No-signaling bounds for quantum cloning and metrology

The impossibility of superluminal communication is a fundamental principle of physics. Here we show that this principle underpins the performance of several fundamental tasks in quantum information processing and quantum metrology. In particular, we derive tight no-signaling bounds for probabilistic cloning and superreplication that coincide with the corresponding optimal achievable fidelities and rates known. In the context of quantum metrology, we derive the Heisenberg limit from the no-signaling principle for certain scenarios including reference frame alignment and maximum likelihood state estimation. We elaborate on the equivalence of assymptotic phase-covariant cloning and phase estimation for different figures of merit.

Figure 1

Generic setting for faster-than-light communication. Alice and Bob share an entangled state ${|\mathrm{\Psi }\rangle }_{AB}$. By applying $U_{\theta }^{\otimes N}$ followed by a suitable measurement Alice can prepare any ensemble $\left\{p_k|{\mathrm{\Phi }}_{\theta |k}\rangle \langle {\mathrm{\Phi }}_{\theta |k}|\right\}$, which Bob processes into $\mathcal{O}\left(\theta \right)={\sum }_k{p}_k\mathcal{P}\left(\theta \right|k)$. The no-signaling condition imposes that $\mathcal{O}\left(\theta \right)$ is independent of $\theta$ chosen by Alice.

Figure 2

Equivalence between probabilistic PCC and deterministic PCC using arbitrary states. The filter in the probabilistic cloning can be viewed as part of a probabilistic preparation of a general state from a separable $N$-qubit state. Allowing for arbitrary input states makes the preparation process deterministic.

Figure 3

The lower bound on the asymptotically achievable error $\frac{1}{2}{\vartheta }_4(0,e^{-t})=\frac{6N^2}{{\pi }^2}(1-{\overline{f}}_t)$ (bottom curve) and the error of the prior $\int p(\theta ;t){sin}^2(\theta /2)d\theta$ (top curve) as functions of $t$.

Figure 4

The two ensemble decompositions of ${\rho }_{\varepsilon }$ for a qubit, leading to faster-than-light communication for the probability distribution Eq. (24) (represented by the red semicircle).