Abstract-Over the past 20 years, bright sources of entangled photons have led to a renaissance in quantum optical interferometry. Optical interferometry has been used to test the foundations of quantum mechanics and implement some of the novel ideas associated with quantum entanglement such as quantum teleportation, quantum cryptography, quantum lithography, quantum computing logic gates, and quantum metrology. In this paper, we focus on the new ways that have been developed to exploit quantum optical entanglement in quantum metrology to beat the shot-noise limit, which can be used, e.g., in fiber optical gyroscopes and in sensors for biological or chemical targets. We also discuss how this entanglement can be used to beat the Rayleigh diffraction limit in imaging systems such as in LIDAR and optical lithography.

Hilbert's projective metric in quantum information theory J. Math. Phys. 52, 082201 (2011) The conditional quantum mutual information I(A; B|C) of a tripartite state ρ ABC is an information quantity which lies at the center of many problems in quantum information theory. Three of its main properties are that it is non-negative for any tripartite state, that it decreases under local operations applied to systems A and B, and that it obeys the duality relation I(A; B|C) = I(A; B|D) for a four-party pure state on systems ABCD. The conditional mutual information also underlies the squashed entanglement, an entanglement measure that satisfies all of the axioms desired for an entanglement measure. As such, it has been an open question to find Rényi generalizations of the conditional mutual information, that would allow for a deeper understanding of the original quantity and find applications beyond the traditional memoryless setting of quantum information theory. The present paper addresses this question, by defining different α-Rényi generalizations I α (A; B|C) of the conditional mutual information, some of which we can prove converge to the conditional mutual information in the limit α → 1. Furthermore, we prove that many of these generalizations satisfy non-negativity, duality, and monotonicity with respect to local operations on one of the systems A or B (with it being left as an open question to prove that monotonicity holds with respect to local operations on both systems). The quantities defined here should find applications in quantum information theory and perhaps even in other areas of physics, but we leave this for future work. We also state a conjecture regarding the monotonicity of the Rényi conditional mutual informations defined here with respect to the Rényi parameter α. We prove that this conjecture is true in some special cases and when α is in a neighborhood of one. C 2015 AIP Publishing LLC. [http://dx

This paper defines the fidelity of recovery of a tripartite quantum state on systems A, B, and C as a measure of how well one can recover the full state on all three systems if system A is lost and a recovery operation is performed on system C alone. The surprisal of the fidelity of recovery (its negative logarithm) is an information quantity which obeys nearly all of the properties of the conditional quantum mutual information I(A; B|C), including non-negativity, monotonicity with respect to local operations, duality, invariance with respect to local isometries, a dimension bound, and continuity. We then define a (pseudo) entanglement measure based on this quantity, which we call the geometric squashed entanglement. We prove that the geometric squashed entanglement is a 1-LOCC monotone (i.e., monotone non-increasing with respect to local operations and classical communication from Bob to Alice), that it vanishes if and only if the state on which it is evaluated is unentangled, and that it reduces to the geometric measure of entanglement if the state is pure. We also show that it is invariant with respect to local isometries, subadditive, continuous, and normalized on maximally entangled states. We next define the surprisal of measurement recoverability, which is an information quantity in the spirit of quantum discord, characterizing how well one can recover a share of a bipartite state if it is measured. We prove that this discord-like quantity satisfies several properties, including nonnegativity, faithfulness on classical-quantum states, invariance with respect to local isometries, a dimension bound, and normalization on maximally entangled states. This quantity combined with a recent breakthrough of Fawzi and Renner allows to characterize states with discord nearly equal to zero as being approximate fixed points of entanglement breaking channels (equivalently, they are recoverable from the state of a measuring apparatus). Finally, we discuss a multipartite fidelity of recovery and several of its properties.

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In the lore of quantum metrology, one often hears (or reads) the following no-go theorem: If you put vacuum into one input port of a balanced Mach-Zehnder Interferometer, then no matter what you put into the other input port, and no matter what your detection scheme, the sensitivity can never be better than the shot noise limit (SNL). Often the proof of this theorem is cited to be in Ref. [C. Caves, Phys. Rev. D 23, 1693(1981], but upon further inspection, no such claim is made there. A quantum-Fisher-information-based argument suggestive of this no-go theorem appears in Ref. [M. Lang and C. Caves, Phys. Rev. Lett. 111, 173601 (2013)], but is not stated in its full generality. Here we thoroughly explore this no-go theorem and give the rigorous statement: the nogo theorem holds whenever the unknown phase shift is split between both arms of the interferometer, but remarkably does not hold when only one arm has the unknown phase shift. In the latter scenario, we provide an explicit measurement strategy that beats the SNL. We also point out that these two scenarios are physically different and correspond to different types of sensing applications.Introduction.-In the field of quantum metrology [1][2][3], a Mach-Zehnder interferometer (MZI) is a tried and true workhorse that has the additional advantage that any result obtained for it also applies to a Michelson interferometer (MI) and hence has a potential application to gravitational wave detection. In most current implementations of gravitational wave detectors, the MI is fed with a strong coherent state of light in one input port and vacuum in the other (Fig. 1). It was in this context that Caves in 1981 [4] showed that such a design would always only ever achieve the shotnoise limit (SNL). Then he showed if you put squeezed vacuum into the unused port, you could beat the SNL. Several implementations of this squeezed vacuum scheme have already been demonstrated in the GEO 600 gravitational detector, and plans are underway to utilize this approach in the LIGO and VIRGO detectors in the future [5,6].It then appeared, that in the lore of quantum metrology, this result was extended -without proof -to the following no-go theorem: If you put quantum vacuum into one input port of a balanced MZI, then no matter what quantum state of light you put into the other input port, and no matter what your detection scheme, the sensitivity can never be better than the SNL. Often the proof of this theorem is cited to be the original 1981 paper by Caves [4], but upon further inspection, no such general claim is made there. A quantum-Fisherinformation-based argument suggestive of this no-go theorem appeared in Ref. [7] by Lang and Caves, but it does not explore the statement in adequate generality.In this work, we give a full statement of the no-go theorem. The statement proved here is the following: if the unknown phase shifts are in both of the two arms of the MZI, then the no-go theorem holds no matter whether the MZI is balanced or not. However, in the case where the unknown phas...