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arXiv:1109.0206 · Physical Review A

Sub–shot-noise sensitivities without entanglement

Fabio Benatti, Daniel Braun

Published Jan 31, 2013• 34 citations
arXivHTMLPDF

Abstract

We critically analyze the commonly maintained statement that entanglement is necessary to beat the shot-noise limit or standard quantum limit (SQL) in the sensitivity with which certain parameters can be measured in interferometric experiments. We consider in detail three different physical realizations of a beam splitter and a Mach-Zehnder interferometer using photons, massive bosons, and distinguishable spins or qubits. We study several input states and the corresponding definitions of entanglement. We show that mode entanglement is not required for photons or massive bosons for beating the SQL. In particular, with a fluctuating number of two-mode bosons, the shot-noise limit can be beaten by nonentangled bosonic states with all bosons in one mode. As a consequence, the trivially present entanglement due to symmetrization that appears when writing bosonic states in terms of individual particles is a useful resource for precision measurements.

DOI
10.1103/physreva.87.012340

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We critically analyze the commonly maintained statement that entanglement is necessary to beat the shot-noise limit or standard quantum limit (SQL) in the sensitivity with which certain parameters can be measured in interferometric experiments. We consider in detail three different physical realizations of a beam splitter and a Mach-Zehnder interferometer using photons, massive bosons, and distinguishable spins or qubits. We study several input states and the corresponding definitions of entanglement. We show that mode entanglement is not required for photons or massive bosons for beating the SQL. In particular, with a fluctuating number of two-mode bosons, the shot-noise limit can be beaten by nonentangled bosonic states with all bosons in one mode. As a consequence, the trivially present entanglement due to symmetrization that appears when writing bosonic states in terms of individual particles is a useful resource for precision measurements.
Phase measurement using a lossless Mach–Zehnder interferometer with certain entangled N-photon states can lead to a phase sensitivity of the order of 1/N, the Heisenberg limit. However, previously considered output measurement schemes are different for different input states to achieve this limit. We show that it is possible to achieve this limit just by the parity measurement for all the commonly proposed entangled states. Based on the parity measurement scheme, the reductions of the phase sensitivity in the presence of photon loss are examined for the various input states.
Abstract These two accompanying papers are concerned with two mode entanglement for systems of identical massive bosons and the relationship to spin squeezing and other quantum correlation effects. Entanglement is a key quantum feature of composite systems in which the probabilities for joint measurements on the composite sub-systems are no longer determined from measurement probabilities on the separate sub-systems. There are many aspects of entanglement that can be studied. This two-part review focuses on the meaning of entanglement, the quantum paradoxes associated with entangled states, and the important tests that allow an experimentalist to determine whether a quantum state—in particular, one for massive bosons is entangled. An overall outcome of the review is to distinguish criteria (and hence experiments) for entanglement that fully utilize the symmetrization principle and the super-selection rules that can be applied to bosonic massive particles. In the first paper (I), the background is given for the meaning of entanglement in the context of systems of identical particles. For such systems, the requirement is that the relevant quantum density operators must satisfy the symmetrization principle and that global and local super-selection rules prohibit states in which there are coherences between differing particle numbers. The justification for these requirements is fully discussed. In the second quantization approach that is used, both the system and the sub-systems are modes (or sets of modes) rather than particles, particles being associated with different occupancies of the modes. The definition of entangled states is based on first defining the non-entangled states—after specifying which modes constitute the sub-systems. This work mainly focuses on the two mode entanglement for massive bosons, but is put in the context of tests of local hidden variable theories, where one may not be able to make the above restrictions. The review provides the detailed arguments necessary for the conclusions of a recent paper, where the question of how to rigorously demonstrate the entanglement of a two-mode Bose–Einstein condensate (BEC) has been examined. In the accompanying review paper (II), we consider spin squeezing and other tests for entanglement that have been proposed for two-mode bosonic systems. We apply the approach of review (I) to determine which tests, and which modifications of the tests, are useful for detecting entanglement in massive bosonic (BEC), as opposed to photonic, systems. Several new inequalities are derived, a theory for the required two-mode interferometry is presented, and key experiments to date are analyzed.
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 NNN particles if they can be optimized with local operations. Results on the optimal local transformations allow us to show that for N=2N=2N=2 all pure entangled states can be made useful for sub-shot-noise interferometry while for N>2N>2N>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.
In 1981, Caves pointed out that the phase sensitivity of a Mach-Zehnder interferometer with single-mode inputs is bounded by the shot-noise limit. The quantum Fisher information analysis shows that this statement holds true for the scenario where two antisymmetric phase shifts occur in two arms, but it is invalid for the scenario where an unknown phase is embedded in one of two arms. In this paper, we focus on the phase sensitivity directed against the latter scenario. The optimal single-mode input is discussed by analyzing common states, including displaced squeezed states, displaced number states, squeezed number states, Schrödinger cat states and completely mixed states. We find that the best choice is a squeezed vacuum state and show the specific measurement scheme which is capable of saturating the corresponding phase sensitivity limit. In addition, we study the effects of several realistic factors-anti-squeezing noise, photon loss and dark counts-on the phase sensitivity. Our results suggest that sub-shot-noise-limited phase sensitivity is attainable with low noise or loss, which paves the way for practical metrology.
We consider the situation when the signal propagating through each arm of an interferometer has a complicated multimode structure. We demonstrate that the shot-noise level for such a setup is the same as for the common two-mode case for as long as the interferometric transformation treats each arm as a whole and does not penetrate its inner structure. We find the relation between the particle entanglement and the possibility to surpass the shot-noise limit of the phase estimation. Our results are general---they apply to pure and mixed states of identical and distinguishable particles (or combinations of both) for a fixed and fluctuating number of particles. An important result is that the method for detecting the entanglement often used in a two-mode system---based on the measurement of the visibility of fringes and the population imbalance fluctuations---can give misleading results when applied to the multimode case. Our main result is that the additional modes do not boost the sensitivity for as long as they are not interacting. Therefore, the improvement above the shot-noise level established for the two-mode systems can safely be treated as an entanglement criterion also in the complex multimode case.
Whenever the value of an unknown parameter $\ensuremath{\theta}$ is extracted from a series of experiments, the result is inevitably burdened by the uncertainty $\ensuremath{\Delta}\ensuremath{\theta}$. If the system that is the subject of measurement consists of unentangled particles, this uncertainty is bounded by the shot-noise limit. To overcome this limitation, it is necessary to use a properly entangled state, which is usually prepared in a dedicated procedure. We show that quantum correlations arising from the indistinguishability of bosons are a sufficient resource for the sub-shot-noise interferometry. To this end, we consider an interferometer, which operates on two independently prepared Bose-Einstein condensates with fluctuating numbers of particles. We calculate the sensitivity obtained from the measurement of the number of atoms and compare it with the ultimate achievable bound. Our main conclusion is that even in the presence of major atom number fluctuations, an interferometer operating on two independently prepared condensates can give very high precision. These observations indicate a new possibility for an interferometer operating below the shot-noise limit.