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arXiv:1712.03475 · Journal of the Optical Society of America B

Intrinsic degree of coherence of classical and quantum states

Abu Musa Patoary, Girish Ashok Kulkarni, Anand K. Jha

Published Sep 16, 2019• 15 citations
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Abstract

In the context of the two-dimensional (2D) polarization states of light, the degree of polarization P2 is equal to the maximum value of the degree of coherence over all possible bases. Therefore, P2 can be referred to as the intrinsic degree of coherence of a 2D state. In addition to (i) the maximum degree of coherence interpretation, P2 also has the following interpretations: (ii) it is the Frobenius distance between the state and the maximally incoherent identity state, (iii) it is the norm of the Bloch vector representing the state, (iv) it is the distance to the center of mass in a configuration of point masses with magnitudes equal to the eigenvalues of the state, (v) it is the visibility in a polarization interference experiment, and (vi) it is the weightage of the pure part of the state. Among these six interpretations of P2, the Bloch vector norm, Frobenius distance, and center-of-mass interpretations have previously been generalized to derive an analogous basis-independent measure PN for N-dimensional (ND) states. In this paper, by extending the concepts of visibility, degree of coherence, and weightage of the pure part to ND spaces, we show that these three remaining interpretations of P2 also generalize to the same quantity PN, establishing PN as the intrinsic degree of coherence of ND states. We then extend PN to the N→∞ limit to quantify the intrinsic degree of coherence P∞ of infinite-dimensional states in the orbital angular momentum, photon number, and position-momentum degrees of freedom.

DOI
10.1364/josab.36.002765

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In the context of the two-dimensional (2D) polarization states of light, the degree of polarization P2 is equal to the maximum value of the degree of coherence over all possible bases. Therefore, P2 can be referred to as the intrinsic degree of coherence of a 2D state. In addition to (i) the maximum degree of coherence interpretation, P2 also has the following interpretations: (ii) it is the Frobenius distance between the state and the maximally incoherent identity state, (iii) it is the norm of the Bloch vector representing the state, (iv) it is the distance to the center of mass in a configuration of point masses with magnitudes equal to the eigenvalues of the state, (v) it is the visibility in a polarization interference experiment, and (vi) it is the weightage of the pure part of the state. Among these six interpretations of P2, the Bloch vector norm, Frobenius distance, and center-of-mass interpretations have previously been generalized to derive an analogous basis-independent measure PN for N-dimensional (ND) states. In this paper, by extending the concepts of visibility, degree of coherence, and weightage of the pure part to ND spaces, we show that these three remaining interpretations of P2 also generalize to the same quantity PN, establishing PN as the intrinsic degree of coherence of ND states. We then extend PN to the N→∞ limit to quantify the intrinsic degree of coherence P∞ of infinite-dimensional states in the orbital angular momentum, photon number, and position-momentum degrees of freedom.
For light fields, the manifestation of correlations between fluctuating electric field components at different space–time points is referred to as coherence, whereas these correlations appearing between orthogonal electric field components at a single space–time point are referred to as polarization. In this context, a natural question is as follows: how are coherence and polarization interconnected? Very recently, a tight equality P2=V2+D2, namely, the “polarization coherence theorem” (PCT) connecting polarization P with interference visibility V (measure of coherence) and distinguishability D (measure of which-path information) has been proposed [Optica4, 1113 (2017)OPTIC82334-253610.1364/OPTICA.4.001113]. We here report a direct observation of the PCT for classical light fields using a Mach–Zehnder interferometer along with a synthesized source producing a complete gamut of degrees of polarizations. Our experimental demonstration could motivate ongoing experimental efforts toward probing the hidden coherences and complementarity features.
Recently, a basis-invariant measure of coherence known as the intrinsic degree of coherence has been established for classical and single-particle quantum states [ J. Opt. Soc. Am. B 36 , 2765 ( 2019 ) JOBPDE 0740-3224 10.1364/JOSAB.36.002765 ]. In this article, we use the same mathematical construction to define the intrinsic degree of coherence of two-qubit states and demonstrate its usefulness in quantifying two-particle quantum correlations and entanglement. In this context, we first show that the intrinsic degree of coherence of a two-qubit state puts an upper bound on the violations of Bell inequalities that can be achieved with the state and that a two-qubit state with an intrinsic degree of coherence less than 1 / 3 cannot violate Bell inequalities. We then show that the quantum discord of a two-qubit state, which quantifies the amount of quantum correlations available in the two-qubit state for certain tasks, is bounded from above by the intrinsic degree of coherence of the state. Next, in the context of two-particle entanglement, we show that the range of values that the concurrence of a two-qubit state can take is decided by the intrinsic degree of coherence of the two-qubit state together with that of the individual qubits. Finally, for the polarization two-qubit states generated by the parametric down-conversion of a pump photon, we propose an experimental scheme to measure the intrinsic degree of coherence of two-qubit states. We also present our theoretical study that shows how the intrinsic degree of coherence of a pump photon dictates the maximum intrinsic degree of coherence of the generated two-qubit state.
We introduce a quantification of genuine three-party pure-state coherence for wave fields, classical and quantum, by borrowing concepts from classical optics. The tensor structure of a classical paraxial light beam composed of three principle degrees of freedom is shown to be equivalent to that of a three-qubit quantum state. The traditional basis-independent optical coherence quantity called degree of polarization is then determined to be the desired quantitative two-party coherence measure. When appropriately generalized, a set of fundamental constraint relations is derived among three two-party coherences. The constraint relations can be geometrically interpreted and visualized as tetrahedra nested within a coherence cube. A measure of three-party coherence is defined based on the constraints. We are reporting completed experimental tests and confirmations of the constraints as well as measurement of three-party coherence in the optical context. Our approach based on classical optics also opens an alternative way to analyze quantum coherence.
The dual wave–particle nature of light and the degree of polarization are fundamental concepts in quantum physics and optical science, but their exact relation has not been explored within a full vector-light quantum framework that accounts for interferometric polarization modulation. Here, we consider vector-light quantum complementarity in double-pinhole photon interference and derive a general link between the degree of polarization and wave–particle duality of light. The relation leads to an interpretation for the degree of polarization as a measure describing the complementarity strength between photon path predictability and so-called Stokes visibility, the latter taking into account both intensity and polarization variations in the observation plane. It also unifies results advanced in classical studies by showing that the degree of polarization can be viewed as the ability of a light beam to exhibit intensity and polarization-state fringes. The framework we establish thus provides novel aspects and deeper insights into the role of the degree of polarization in quantum-light complementarity and photon interference.
We explore the concepts of coherence and entanglement as they apply to both the classical and quantum natures of light. In the classical domain, we take inspiration from the tools and concepts developed in foundational quantum mechanics and quantum information science to gain a better understanding of classical coherence theory of light with multiple degrees of freedom (DoFs). First, we use polarization and spatial parity DoFs to demonstrate the notion of classical entanglement, and show that Bell's measure can serve as a useful tool in distinguishing between classical optical coherence theory. Second, we establish a methodical yet versatile approach called 'optical coherency matrix tomography' for reconstructing the coherency matrix of an electromagnetic beam with multiple DoFs. This technique exploits the analogy between this problem in classical optics and that of tomographically reconstructing the density matrix associated with multipartite quantum states in quantum information science. Third, we report the first experimental measurements of the 4 x 4 coherency matrix associated with an electromagnetic beam in which polarization and a spatial DoF are relevant, ranging from the traditional two-point Young's double slit to spatial parity and orbital angular momentum modes. In the quantum domain, we use the modal structure of classical fields to develop qubits and structure Hilbert spaces for use in quantum information processing. Advancing to three-qubit logic gates is an important step towards the success of optical schemes for quantum computing. We experimentally implement a variety of two- and three- qubit, linear and deterministic, single-photon, controlled, quantum logic gates using polarization and spatial parity qubits. Lastly, we demonstrate the implementation of two-qubit single-photon logic using polarization and orbital angular momentum qubits.
We report that the true intrinsic degree of freedom of the photon is neither the polarization nor the spin. It describes a local property in momentum space and is represented in the local representation by the Pauli matrices. This result is achieved by treating the transversality condition on the vector wavefunction as a nonholonomic quantum constraint. We find that the quantum constraint makes it possible to generalize the Stokes parameters to characterize the polarization of a general state. Unexpectedly, the generalized Stokes parameters are specified in a momentum-space local reference system that is fixed by another degree of freedom, called Stratton vector. Only constant Stokes parameters in one particular local reference system can convey the intrinsic degree of freedom of the photon. We show that the optical rotation is one of such processes that change the Stratton vector with the intrinsic quantum number remaining fixed. Changing the Stratton vector of the eigenstate of the helicity will give rise to a Berry's phase.