Optimized entanglement for quantum parameter estimation from noisy qubits

Abstract

For parameter estimation from an N -component composite quantum system, it is known that a separable preparation leads to a mean-squared estimation error scaling as 1/N while an entangled preparation can in some conditions afford a smaller error with 1/N 2 scaling. This quantum superefficiency is however very fragile to noise or decoherence, and typically disappears with any small amount of random noise asymptotically at large N . To complement this asymptotic characterization, here we characterize how the estimation efficiency evolves as a function of the size N of the entangled system and its degree of entanglement. We address a generic situation of qubit phase estimation, also meaningful for frequency estimation. Decoherence is represented by the broad class of noises commuting with the phase rotation, which includes depolarizing, phase-flip, and thermal quantum noises. In these general conditions, explicit expressions are derived for the quantum Fisher information quantifying the ultimate achievable efficiency for estimation. We confront at any size N the efficiency of the optimal separable preparation to that of an entangled preparation with arbitrary degree of entanglement. We exhibit the 1/N 2 superefficiency with no noise, and prove its asymptotic disappearance at large N for any non-vanishing noise configuration. For maximizing the estimation efficiency, we characterize the existence of an optimum N opt of the size of the entangled system along with an optimal degree of entanglement. For nonunital noises, maximum efficiency is usually obtained at partial entanglement. Grouping the N qubits into independent blocks formed of N opt entangled qubits restores at large N a nonvanishing efficiency that can improve over that of N independent qubits optimally prepared. Also, one inactive qubit included in the entangled probe sometimes stands as the most efficient setting for estimation. The results further attest with new characterizations the subtlety of entanglement for quantum information in the presence of noise, showing that when entanglement is beneficial maximum efficiency is not necessarily obtained by maximum entanglement but instead by a controlled degree and finite optimal amount of it.

Indefinite causal order for quantum metrology with quantum thermal noise

“…Two entangled probe qubits, although more complicated to handle, could even be envisaged to probe the standard channel of Fig. 2, to benefit from the so-called Heisenberg enhanced performance, but the optimal configurations in the presence of thermal noise are not fully characterized, and it is known that the Heisenberg enhanced performance is very fragile to noise [19,49,38,50]. In the high range of the noise parameters (p, γ) in Fig.…”

Quantum parameter estimation on coherently superposed noisy channels

“…( 57)-( 58). With two entangled probing qubits, the performance of conventional schemes assessed by the Fisher information can be super-additive, benefiting from Heisenberg enhancement, which is nevertheless very fragile in the presence of noise [15,16,42]. But in any case, in the presence of a fully depolarizing noise N(•), a standard interaction of the noisy unitary of Fig.…”

An Architecture Superposing Indefinite Causal Order and Path Superposition Improving Pauli Channels’ Parameter Estimation

“…Even when one of the entangled qubits does not interact with the process being estimated, the presence of entanglement leads to improved performance. This enhancement holds for both partially and maximally entangled qubit pairs, depending on the level of noise [58]. Another example, in the context of quantum metrology, several strategies including independent channels, sequential channels, or channels in parallel using a general entangled state and superposition of causal order have been probed, where those strategies provide no asymptotic advantage over the case where an entangled probe state has been used [11].…”