Not all pure entangled states are useful for sub-shot-noise interferometry

We investigate the connection between the shot-noise limit in linear interferometers and particle entanglement. In particular, we ask whether sub-shot-noise sensitivity can be reached with all pure entangled input states of $N$ particles if they can be optimized with local operations. Results on the optimal local transformations allow us to show that for $N=2$ all pure entangled states can be made useful for sub-shot-noise interferometry while for $N>2$ this is not the case. We completely classify the useful entangled states available in a bosonic two-mode interferometer. We apply our results to several states, in particular to multiparticle singlet states and to cluster states. The latter turn out to be practically useless for sub-shot-noise interferometry. Our results are based on the Cramer-Rao bound and the Fisher information.

Figure 1

Systems that can be used for linear two-state interferometry: (a) Archetypical optical Mach-Zehnder interferometer as in Refs. [22, 23], (b) double-well system as implemented in recent experiments on squeezing in BECs [28, 29, 30], and (c) system of single wells as in ion traps [25, 42]. In the first two cases, each of the $N$ particles lives in the subspace of the two states labeled $a$ and $b$, corresponding to momentum states in case (a) and to the left and the right well in case (b). In case (c), there is one particle per well, and particle $k$ in trap $k$ has the two internal degrees of freedom $a_k$ and $b_k$ (displayed are trap states, while in ion traps typically internal states of the ions are used [42]). The interferometer operations acts on the $a$-$b$ subspace in the cases (a) and (b) and identically on the subspaces $a_k$-$b_k$ in case (c). In the latter case, the particles are accessible individually via the different traps in principle. They can be treated as distinguishable particles labeled by the trap number $k$ if the spacial wave functions of the particles in the different traps do not overlap [43].

Figure 2

A general phase estimation scheme consisting of (i) the input state, (ii) the phase transformation, (iii) the measurement, and (iv) the data-processing stage.

Figure 3

Examples of symmetric states on which we apply Observation 1 given in the basis of symmetric Dicke states. (a) $N00N$ state, (b) twin-Fock state, (c) state considered in Ref. [9]. Plotted are the squared absolute values of the weights of the symmetric Dicke states in the superpositions.

Figure 4

Examples for graphs describing important graph states: (a) a GHZ or $N00N$ state (up to LU operations) [77, 78], (b) a linear cluster graph, (c) a ring cluster graph, (d) a cluster graph in two dimensions.