf satisfies c1 = −c2 = c. Thus producing EPR resource with CM in a standard form significantly simplifies the procedure for remote generation of Wigner negativity and makes the resulting non-Gaussian state readily available for further applications.

From the standard form σAB,sf, the Schur complement of σB is given by

$${V}_{A| B}=\left(\begin{array}{ll}a-{c}_{1}^{2}/b&0\\ 0&a-{c}_{2}^{2}/b\end{array}\right),$$
(18)

whose symplectic eigenvalue is $$v=\sqrt{(a-{c}_{1}^{2}/b)(a-{c}_{2}^{2}/b)}$$. When there exists Gaussian steering $${{{{\mathcal{G}}}}}^{B\to A}$$, the symplectic eigenvalue v must be smaller than 142. Without any local Gaussian transformation RAB prior to the photon subtraction on mode A, the condition for $${W}_{B}\left({{{{\boldsymbol{\beta }}}}}_{B}\right) \,<\, 0$$ should be $${{\mathrm{tr}}}\,\left[{V}_{A| B}\right] \,< \,2$$. Note that every CM σAB that corresponds to a physical quantum state has to satisfy the bona fide condition $$a-{c}_{1}^{2}/b\, > \,0$$ and $$a-{c}_{2}^{2}/b\, > \,0$$42, then we have

$${{\mathrm{Tr}}}\,\left[{V}_{A| B}\right]=\left(a-\frac{{c}_{1}^{2}}{b}\right)+\left(a-\frac{{c}_{2}^{2}}{b}\right)\geqslant 2\sqrt{\left(a-\frac{{c}_{1}^{2}}{b}\right)\left(a-\frac{{c}_{2}^{2}}{b}\right)}=2v.$$
(19)

The above inequality can be saturated if and only if $${c}_{1}^{2}={c}_{2}^{2}$$. With this condition, $${{\mathrm{tr}}}\,\left[{V}_{A| B}\right]\, < \,2$$ is equivalent to v < 1, i.e., the photon subtraction on mode A can always generate Wigner negativity in mode B as long as $${{{{\mathcal{G}}}}}^{B\to A}\, > \,0$$ without any prior local Gaussian transformation. Otherwise, if $${c}_{1}^{2}\,\ne \,{c}_{2}^{2}$$, then $${{\mathrm{tr}}}\,\left[{V}_{A| B}\right]\, > \,2v$$, which means an additional local Gaussian transformation RAB is necessary to make EPR steering sufficient for remotely generating Wigner negativity. This complements the results of ref. 18, where it was shown that an additional local Gaussian transformation prior to photon subtraction is necessary to make EPR steering sufficient for remotely generating Wigner negativity. This Gaussian transformation requires inline squeezing and is experimentally challenging. Our results can be used as a recipe to prepare resourceful Gaussian states for the remote generation of Wigner negativity without the need for inline squeezing.

### Distribution of Wigner negativity in the other direction

Steering is a directional form of nonlocality, related to the Einstein “spooky” paradox, which is fundamentally defined differently from entanglement. In previous work36, we have derived the CKW-type monogamy inequalities for multipartite Gaussian steering in two directions. We then wonder whether the CKW-type monogamy inequality holds for the distribution of Wigner negativity created in the opposite direction, i.e., $${{{{\mathcal{N}}}}}_{A}({{{{\mathcal{L}}}}}_{{B}_{1}{B}_{2}| A})\ge {{{{\mathcal{N}}}}}_{A}({{{{\mathcal{L}}}}}_{{B}_{1}| A})+{{{{\mathcal{N}}}}}_{A}({{{{\mathcal{L}}}}}_{{B}_{2}| A})$$.

We present a three-mode entangled Gaussian state that is similar to the case shown in Fig. 2a, but with the first beamsplitter being adjustable R1: (1 − R1) and the second fixed as a balanced one. In particular, when the first beamsplitter is adjusted at R1 = 1/3, the initial Gaussian state is produced as a GHZ-like state. For the direction where mode A acts as the steering party to steer the modes B1, B2, the Gaussian steerability distributed among three modes and the corresponding Wigner negativities remotely created by a single-photon subtraction on the individual or joint modes B1, B2 are denoted in Fig. 6a, b. It is clear that the two-mode and three-mode Gaussian steerability $${{{{\mathcal{G}}}}}^{A\to {B}_{1}({B}_{2})}\, > \,0$$ and $${{{{\mathcal{G}}}}}^{A\to {B}_{1}{B}_{2}}\, > \,0$$ are necessary to induce negativities $${{{{\mathcal{N}}}}}_{A}({{{{\mathcal{S}}}}}_{{B}_{1}({B}_{2})| A})$$ and $${{{{\mathcal{N}}}}}_{A}({{{{\mathcal{S}}}}}_{{B}_{1}{B}_{2}| A})$$ in the Wigner functions of the steering mode A, respectively. Interestingly, we observe that $${{{{\mathcal{N}}}}}_{A}({{{{\mathcal{S}}}}}_{{B}_{1}{B}_{2}| A})\, < \,{{{{\mathcal{N}}}}}_{A}({{{{\mathcal{S}}}}}_{{B}_{1}| A})+{{{{\mathcal{N}}}}}_{A}({{{{\mathcal{S}}}}}_{{B}_{2}| A})$$ when R1 approaches to 1 (Fig. 6d), even though the Gaussian steerability still follows the monogamy constraint, i.e., $${{{{\mathcal{G}}}}}^{A\to {B}_{1}{B}_{2}}-{{{{\mathcal{G}}}}}^{A\to {B}_{1}}-{{{{\mathcal{G}}}}}^{A\to {B}_{2}}\, > \,0$$ presented in Fig. 6c. This settles an open question for the shareability of generated Wigner negativity in this direction.

## Data availability

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

## Code availability

The codes for numerical simulation and data processing are available from the corresponding authors upon reasonable request.

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## Acknowledgements

M.W. acknowledges enlightening discussions with C. Dross regarding Section “Distillation of nonlocal Wigner negativity”. This work is supported by the National Natural Science Foundation of China (Grants No. 12004011, No. 11975026, and No. 12125402), the National Key R&D Program of China (Grant No. 2019YFA0308702). Q.H. also thanks partial support from Beijing Natural Science Foundation (Grant No. Z190005) and the Key R&D Program of Guangdong Province (Grant No. 2018B030329001). N.T. and M.W. received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No 899587.

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Y.X. and S.L. contributed equally to this work. All authors contributed to the research and the preparation of the manuscript.

### Corresponding author

Correspondence to Qiongyi He.

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Xiang, Y., Liu, S., Guo, J. et al. Distribution and quantification of remotely generated Wigner negativity. npj Quantum Inf 8, 21 (2022). https://doi.org/10.1038/s41534-022-00533-3

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