Networking plays a ubiquitous role in quantum technology [1, 2]. It is an integral part of quantum communication and has significant potential for upscaling quantum computer technologies that are otherwise not scalable [3]. Recently, it was realized that sensing of multiple spatially distributed parameters may also benefit from an entangled quantum network [4][5][6][7][8][9]. Here we experimentally demonstrate how sensing of an averaged phase shift among four distributed nodes benefits from an entangled quantum network. Using a four-mode entangled continuous variable (CV) state, we demonstrate deterministic quantum phase sensing with a precision beyond what is attainable with separable probes. The techniques behind this result can have direct applications in a number of primitives ranging from biological imaging to quantum networks of atomic clocks.Quantum noise associated with quantum states of light and matter ultimately limits the precision by which measurements can be carried out [10][11][12]. However, by carefully designing the coherence of this quantum noise to exhibit properties such as entanglement and squeezing, it is possible to measure various physical parameters with significantly improved sensitivity compared to classical sensing schemes. Numerous realizations of quantum sensing utilizing non-classical states of light [2,13,15] and matter [16] have been reported, while only a few applications have been explored. Examples are quantum-enhanced gravitational waves interferometry [17], detection of magnetic fields [18][19][20] and sensing of the viscous-elasticity parameter of yeast cells [21]. All these implementations are, however, restricted to the sensing of a single parameter at a single location.Spatially distributed sensing of parameters at multiple locations in a network is relevant for applications from local beam tracking [22] to global scale clock synchronization [4]. The development of quantum networks [1, 2,23,24] enables new strategies for enhanced performance in such scenarios. Theoretical works [5][6][7][8][25][26][27][28] have shown that entanglement can improve sensing capabilities in a network using either twin-photons * or Greenberger-Horne-Zeilinger (GHZ) states combined with photon number resolving detectors [6,7] or using CV entanglement for the detection of distributed phase space displacements [8]. In this Letter, we experimentally demonstrate an entangled CV network for sensing of multiple phase shifts inspired by the theoretical proposal of Ref. [8]. Moreover, for the first time in any system, we demonstrate deterministic distributed sensing in a network of four nodes with a sensitivity beyond that achievable with a separable approach using similar quantum states. BSN … vaccum vaccum S a b c d FIG. 1.Distributed phase sensing scheme. The task is to estimate the average value of M spatially distributed phase shifts φ1, . . . , φM . (a) Without a network, the average phase shift must be estimated by probing each sample individually. This can be done with homodyne detection of the...

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...

Quantum receiver is an important tool for overcoming the standard quantum limit (SQL) of discrimination errors in optical communication. We theoretically study the quantum receivers for discriminating ternary and quaternary phase shift keyed coherent states in terms of average error rate and mutual information. Our receiver consists of on/off-type photon detectors and displacement operations w/o electrical feedforward operations. We show that for the ternary signals, the receiver shows a reasonable gain from the SQL even without feedforward. This scheme is realizable with the currently available technology. For the quaternary signals feedforward operation is crucial to overcome the SQL with imperfect devices. We also analytically examine the asymptotic limit of the performance of the proposed receiver with respect to the number of feedforward steps