Resource Theory of Steering

We present an operational framework for Einstein-Podolsky-Rosen steering as a physical resource. For arbitrary-dimensional bipartite systems composed of a quantum subsystem and a black-box device, we show that local operations assisted by one-way classical communication (1W-LOCCs) from the quantum part to the black box cannot create steering. Based on this, we build a resource theory of steering with 1W-LOCCs as the free operations. We introduce the notion of convex steering monotones as the fundamental axiomatic quantifiers of steering. As a convenient example thereof, we present the relative entropy of steering. In addition, we prove that two previously proposed quantifiers, the steerable weight and the robustness of steering, are also convex steering monotones. To end up with for minimal-dimensional systems, we establish, on the one hand, necessary and sufficient conditions for pure-state steering conversions under stochastic 1W-LOCCs and prove, on the other hand, the nonexistence of steering bits, i.e., measure-independent maximally steerable states from which all states can be obtained by means of the free operations. Our findings reveal unexpected aspects of steering and lay the foundations for further research, with potential implications in Bell nonlocality.

Popular Summary

Einstein-Podolsky-Rosen (EPR) steering, an exotic form of nonlocal quantum effect, can best be understood by considering a hypothetical situation: Imagine a large bank interested in offering its clients an unbreakable quantum cryptographic service to encrypt the data exchanged in all transactions. The bank has enough resources to afford sophisticated, secure quantum installations. However, to make the service attractive, each client needs to have his or her own device that is simpler to control and more practical for household or office use. These devices do not require characterization or maintenance and are therefore untrusted. Can the bank implement an unconditionally secure quantum-cryptographic protocol that is independent of the behavior of the untrusted devices? The answer is yes, and the physical property that makes such protocols possible is called EPR steering.

In EPR steering, ensembles of quantum states can be remotely prepared by local measurements in a distant lab. From a fundamental point of view, EPR steering represents an intermediate type of quantum correlation between entanglement and Bell nonlocality, where the latter refers to both users possessing untrusted devices. From a practical point of view, it has recently been identified as a physical resource for one-sided, device-independent, information-theoretic tasks (e.g., the example of the bank and its clients). However, a formal framework of steering as a physical resource has been, until now, missing. This framework, which we develop here, is given by a resource theory of steering that mathematically quantifies steering, identifies its nonequivalent classes, and addresses steering conversion problems (e.g., dilution and distillation). We study the relative entropy of steering, and we develop necessary and sufficient conditions for steering.

We expect that our results will inform future studies of quantum processes such as entanglement.

Figure 1

Schematic representation of a 1W-LOCC map $\mathcal{M}$: The initial assemblage ${\rho }_{A|X}$ consists of a black box, with inputs $x$ and outputs $a$, governed by the probability distribution $P_{A|X}$, in Alice’s hand, and a quantum subsystem in one of the states $\{\varrho (a,x){\}}_{a,x}$, in Bob’s hands. The final assemblage ${\rho }_{A_f|X_f}=\mathcal{M}({\rho }_{A|X})$ is given by a final black box, represented by the dashed-lined rectangle, of inputs $x_f$ and outputs $a_f$, and a final subsystem, outside the dashed-lined rectangle, in the state $\varrho (a_f,x_f)={\mathcal{E}}_{\omega }\mathbf{(}\varrho (a,x)\mathbf{)}$. To implement $\mathcal{M}$, first Bob applies, with a probability $P_{\mathrm{\Omega }}(\omega )$, a stochastic quantum operation ${\mathcal{E}}_{\omega }$ that leaves his subsystem in the state ${\mathcal{E}}_{\omega }\mathbf{(}\varrho (a,x)\mathbf{)}$. He communicates $\omega$ to Alice. Then, Alice generates $x$ by processing the classical bits $\omega$ and $x_f$ with a local wiring described by a conditional distribution $P_{X|X_f,\mathrm{\Omega }}$. She inputs $x$ to her initial device, upon which the bit $a$ is produced. Finally, Alice generates the output $a_f$ of the final device by processing $x_f$, $\omega$, $x$, and $a$, with a local wiring described by a distribution $P_{A_f|A,X,\mathrm{\Omega },X_f}$.