General Optimality of the Heisenberg Limit for Quantum Metrology

Abstract

Quantum metrology promises improved sensitivity in parameter estimation over classical procedures. However, there is an extensive debate over the question how the sensitivity scales with the resources (such as the average photon number) and number of queries that are used in estimation procedures. Here, we reconcile the physical definition of the relevant resources used in parameter estimation with the information-theoretical scaling in terms of the query complexity of a quantum network. This leads to a completely general optimality proof of the Heisenberg limit for quantum metrology. We give an example how our proof resolves paradoxes that suggest sensitivities beyond the Heisenberg limit, and we show that the Heisenberg limit is an information-theoretic interpretation of the Margolus-Levitin bound, rather than Heisenberg's uncertainty relation. 03.65.Ta, 42.50.Lc Parameter estimation is a fundamental pillar of science and technology, and improved measurement techniques for parameter estimation have often led to scientific breakthroughs and technological advancement. Caves [1] showed that quantum mechanical systems can in principle produce greater sensitivity over classical methods, and many quantum parameter estimation protocols have been proposed since [2]. The field of quantum metrology started with the work of Helstrom [3,4], who derived the minimum value for the mean square error in a parameter in terms of the density matrix of the quantum system and a measurement procedure. This was a generalisation of a known result in classical parameter estimation, called the Cramér-Rao bound. Braunstein and Caves [5] showed how this bound can be formulated for the most general state preparation and measurement procedures. While it is generally a hard problem to show that the Cramér-Rao bound can be attained in a given setup, at least it gives an upper limit to the precision of quantum parameter estimation.The quantum Cramér-Rao bound is typically formulated in terms of the Fisher information, an abstract quantity that measures the maximum information about a parameter ϕ that can be extracted from a given measurement procedure. One of the central questions in quantum metrology is how the Fisher information scales with the physical resources used in the measurement procedure. We usually consider two scaling regimes: First, in the standard quantum limit (SQL) [6] or shot-noise limit the Fisher information is constant, and the error scales with the inverse square root of the number of times T we make a measurement. Second, in the Heisenberg limit [7] the error is bounded by the inverse of the physical resources. Typically, these are expressed in terms of the size N of the probe system, e.g., (average) photon number. However, it has been clearly demonstrated that this form of the limit is not universally valid. For example, Beltrán and Luis [8] showed that the use of classical optical nonlinearities can lead to an error with average photon number scaling N −3/2 . Boixo et al. [9] devised a parameter estimation procedur...

Advances in quantum metrology

“…Ref. [97]. Ultimately the idea of these proposals is to consider settings where the unitary transformation that "writes" the unknown parameter x into the probing signals, is characterized by many-body Hamiltonian generators which are no longer extensive functions of the number of probes employed in the estimation [83][84][85][86][87][88][89][90][91][92] or, for the optical implementations which yielded the inequality (6), in the photon number operator of the input signals [80][81][82][83][93][94][95][96].…”

Entanglement-enhanced lidars for simultaneous range and velocity measurements

“…When the M probes are in a product state, the standard quantum limit (SQL)-with a root-meansquare (rms) estimation error proportional to 1/ √ M-can be achieved. Entangled probes, however, can realize the Heisenberg limit (HL) [2,3], viz., an rms estimation error that is proportional to 1/M [2][3][4][5][6][7]. SQL vs HL behavior for single-parameter estimation can arise, e.g., in measuring time delays [5], point-source separations [8][9][10][11], displacements [12][13][14], or magnetic fields [15].…”

Quantum-limited metrology with nonlinear detection schemes

“…It might be argued that for different generators the Heisenberg limit should be different [24]. Leaving aside terminology, we stress that the key point is the different dependence on the number of particles N in Eqs.…”

“…Typical effective generators for light propagation in nonresonant nonlinear media are given by powers and products of photon-number operators of the form G =n k in a single-mode approach [4,5,7,8,16,24], G = J 2 z in a two-mode scheme [4], G = ⊗ k jn j in a multimode configuration [7], or even G = jn k j [23].…”