Observable Measure of Quantum Coherence in Finite Dimensional Systems

Abstract

Quantum coherence is the key resource for quantum technology, with applications in quantum optics, information processing, metrology and cryptography. Yet, there is no universally efficient method for quantifying coherence either in theoretical or in experimental practice. I introduce a framework for measuring quantum coherence in finite dimensional systems. I define a theoretical measure which satisfies the reliability criteria established in the context of quantum resource theories. Then, I present an experimental scheme implementable with current technology which evaluates the quantum coherence of an unknown state of a d-dimensional system by performing two programmable measurements on an ancillary qubit, in place of the O(d 2 ) direct measurements required by full state reconstruction. The result yields a benchmark for monitoring quantum effects in complex systems, e.g. certifying non-classicality in quantum protocols and probing the quantum behaviour of biological complexes. Introduction -While harnessing quantum coherence is matter of routine in delivering quantum technology [1][2][3][4][5], and the quantum optics rationale rests on creation and manipulation of coherence [6], there is no universally efficient route to measure the amount of quantum coherence carried by the state of a system in dimension d > 2. It is customary to employ quantifiers tailored to the scenario of interest, i.e. of not general employability, expressed in terms of ad hoc entropic functions, correlators, or functions of the off-diagonal density matrix coefficients (if available) [7][8][9]. Quantum information theory provides the framework to address the problem. Physical laws are interpreted as restrictions on the accessible quantum states and operations, while the properties of physical systems are the resources that one must consume to perform a task under such laws [10]. An algorithmic characterization of quantum coherence as a resource and a set of bona fide criteria for coherence monotones have been identified [7, 11, 12]. Also, coherence has been shown to be related to the asymmetry of a quantum state [13, 14]. On the experimental side, the scalability of the detection scheme is a major criterion in developing witnesses and measures of coherence, as we are interested in exploring the quantum features of highly complex macrosystems, e.g. multipartite quantum registers and networks. Therefore, it is desirable to have a coherence measure which is both theoretically sound and experimentally appealing.

Quantum modeling of ultrafast photoinduced charge separation

“…In fact, in the case of pure states, quantum fluctuations are the only source of uncertainty in the measurement outcome, whereas mixing represents an additional source of uncertainty in the case of mixed states. The above arguments qualitatively justify the use of the WignerYanase-Dyson skew information I(ρ,K) as a coherence quantifier [165]. The skew information I, which is meant to single out the uncertainties resulting from quantum fluctuations, is given by:…”

Quantum coherence: Reciprocity and distribution

“…Measures that satisfy the above conditions, include l 1 norm and relative entropy of coherence [15] and the skew information [17]. Coherence can also be quantified through entanglement.…”

“…Moreover, combined with the tensor product structure of quantum state space, it gives rise to the novel concepts such as entanglement and quantum correlations. It, being the premise of quantum correlations in multipartite systems, has attracted the attention of quantum information community significantly, and * asukumar@hri.res.in in addition to its quantification [17][18][19], other developments like the freezing phenomena [20], the coherence transformations under incoherent operations [21], establishment of geometric lower bound for a coherence measure [22], the complementarity between coherence and mixedness [23], its relation with other measures of quantum correlations and creation of coherence using unitary operations [24,25], erasure of quantum coherence [26], and catalytic transformations of coherence [27] have been reported recently.…”

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

“…Skew Information: If an observable X is measured on the state ρ, the skew information is given by [25], [32],…”

“…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. However, as is the case with entanglement, there exists no unique quantifier of coherence.…”