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Journal of Optics

Relationship between entanglement and polarization in tripartite states

C. Montenegro La Torre, Yonny Yugra, F. De Zela

Published Aug 18, 2022• 2 citations
Publisher

Abstract

Abstract Entanglement and polarization of pure, two-qubit systems are known to be constrained by the relationship C 2 + P 2 = 1 . Here, C stands for concurrence, a standard measure of entanglement, and P refers to the degree of polarization of either of the two, single-qubit subsystems. The above constraint may be understood as reflecting that a system cannot be in a pure state, if it is entangled with another one. In order to explore the connection between entanglement and polarization beyond two-qubit systems, we addressed states in which at least one of the parties has dimensionality greater than two. The case of pure, tripartite states offered itself as a timely candidate. We focussed on three qubits and studied the connection between one qubit’s polarization and its entanglement with the other two taken together. We derived new constraints, akin to the above one, and submitted them to experimental tests using single photons entangled in polarization. The latter provided two qubits, each one attached to one photon. The third qubit was chosen to be the path (momentum) of one of the photons. Our experimental results confirmed the validity of the new constraints. Our theoretical results hold also for classical light. Possible experimental tests could be done with so-called structured light beams.

DOI
10.1088/2040-8986/ac8aab

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Abstract Entanglement and polarization of pure, two-qubit systems are known to be constrained by the relationship C 2 + P 2 = 1 . Here, C stands for concurrence, a standard measure of entanglement, and P refers to the degree of polarization of either of the two, single-qubit subsystems. The above constraint may be understood as reflecting that a system cannot be in a pure state, if it is entangled with another one. In order to explore the connection between entanglement and polarization beyond two-qubit systems, we addressed states in which at least one of the parties has dimensionality greater than two. The case of pure, tripartite states offered itself as a timely candidate. We focussed on three qubits and studied the connection between one qubit’s polarization and its entanglement with the other two taken together. We derived new constraints, akin to the above one, and submitted them to experimental tests using single photons entangled in polarization. The latter provided two qubits, each one attached to one photon. The third qubit was chosen to be the path (momentum) of one of the photons. Our experimental results confirmed the validity of the new constraints. Our theoretical results hold also for classical light. Possible experimental tests could be done with so-called structured light beams.
We present an experimental proof-of-principle for the generation and detection of pure two-qubit states that have been encoded in degrees of freedom that are common to both classical-light beams and single photons. Our protocol requires performing polarization tomography on a single qubit from a qubit pair. The degree of entanglement in the qubit pair is measured by concurrence, which can be directly extracted from intensity measurements-or photon counting-entering single-qubit polarization tomography.
We establish a quantitative relation between local spin polarization and quantum entanglement in two-qubit systems by mathematically proving a closed-form upper bound on the concurrence at fixed local polarization magnitudes. The bound shows how the maximal concurrence is jointly constrained by the two local polarizations. We further demonstrate that this bound is saturated by pure states in certain cases with identical polarizations. As a concrete physical application, we consider the parity-violating process e+e−→Z0→qqˉe^+e^- \to Z^0 \to q\bar{q}e+e−→Z0→qqˉ​, which generates final-state spin polarization. We show that the maximal concurrence is attained in specific kinematic regions and is significantly reduced relative to the unpolarized case. These results establish a general, process-independent framework connecting local polarization, maximal entanglement, and pure states.
Given a multipartite quantum system that consists of two-level particles (qubits), one may or may not have access to all the subsystems. What can we know about the entanglement of the multiqubit system and residual correlations beyond two-tangle if we have access only to two-qubits at a time? Algebraic analysis of two-qubit states yields monogamy constraints on distribution of entanglement between sub-systems of an N-qubit state and criterion to determine if the state has multipartite entanglement. Monogamy constraints, reported in this letter, are relations between well known entanglement measures such as one-tangle, two-tangles and three-tangles of an N-qubit pure state.
Photonic multidimensional quantum networks (MDQN), where individual subsystems are encoded using multiple degrees of freedom and photons, are an emerging platform for quantum algorithms because they offer high scalability. The distribution of non-classical and non-local correlations between the individual subsystems in an MDQN is of fundamental interest for many quantum protocols. Interestingly in an MDQN, the inseparability of two subsystems underlying entanglement can occur both between multiple distinct photons as well as between individual degrees of freedom associated with a single photon. In this work, we investigate the entanglement transfer enabled by the interplay of both entanglement between two distinct photons as well as inseparability between multiple degrees of freedom. For this purpose, we subject one photon of a polarization entangled two-photon pair to a discrete-time quantum walk introducing the position subsystem of the quantum walk as a third subsystem with qudit encoding. Here we study the resulting transfer of entanglement from the polarization degree of freedom, representing qubit encoding, towards the position degree of freedom, representing quidt encoding, via partial state tomography and correlation measurements.
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 propose and experimentally demonstrate an asymmetric key-generation scheme in which public and private keys are derived from a six-qubit photonic entangled state. A six-qubit entangled state is generated from two photons, each carrying one polarization and two path qubits. The state is prepared in an equal superposition of eight of the sixty-four six-qubit basis states, in which each photon's three qubits form a codeword of the classical [3,2,2] even-parity code, and are distributed between the two parties, Alice and Bob. This ensures, each triple is a uniformly random codeword and the two codewords are locked together through the correlation. The parity constraint is a local stabilizer, so each party verifies it on its own data every round, and eliminates the bit flip errors arising from imperfections, detector dark counts and multiple occupancy before it enters the raw key. A single publicly announced bit, chosen at random from either of the path qubits, allows each party to reconstruct the other's full triple while leaking no information to third party. We realize the desired six-qubit state using a Sagnac-based type-II SPDC source, reporting a Bell-CHSH parameter S = 2.798 +/- 0.008, by implementing gate operation on polarization and path degree's of freedom. This approach provides a physics-based alternative to public-key cryptosystems built on computational hardness assumptions. A t-bit string generated from multiple rounds for encryption will be intrinsically random. Without the private key, one needs at least O(2^t) runs on an t-qubit quantum system to find the matching bit to decrypt.