Quantum Steering Beyond Instrumental Causal Networks

We theoretically predict, and experimentally verify with entangled photons, that outcome communication is not enough for hidden-state models to reproduce quantum steering. Hidden-state models with outcome communication correspond, in turn, to the well-known instrumental processes of causal inference but in the one-sided device-independent scenario of one black-box measurement device and one well-characterized quantum apparatus. We introduce one-sided device-independent instrumental inequalities to test against these models, with the appealing feature of detecting entanglement even when communication of the black box’s measurement outcome is allowed. We find that, remarkably, these inequalities can also be violated solely with steering, i.e., without outcome communication. In fact, an efficiently computable formal quantifier—the robustness of noninstrumentality—naturally arises, and we prove that steering alone is enough to maximize it. Our findings imply that quantum theory admits a stronger form of steering than known until now, with fundamental as well as practical potential implications.

Figure 1

Hybrid (classical-quantum) instrumental processes. Causal structures are specified by directed acyclic graphs (DAGs). Each node encodes either a classical random variable or a quantum system and each directed edge a causal influence. The term acyclic refers to the physical requirement of no causal loops. (a) One-sided quantum instrumental ($1\mathsf{SQI}$) causal networks generalize the classical instrumental causal structure to the case where node $B$ is a quantum system. Removing the red dashed edge, in turn, leads to local hidden-state ($\mathsf{LHS}$) models. (b) Quantizing also $\mathrm{\Lambda }$ gives what we here refer to as quantum instrumental ($\mathsf{QI}$) processes: $\mathrm{\Lambda }$ is now a bipartite system in a possibly entangled quantum state, with one subsystem causally influencing $A$ and the other one $B$. Removing the red dashed edge here leads to quantum ($\mathsf{Q}$) models. (c) Pictorial representation of the inner geometry of the set $\mathsf{QI}$. $1\mathsf{SQI}$ is a strict subset of $\mathsf{QI}$. The lower dimensional manifold $\mathsf{NS}$ of nonsignaling assemblages coincides (for the bipartite case) with $\mathsf{Q}$, in turn containing $\mathsf{LHS}$. $\mathsf{LHS}$ and $\mathsf{Q}$ are the sets studied in steering theory. Surprisingly, $\mathsf{Q}$ is not contained in $1\mathsf{SQI}$. Furthermore, for any assemblage in $\mathsf{QI}$ and outside of $1\mathsf{SQI}$, there is an assemblage in $\mathsf{Q}$ as far away from $1\mathsf{SQI}$ as the former.

Figure 2

Robustnesses of steering and of noninstrumentality. The curves and points correspond, respectively, to theory and experiment. An assemblage is produced by three projective local measurements by Alice on ${\varrho }_{\mathrm{\Lambda }}=V|{\mathrm{\Phi }}^{+}⟩⟨{\mathrm{\Phi }}^{+}|+[(1-V)/2](|00⟩⟨00|+|11⟩⟨11|)$, generated by local dephasing on the maximally entangled state $|{\mathrm{\Phi }}^{+}⟩=(1/\sqrt{2})(|00⟩+|11⟩)$ with strength given by the visibility $V$. The assemblage is steerable in the usual sense for all $V>0$ and is compatible with 1SQI models up to $V<0.24$ (white region). That point is the onset of a stronger form of quantum steering than explainable by 1SQI models (light and dark green). Finally, for $V>(9-4\sqrt{2})/7\simeq 0.48$ (dark green), the assemblage produces (under the optimal [61] $a$-dependent measurements on Bob’s qubit) black-box correlations that violate the DI instrumental inequality of [35, 36] (see Appendix III in the SM [52]). That is, the assemblage’s incompatibility with $1\mathsf{SQI}$ can be verified in the 1S DI setting for a broader range of visibilities than in the fully DI one. Error bars are obtained assuming Poissonian distributions for the photon counts.

Figure 3

Experimental setup. (a) The BBO crystal is pumped by a vertically polarized laser beam at 325 nm from a He-Cd source, so that a pair of down-converted photons, at 650 nm, emerge in an entangled state of momenta due to momentum conservation. The momenta encode the two common-cause qubits. All but two momentum modes, corresponding to $|00⟩$ and $|11⟩$, are filtered out. The produced momentum state is the two-qubit maximally entangled state $|{\mathrm{\Phi }}^{+}⟩⟨{\mathrm{\Phi }}^{+}|$. One qubit is sent to Alice and the other one to Bob. (b) Side view of each party’s device, depicting the preparation (by Alice) and measurement (by Bob) of the assemblage. Beam displacers $B{D}_A$ and $B{D}_B$ and half-wave plates (HWP) map the momenta of each photon into its polarizations. A glass slide introduces dephasing between Bob’s two paths, allowing us to tune the visibility $V$ of the resulting dephased state ${\varrho }_{\mathrm{\Lambda }}$ from $V\sim 0$ to $V\sim 94,5\%$. Quarter-wave plates ($Q_A$ and $Q_B$), half-wave plates ($H_A$ and $H_B$), polarized beam splitters (PBS), and a coincidence-photon detector implement projective polarization measurements.