Extending Noether’s theorem by quantifying the asymmetry of quantum states

Abstract

Noether's theorem is a fundamental result in physics stating that every symmetry of the dynamics implies a conservation law. It is, however, deficient in several respects: for one, it is not applicable to dynamics wherein the system interacts with an environment; furthermore, even in the case where the system is isolated, if the quantum state is mixed then the Noether conservation laws do not capture all of the consequences of the symmetries. Here we address these deficiencies by introducing measures of the extent to which a quantum state breaks a symmetry. Such measures yield novel constraints on state transitions: for nonisolated systems they cannot increase, whereas for isolated systems they are conserved. We demonstrate that the problem of finding non-trivial asymmetry measures can be solved using the tools of quantum information theory. Applications include deriving model-independent bounds on the quantum noise in amplifiers and assessing quantum schemes for achieving high-precision metrology.

Practical quantum metrology in noisy environments

“…As we illustrated, acquiring partial information about the geometry of the parameter-imprinting process allows one to optimize the estimation protocol at the single-probe level by simply adjusting the sampling time or number of rounds. Such a sequential estimation protocol relies on the initial amount of "unspeakable" coherence [21,22], which is a genuinely quantum feature [20], and is here confirmed as the key resource for estimating parameters encoded in incoherent operations, which include all phase-covariant channels. However, the estimation performance only scales linearly or "classically" in the probe size, whereby scaling up the probe size is intended as repeating the optimized sequential procedure M 1 times using independent probe qubits, all initialized in a maximally coherent state.…”

“…In the absence of noise, this sequential setting is formally equivalent to the parallel one [19], the only difference being that quantum coherence [20][21][22][23] takes over the instrumental role of entanglement. The sequential scheme seems more appealing from a practical viewpoint, as only a single probe needs to be addressed in both state preparation and final interrogation [24].…”

Witnessing Multipartite Entanglement by Detecting Asymmetry

“…In this context, the degree of coherent superposition of a state ∑ i c i |i i|, ∑ i |c i | 2 = 1, i.e., coherence (we omit the quantum label, from now on) in a reference basis {i}, is a resource. The crucial question is to determine how to obtain a computational advantage powered by coherence [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. The coherence of a finite-dimensional quantum state ρ has been defined as its distinguishability from the sets of states which are diagonal in a given basis [14][15][16][17][18][19].…”

“…Yet, to date, there is no operational interpretation for such definition of coherence. A concurrent body of work has linked the coherence of ρ in a basis {h} to the degree of uncertainty in a measurement of an observable H = ∑ h h|h h| on ρ [2][3][4][5][6][7][8][9][10][11][12][13]. Such genuinely quantum uncertainty is due to the non-commutativity between state and observable.…”

Preservation of quantum coherence under Lorentz boost for narrow uncertainty wave packets

“…Defined in a quantitative manner based on the framework of resource theory [16], [17], [18][19][20][21][22][23][24] quantum coherence may be exploited to perform quantum tasks. Several operational measures of quantum coherence have been proposed [25], [26], enabling it to be used for detection of genuine non-classicality in physical states.…”