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.
We study the time evolution of two two-state systems (two qubits) initially in the pure entangled states or the maximally entangled mixed states interacting with the individual environmental noise. It is shown that due to environment noise, all quantum entangled states are very fragile and become a classical mixed state in a short-time limit. But the environment can affect entanglement in very different ways. The type of decoherence process for certain entangled states belongs to amplitude damping, while the others belong to dephasing decoherence.
Why do we not see large macroscopic objects in entangled states? There are two ways to approach this question. The first is dynamic. The coupling of a large object to its environment cause any entanglement to decrease considerably. The second approach, which is discussed in this paper, puts the stress on the difficulty of observeing a large-scale entanglement. As the number of particles
n grows we need an ever more precise knowledge of the state and an ever more carefully designed experiment, in order to recognize entanglement. To develop this point we consider a family of observables, called witnesses, which are designed to detect entanglement. A witness
W distinguishes all the separable (unentangled) states from some entangled states. If we normalize the witness
W to satisfy $\ensuremath{\mid}\mathrm{tr}(W\ensuremath{\rho})\ensuremath{\mid}\ensuremath{\leqslant}1$ for all separable states $\ensuremath{\rho}$, then the efficiency of
W depends on the size of its maximal eigenvalue in absolute value; that is, its operator norm $\ensuremath{\parallel}W\ensuremath{\parallel}$. It is known that there are witnesses on the space of
n qubits for which $\ensuremath{\parallel}W\ensuremath{\parallel}$ is exponential in
n. However, we conjecture that for a large majority of
n-qubit witnesses $\ensuremath{\parallel}W\ensuremath{\parallel}\ensuremath{\leqslant}O(\sqrt{n\phantom{\rule{0.2em}{0ex}}\mathrm{log}\phantom{\rule{0.2em}{0ex}}n})$. Thus, in a nonideal measurement, which includes errors, the largest eigenvalue of a typical witness lies below the threshold of detection. We prove this conjecture for the family of extremal witnesses introduced by Werner and Wolf [Phys. Rev. A 64, 032112 (2001)].
We study the dynamics of quantum dissension for three qubit states in various dissipative channels such as amplitude damping, dephasing and depolarizing. Our study is solely based on Markovian environments where quantum channels are without memory and each qubit is coupled to its own environment. We start with mixed GHZ, mixed W, mixture of separable states, a mixed biseparable state, as the initial states and mostly observe that the decay of quantum dissension is asymptotic in contrast to sudden death of quantum entanglement in similar environments. This is a clear indication of the fact that quantum correlation in general is more robust against the effect of noise. However, for a given class of initial mixed states we find a temporary leap in quantum dissension for a certain interval of time. More precisely, we observe the revival of quantum correlation to happen for certain time period. This signifies that the measure of quantum correlation such as quantum discord, quantum dissension, defined from the information theoretic perspective is different from the correlation defined from the entanglement-separability paradigm and can increase under the effect of the local noise. We also study the effects of these channels on the monogamy score of each of these initial states. Interestingly, we find that for certain class of states and channels, there is change from negative values to positive values of the monogamy score with classical randomness as well as with time. This gives us an important insight in obtaining states which are freely sharable (polygamous state) from the states which are not freely sharable (monogamous). This is indeed a remarkable feature, as we can create monogamous states from polygamous states Monogamous states are considered to have more signatures of quantum ness and can be used for security purpose.
We study the robustness of the GHZ (or ``cat'') class of multi-partite states under decoherence. The noise model is described by a general completely positive map for qubits independently coupled to the environment. In particular, the robustness of N-party entanglement is studied in the large N limit when (a) the number of spatially separated subsystems is fixed but the size of each subsystem becomes macroscopic (b) the size of the subsystems is fixed while their number becomes arbitrarily large. We obtain threshold conditions for entanglement in these two cases. Among our other results, we show that the parity of an entangled state (i.e., whether it contains an even or odd number of qubits) can lead to qualitatively different robustness of entanglement under certain conditions.
We investigate the decay of entanglement, due to decoherence, of multiqubit systems that are initially prepared in highly (in some cases maximally) entangled states. We assume that during the decoherence processes each qubit of the system interacts with its own independent environment. We determine, for systems with a small number of qubits and for various decoherence channels, the initial states exhibiting the most robust entanglement. We also consider a restricted version of this robustness-optimization problem that only involves states equivalent, under local unitary transformations, to the Greenberger-Horne-Zeilinger
∣GHZ⟩ state.