We ask whether the optimal probe is entangled, and if so, what is its character and amount, for estimating the noise parameter of a large class of local quantum encoding processes that we refer to as vector encoding, examples of which include the local depolarizing and bit-flip channels. We first establish that vector encoding is invariably ``continuously commutative'' for optimal probes. We utilize this result to deal with the queries about entanglement in the optimal probe. We show that for estimating noise extent of the two-party arbitrary-dimensional local depolarizing channel, there is a descending staircase of optimal-probe entanglement for increasing depolarizing strength. For the multi-qubit case, the analysis again leads to a staircase, but which can now be monotonic or not, depending on the multiparty entanglement measure used. We also find that when sufficiently high depolarizing noise is to be estimated, fully product multiparty states are the only choice for being optimal probes. In many cases, for even moderately high depolarizing noise, fully product states are optimal. For two-qubit local bit-flip channels, the continuous commutativity of the channel and optimal probe implies that a product state suffices for obtaining the optimal precision.
We ask whether the optimal probe is entangled, and if so, what is its character and amount, for estimating the noise parameter of a large class of local quantum encoding processes that we refer to as vector encoding, examples of which include the local depolarizing and bit-flip channels. We first establish that vector encoding is invariably ``continuously commutative'' for optimal probes. We utilize this result to deal with the queries about entanglement in the optimal probe. We show that for estimating noise extent of the two-party arbitrary-dimensional local depolarizing channel, there is a descending staircase of optimal-probe entanglement for increasing depolarizing strength. For the multi-qubit case, the analysis again leads to a staircase, but which can now be monotonic or not, depending on the multiparty entanglement measure used. We also find that when sufficiently high depolarizing noise is to be estimated, fully product multiparty states are the only choice for being optimal probes. In many cases, for even moderately high depolarizing noise, fully product states are optimal. For two-qubit local bit-flip channels, the continuous commutativity of the channel and optimal probe implies that a product state suffices for obtaining the optimal precision.
Among the intriguing features of quantum theory, the problem of distinguishing quantum channels is of fundamental interest. In this paper, we focus on the single-shot discrimination of two noisy quantum channels using two distinct classes of probes: single-system (product) probes and entangled probes. Our aim is to identify optimal probing state for specific discrimination tasks and to analyze the necessity and role of entanglement in enhancing channel distinguishability. We show that maximally entangled probes are optimal for discriminating two qubit depolarizing channels, with any nonzero entanglement providing an advantage over single-system probes. In contrast, for dephasing channels in arbitrary dimensions, we prove that single-system probe can be optimal and that entanglement offers no improvement, even when the dephasing unitary is generalized. For qubit amplitude-damping channels, we identify distinct noise-dependent regimes in which either single-system probe outperforms maximally entangled probes and vice-versa. Moreover, we demonstrate that non-maximally entangled probes can act as the optimum probe if the noise parameters restricted to certain values in this task. We also present examples of noisy unitary channels for which discrimination is possible using non-maximally entangled probe, while both single-systems and maximally entangled probes fail. We introduce another class of noisy unitary channels for which perfect discrimination is achievable with a single system, while maximally entangled probes are insufficient. Finally, we show that two erasure channels can be optimally discriminated using any pure single-system probe, with no advantage gained from entanglement.
Estimation of local quantum fields is a crucial aspect of quantum metrology applications, and often also forms the test-bed to analyze the utility of quantum resources, like entanglement. However, so far, this has been analyzed using the same local field for all the probes, and so, although the encoding process utilizes a local Hamiltonian, there is an inherent "nonlocality" in the encoding process in the form of a common local field applied on all the probes. We show that estimation of even independent multiple field strengths of a local Hamiltonian, i.e., one formed by a sum of single-party terms, necessitates the utility of genuine multipartite entangled state as the input probe. The feature depends on the choice of the weight matrix considered, which is full-rank and contains non-vanishing off-diagonal terms. We obtain this result by providing a lower bound on the precision of multiparameter estimation, optimized over input probes, for an arbitrary positive semi-definite weight matrix. We show that there exists a weight matrix for which this bound is always attainable by the Greenberger-Horne-Zeilinger (GHZ) state, chosen in a certain basis. Furthermore, we find the parametric form of the most general optimal state for three parties. We also show that no pure product state can achieve the lower bound. Finally, for an arbitrary weight matrix and an arbitrary multiparty local encoding Hamiltonian, we prove that using a probe that is in any mixed state provides a precision lower than that obtainable using pure states. To emphasize the importance of the weight matrix considered, we also prove that the choice of identity operator as the same - thereby ignoring the "off-diagonal" covariances in the precision matrix - does not require the use of genuine multiparty entanglement in input probes for attaining the best precision, and the optimal probe can be a pure product.
For parameter estimation from an [Formula: see text]-component composite quantum system, it is known that a separable preparation leads to a mean-squared estimation error scaling as [Formula: see text] while an entangled preparation can in some conditions afford a smaller error with [Formula: see text] 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 [Formula: see text]. To complement this asymptotic characterization, here we characterize how the estimation efficiency evolves as a function of the size [Formula: see text] 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 [Formula: see text] the efficiency of the optimal separable preparation to that of an entangled preparation with arbitrary degree of entanglement. We exhibit the [Formula: see text] superefficiency with no noise, and prove its asymptotic disappearance at large [Formula: see text] for any nonvanishing noise configuration. For maximizing the estimation efficiency, we characterize the existence of an optimum [Formula: see text] 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 [Formula: see text] qubits into independent blocks formed of [Formula: see text] entangled qubits restores at large [Formula: see text] a nonvanishing efficiency that can improve over that of [Formula: see text] 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.
Quantum multiparameter estimation offers a framework for the simultaneous estimation of multiple parameters, pertaining to possibly noncommutating observables. While the optimal probe for estimating a single unitary phase is well understood - being a pure state that is an equal superposition of the eigenvectors of the encoding Hamiltonian corresponding to its maximum and minimum eigenvalues - the structure of optimal probes in the multiparameter setting remains more intricate. We investigate the simultaneous estimation of two phases, each encoded through arbitrary qubit Hamiltonians, using arbitrary weight matrices, and considering single-qubit probes. We also consider single-qutrit probes, for which the encoding Hamiltonians are chosen as SU(2) generators. We find that in both the qubit and qutrit scenarios, the optimal probe is a coherent superposition of the eigenstates corresponding to the largest and smallest eigenvalues of the total encoding Hamiltonian, with a fixed - and not arbitrary - relative phase. Remarkably, this optimal probe is independent of the specific choice of weight matrix, making it a universally optimal input state for the estimation of any pair of SU(2) parameters, which can be reparameterized to the phase estimation problem. Furthermore, we show that this probe also maximizes the determinant of the quantum Fisher information matrix, providing a handy tool for identifying the optimal probe state. We also examine the role of commutativity between the generators of the unitary encodings. Our results demonstrate that a high degree of commutativity degrades the achievable precision, rendering estimation infeasible in the extreme case. However, maximal noncommutativity does not necessarily yield optimal precision.
We investigate different quantum parameter estimation scenarios in the presence of noise, and identify optimal probe states. For frequency estimation of local Hamiltonians with dephasing noise, we determine optimal probe states for up to 70 qubits, and determine their key properties. We find that the so-called one-axis twisted spin-squeezed states are only almost optimal, and that optimal states need not to be spin-squeezed. For different kinds of noise models, we investigate whether optimal states in the noiseless case remain superior to product states also in the presence of noise. For certain spatially and temporally correlated noise, we find that product states no longer allow one to reach the standard quantum limit in precision, while certain entangled states do. Our conclusions are based on numerical evidence using efficient numerical algorithms which we developed in order to treat permutational invariant systems.