Unlimited One-Way Steering

Nonlocality is among the central features of quantum theory. This effect manifests at different levels. On the one hand, in the mathematical structure of the theory (i.e. the Hilbert space) via the notion of entanglement Horodecki et al. (2009); Gühne and Tóth (2009). On the other hand, the measurement statistics of local measurements performed on entangled states can feature strong nonlocal correlations, which are incompatible with any local model Brunner et al. (2014). Initially, believed to be two facets of the same phenomenon, it is now clear that entanglement and quantum Bell nonlocality are in fact inherently different, see e.g. Werner (1989); Barrett (2002).

Another perspective on quantum nonlocality is provided by the notion of quantum steering, see Cavalcanti and Skrzypczyk (2016); Uola et al. (2020) for reviews. This effect, formalized by Wiseman, Jones and Doherty Wiseman et al. (2007), takes its roots in the early works of Einstein-Podolsky-Rosen Einstein et al. (1935) and Schrödinger Schrödinger (1935, 1936). Here, an untrusted party (Bob) wants to convince another party (Alice) that they share an entangled state. By demonstrating his ability to remotely steer Alice’s local state in different measurement basis, Bob can convince Alice. Steering is thus an inherently asymmetrical task, contrary to entanglement and Bell nonlocality.

Interestingly, the asymmetry of steering is also observed at the level of quantum states. Specifically, there exist entangled states that lead to steering from Bob to Alice, but not the other way around Bowles et al. (2014); Bob can convince Alice that the shared state is entangled, while Alice can never convince Bob, even if she would use all possible local measurements. This effect, coined “one-way steering” has attracted considerable attention in recent years, with many examples in low-dimensional (mostly two-qubit) Bowles et al. (2014); Skrzypczyk et al. (2014); Quintino et al. (2015); Bowles et al. (2016); Zeng (2022); Márton et al. (2021) and continuous variable Gaussian systems Midgley et al. (2010); Olsen (2013) as well as experimental demonstrations, see e.g. Händchen et al. (2012); Wollmann et al. (2016); Sun et al. (2016); Tischler et al. (2018).

A relevant question is whether there exists different forms of one-way steering, and whether some are stronger than others. So far, this question has not been discussed, due to the lack of an appropriate measure for steering in this context. Here, we tackle this problem, taking advantage of the recently introduced notion of genuine high-dimensional steering Designolle et al. (2021), see also Designolle (2022); de Gois et al. (2022). This allows for a dimensional quantification of steering, specifically to lower bound the Schmidt number of an entangled state in a steering scenario. We then ask whether there exist entangled state with the following properties: (i) Alice cannot steer the state of Bob (even when allowing for all possible local measurements), and (ii) Bob can steer Alice’s state strongly, i.e. ensuring the presence of high-dimensional entanglement (Schmidt number at least $d$). We answer this question in the affirmative by constructing a family of entangled states (of dimension $d\times(d+1)$) with the above two properties for any finite dimension $d$. To do so, we exploit the connection between steering and measurement incompatibility Quintino et al. (2014); Uola et al. (2014, 2015); Kiukas et al. (2017), and also its recent generalisation Jones et al. (2022) to high-dimensional steering and the concept of $n$-simulability of measurements Ioannou et al. (2022). Notably, we give a sufficient condition for the joint measurability of the set of all POVMs subjected to the combined effect of noise and loss.

We consider a scenario where two distant parties, Alice and Bob, share an entangled state $\varrho_{AB}$. Each party performs local measurements, represented by sets of positive-operator-valued measures (POVMs). Specifically, Alice’s measurements are described by a set of POVMs $\{A_{a|x}\}$, where $x$ denotes the measurement choice and $a$ the outcome, with the properties that $A_{a|x}\geq 0$ and $\sum_{a}A_{a|x}=\openone$ for all $a$ and $x$. Similarly, for Bob we define the set of POVMs $\{B_{b|y}\}$.

As the shared state is entangled, the effect of each party’s local measurements is to remotely prepare (steer) the other party’s state. This is described via two so-called state assemblages: $\{\sigma^{A}_{b|y}\}$ describe the states of Alice’s system conditioned on Bob’s measurement while $\{\sigma^{B}_{a|x}\}$ are Bob’s states given Alice’s measurement. These are given by

The main question we address here is how different these two state assemblages can become, in other words how asymmetric the effect of steering can be. Loosely speaking we are looking for an entangled state $\varrho_{AB}$ such that one of the state assemblages, say $\{\sigma^{A}_{b|y}\}$, is classical (in the sense that it can never lead to steering), while the other assemblage $\{\sigma^{B}_{a|x}\}$ is highly non-classical (in the sense that it exhibits strong steering, witnessing high entanglement dimensionality).

Before defining the problem more precisely, let us first observe that we are looking for some entangled states that are high-dimensional and asymmetrical. Obviously, if the state $\varrho_{AB}$ would be symmetrical under the exchange of Alice and Bob, any assemblage obtainable in one direction can also be obtained the other way around. Moreover, the state $\varrho_{AB}$ should feature a high entanglement dimensionality, as quantified here via the Schmidt number Terhal and Horodecki (2000); Sanpera et al. (2001): the Schmidt number of $\varrho_{AB}$ is the minimum $n$ such that there exists a decomposition $\varrho_{AB}=\sum_{j}p_{j}\ket{\psi_{j}}\bra{\psi_{j}}$ where all $\ket{\psi_{j}}$ are pure entangled states of Schmidt rank at most $n$.

More formally, we are looking for entangled states with the following two properties:

1. 1.

   The assemblage $\{\sigma^{A}_{b|y}\}$ admits a local hidden state (LHS) model Wiseman et al. (2007):

   $$\sigma^{A}_{b|y}=p(b|y)\sum_{\lambda}p(\lambda|b,y)\sigma_{\lambda}\quad\forall b,y\,.$$

   (3)

   Importantly, this should hold for any set of measurements for Bob $\{B_{b|y}\}$. Thus, no steering from Bob to Alice is possible, implying that Bob can never convince Alice that the shared state $\varrho_{AB}$ is entangled.

2. 2.

   The assemblage $\{\sigma^{B}_{a|x}\}$ is not $(d-1)$-preparable Designolle et al. (2021), i.e.

   $$\sigma^{B}_{a|x}\neq\sum_{\lambda}p(\lambda)\text{tr}_{A}[(M^{\lambda}_{a|x}\otimes\openone)\rho^{\lambda}_{AB}]$$

   (4)

   with $p(\lambda)$ an arbitrary probability distribution, $M^{\lambda}_{a|x}$ arbitrary measurements (in $\mathbb{C}^{d}$), and all $\rho^{\lambda}_{AB}$ being arbitrary states of Schmidt number (at most) $d-1$. Thus, genuine $d$-dimensional steering is demonstrated, so that Alice can convince Bob that the shared state $\varrho_{AB}$ has large Schmidt number $d$. For short, we say that $\varrho_{AB}$ is $d$-steerable.

In the following, we construct a family of entangled states in dimension $d\times(d+1)$ with the above two properties, for any $d$. This shows that one-way steering can become unlimited, in the sense of being maximally asymmetrical. Specifically, we start from a maximally entangled two-qudit state $\ket{\phi^{+}_{d}}=\frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}\ket{k,k}$ (with the notation $\Phi_{d}^{+}=\outerproduct{\phi^{+}_{d}}{\phi^{+}_{d}}$) and apply successively a white noise (depolarizing) channel followed by a loss channel, defined by

The depolarizing (loss) channel replaces the original state with a maximally mixed state (vacuum state $\outerproduct{\textup{\,\o\,}}{\textup{\,\o\,}}$) with probability $1-p$ ($1-\eta$). Note that the vacuum state represents an additional ”level”, orthogonal to the input Hilbert space, so that the output state has dimension $d+1$. After the two channels, we obtain the final state

which is of dimension $d\times(d+1)$.

(i) $\varrho_{AB}^{(\eta,p)}$ is $d$-steerable from Alice to Bob if

(ii) $\varrho_{AB}^{(\eta,p)}$ is unsteerable from Bob to Alice if

Measurement incompatibility is a property of a set of measurements which cannot be performed simultaneously on a single copy of a system. Formally, a set of measurements $\{M_{a|x}\}$ is called incompatible if there exists no ”parent” POVM $\{E_{\lambda}\}$ and classical post-processings $\{p(a|x,\lambda)\}$ such that

Sets of measurements allowing for such a model are called jointly measurable; see e.g. Heinosaari et al. (2016); Gühne et al. (2023) for reviews.

Our focus is on the (in)compatibility of measurements which are noisy and lossy. For any POVM $\{M_{a}\}$ with $m$ outputs $a=1,\dots,m$ acting on a system of dimension $d$, we define its imperfect version $\{\bar{M}_{a}^{(\eta,p)}\}$ as a POVM with $m+1$ elements given by

The noisy measurement apparatus behaves like the ideal one with probability $\eta p$, produces a random outcome with probability $\eta(1-p)$, and does not click with probability $1-\eta$ (formally this corresponds to the no-click outcome $a=\textup{\,\o\,}$). We are interested in knowing for which values of $(\eta,p)$ all measurements in a given dimension $d$ become compatible. The following result gives a sufficient condition.

Proof: To prove the result we present an explicit construction, inspired by Hirsch et al. (2013), able to simulate all imperfect POVMs with noise parameters fulfilling Eq. (13). Note that it is sufficient to show that it can simulate all rank-one POVMs $\{M_{a}=\alpha_{a}\outerproduct{\varphi_{a}}{\varphi_{a}}\}$ with $\sum\alpha_{a}=d$, since they can be post-processed to simulate all other measurements. The post-processing is done by mixing POVM elements, i.e. coarse-graining the corresponding outputs, by linearity it is consistent with the noisification of the measurements defined in Eq. (12).

We take the parent POVM to be the covariant one – the continuous-valued measurement with the density

where $\ket{\bm{z}}=\sum_{k=0}^{d-1}z_{k}\ket{k}$ with $\bm{z}\in\mathds{C}^{d}$ and $|\bm{z}|^{2}=1$, and $\differential\bm{z}$ is the invariant (under unitary transformations) measure over pure quantum states $\ket{\bm{z}}$ such that $\int\differential\bm{z}\outerproduct{\bm{z}}{\bm{z}}=\frac{1}{d}\mathds{1}_{d}$.

To define the response function $p(a|\bm{z})$ simulating a notified rank-one POVM $\{\alpha_{a}\outerproduct{\varphi_{a}}{\varphi_{a}}\}_{a}$ we proceed in two steps. First one samples a possible output $a$ from the probability distribution $\frac{\alpha_{a}}{d}$. Given $a$ and the corresponding state $\ket{\varphi_{a}}$, one simulates a two outcome POVM $\{N_{a}^{(a)},N_{\textup{\,\o\,}}^{(a)}\}$ by using the deterministic response functions

and $p^{(a)}(\textup{\,\o\,}|\bm{z})=1-p^{(a)}(a|\bm{z})$, with a parameter $t\in[0,1]$. Let us now compute the resulting operators

and $N_{\textup{\,\o\,}}^{(a)}=\mathds{1}_{d}-N_{a}^{(a)}$. One notes that $N_{a}^{(a)}=UN_{a}^{(a)}U^{\dagger}$ is invariant under all unitary transformations $U$ which leave the state $\ket{\varphi_{a}}$ unchanged $U\ket{\varphi_{a}}=\ket{\varphi_{a}}$, since both the measure $\differential\bm{z}$ and the response function $p^{(a)}(a|\bm{z})$ are invariant under such transformations. In other words, any such unitary commutes with our operator $[U,N_{a}^{(a)}]=0$. It follows that $N_{a}^{(a)}$ has $\ket{\varphi_{a}}$ for an eigenstate, and is furthermore proportional to identity on the orthogonal subspace. I.e. it is of the form

with the scalar functions that can be computed as

These integrals are straightforward to compute. In the Appendix B we show that they give $\bar{A}_{d}(t)=(1-t)^{d-1}\left((d-1)t+1\right)$ and $\bar{T}_{d}(t)\equiv\bar{A}_{d}(t)+\bar{B}_{d}(t)=d(1-t)^{d-1}$.

Finally, averaging over the sampled value $a$ we see that this strategy simulates $N_{a}=\frac{\alpha_{a}}{d}N_{a}^{(a)}$ and $N_{\textup{\,\o\,}}=\sum_{a}\frac{\alpha_{a}}{d}N_{\textup{\,\o\,}}^{(a)}$, leading to

and $N_{\textup{\,\o\,}}=\mathds{1}_{d}-\sum_{a}N_{a}$. On the other hand from Eq. (12) we obtain $\bar{M}_{a}^{(\eta,p)}=\frac{\alpha_{d}}{d}\big{(}d\,\eta p\outerproduct{\varphi_{a}}{\varphi_{a}}+\eta(1-p)\mathds{1}_{d}\big{)}$. Comparing with Eq. (19) we conclude that for

and $t\in[0,1]$ any POVM $\{\bar{M}_{a}^{(\eta,p)}\}$ can be simulated. Reproducing measurements that are even more noisy is straightforward by adding noise to the above construction. Hence, for any noise level $p=t$ our construction simulates all POVMs $\{\bar{M}_{a}^{(\eta,p)}\}$ if $\eta\leq(1-t)^{d-1}=(1-p)^{d-1}$. $\square$

Considering a bipartite steering scenario based on high-dimensional entanglement, we have investigated how asymmetrical the effect of steering can become. We presented entangled states $\varrho_{AB}$ such that Alice can convince Bob that $\varrho_{AB}$ is of high Schmidt number, while Bob can never convince Alice that the state is even entangled. Thus, one-way steering can become unlimited.

Specifically, we constructed families of entangled states that exhibit genuine $d$-dimensional steering in one direction, while remaining unsteerable (under the most general measurements) in the other direction. We note that this construction can be straightforwardly generalized to states with genuine $n$-dimensional steering in one direction and unsteerable the other way around, for any $1<n\leq d$ ¹¹1For this, one should simply adapt the bounds in Eqs (8) and (10) to demand only $n-$steerability, following the results of Jones et al. (2022) for (8) and Designolle et al. (2021) for (10).. A practical implementation of such states should be feasible, e.g. with the setups of Refs Zeng et al. (2018); Srivastav et al. (2022) for an experimental demonstration of unlimited one-way steering.

Finally, our work also contributed to the characterisation of joint measurability in high-dimensional measurements. In particular, we obtained a criterion for the compatibility of arbitrary measurements subjected to both noise and loss. This result is of independent interest and may find other applications.

Acknowledgments.— We acknowledge financial support from the Swiss National Science Foundation (projects 192244, Ambizione PZ00P2-202179, and NCCR SwissMAP).