Quantum Steering

Quantum steering asks whether the states conditionally prepared at one laboratory by another laboratory’s measurement records admit an explanation using a pre-existing ensemble of local quantum states. It lies between ordinary entanglement certification and a fully untrusted Bell test. The asymmetry matters: Bob trusts his quantum measurement description, while Alice’s response device need not be characterized.

Helpful background. No-Signaling distinguishes conditional remote states from communication.

Assemblages and local hidden states

Given $\rho_{AB}$ and Alice’s effects $E_{a\mid x}$, define subnormalized conditional states

$$\sigma_{a\mid x}=\operatorname{Tr}_A[(E_{a\mid x}\otimes I)\rho_{AB}].$$

Their traces are Alice’s outcome probabilities; their normalized forms are Bob’s states conditional on those outcomes. The collection is an assemblage. No-signaling requires $\sum_a\sigma_{a\mid x}=\rho_B$ independently of $x$.

A local-hidden-state model has the form

$$\sigma_{a\mid x}=\int\mu(d\lambda)\,p(a\mid x,\lambda)\tau_\lambda,$$

where every $\tau_\lambda$ is a normalized quantum state for Bob. If no such decomposition exists for the chosen measurements, the assemblage is steerable from Alice to Bob. Unlike a Bell-local model, Bob’s responses must arise from one quantum state for all his trusted measurements.

A two-axis witness

Let Bob measure $X$ and $Z$ on a qubit, and let Alice announce outcomes $a_X,a_Z\in\{-1,+1\}$. Consider

$$F=\langle A_X\otimes X\rangle+\langle A_Z\otimes Z\rangle.$$

For a hidden qubit state with Bloch components $r_x,r_z$, Alice’s response averages have magnitude at most one. Its contribution is bounded by

$$|r_x|+|r_z|\le\sqrt2\sqrt{r_x^2+r_z^2}\le\sqrt2.$$

Averaging preserves $F\le\sqrt2$. For $|\Phi^+\rangle$ and matching $X,Z$ measurements, both correlations are one, giving $F=2$. The violation excludes a local-hidden-state model given Bob’s trusted measurement assumptions.

With depolarizing noise $\rho_v=v|\Phi^+\rangle\langle\Phi^+|+(1-v)I/4$, this witness gives $F=2v$ and succeeds for $v>1/\sqrt2$. This is a witness threshold, not the optimal steerability boundary for all possible measurement sets.

Distinguishing the three correlation resources

A separable state always yields a local-hidden-state decomposition by using its separable components. Every local-hidden-state model also gives a Bell-local distribution when Bob measures it. Thus Bell nonlocality implies steering in the corresponding setting, and steering implies entanglement. The converses are not automatic, especially for mixed states and restricted measurements.

Steering can be direction-dependent because the trusted and untrusted roles differ. An inability to certify Alice-to-Bob steering with selected measurements does not establish unsteerability with every measurement or in the reverse direction.

The conditional ensembles do not let Alice choose Bob’s individual outcome or change his unconditioned density matrix. Communication of Alice’s labels is required to sort the data into the steered ensembles.

Exercise: remove the labels

For $|\Phi^+\rangle$, write Bob’s ensemble after Alice measures $X$, and after she measures $Z$. What remains when Alice’s labels are forgotten?

Solution

The ensembles are equal mixtures of $|+\rangle,|-\rangle$ and of $|0\rangle,|1\rangle$, respectively. Both sum to $I/2$. Steering tests use the full assemblage and Alice’s declared labels; Bob’s marginal alone cannot distinguish the choices.

References

• Cavalcanti, Eric G., Steven J. Jones, Howard M. Wiseman, and Margaret D. Reid, “Experimental criteria for steering and the Einstein–Podolsky–Rosen paradox,” Physical Review A 80, 032112 (2009), doi:10.1103/PhysRevA.80.032112.
• Wiseman, Howard M., S. J. Jones, and Andrew C. Doherty, “Steering, Entanglement, Nonlocality, and the Einstein-Podolsky-Rosen Paradox,” Physical Review Letters 98, 140402 (2007), doi:10.1103/PhysRevLett.98.140402.