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AIP conference proceedings

Theoretical and Experimental Characterization of Entropic Inequalities

Fabio Antonio Bovino, Concita Sibilia, Diederik Sybolt Wiersma

Published Jan 1, 2007• 1 citation
Publisher

Abstract

Nonlinear inequalities based on the quadratic Renyi entropy for mixed two‐qubit states are characterized on the Entropy‐Concurrence plane. This class of inequalities is stronger than Clauser‐Horne‐Shimony‐Holt (CHSH) inequalities and, in particular, are violated “in toto” by the set of Type I Maximally‐Entangled‐Mixture States (MEMS I). Renyi entropy is experimentally obtained by local measurements on two pairs of polarization‐entangled photons. The novel “phase marking” technique allows the selection of uncorrupted outcomes even with nondeterministic sources of entangled photons. Experimental data demonstrate the violation of entropic inequalities which are an example of nonlinear entanglement witnesses.

DOI
10.1063/1.2821604

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Nonlinear inequalities based on the quadratic Renyi entropy for mixed two‐qubit states are characterized on the Entropy‐Concurrence plane. This class of inequalities is stronger than Clauser‐Horne‐Shimony‐Holt (CHSH) inequalities and, in particular, are violated “in toto” by the set of Type I Maximally‐Entangled‐Mixture States (MEMS I). Renyi entropy is experimentally obtained by local measurements on two pairs of polarization‐entangled photons. The novel “phase marking” technique allows the selection of uncorrupted outcomes even with nondeterministic sources of entangled photons. Experimental data demonstrate the violation of entropic inequalities which are an example of nonlinear entanglement witnesses.
Nonlinear properties of quantum states, such as entropy or entanglement, quantify important physical resources and are frequently used in quantum information science. They are usually calculated from a full description of a quantum state, even though they depend only on a small number of parameters that specify the state. Here we extract a nonlocal and a nonlinear quantity, namely the Renyi entropy, from local measurements on two pairs of polarization entangled photons. We also introduce a "phase marking" technique which allows to select uncorrupted outcomes even with nondeterministic sources of entangled photons. We use our experimental data to demonstrate the violation of entropic inequalities. They are examples of a nonlinear entanglement witness and their power exceeds all linear tests for quantum entanglement based on all possible Bell-CHSH inequalities.
Nonlinear properties of quantum states, such as entropy or entanglement, quantify important physical resources and are frequently used in quantum-information science. They are usually calculated from a full description of a quantum state, even though they depend only on a small number of parameters that specify the state. Here we extract a nonlocal and a nonlinear quantity, namely, the Renyi entropy, from local measurements on two pairs of polarization-entangled photons. We also introduce a "phase marking" technique which allows the selection of uncorrupted outcomes even with nondeterministic sources of entangled photons. We use our experimental data to demonstrate the violation of entropic inequalities. They are examples of nonlinear entanglement witnesses and their power exceeds all linear tests for quantum entanglement based on all possible Bell-Clauser-Horne-Shimony-Holt inequalities.
It is well known that the violation of Bell's inequality in the form given by Clauser, Horne, Shimony, and Holt (CHSH) in two-qubit systems requires entanglement, but not vice versa, i.e., there are entangled states which do not violate the CHSH inequality. Here we compare some standard entanglement measures with violations of the CHSH inequality (as given by the Horodecki measure) for two-qubit states generated by Monte Carlo simulations. We describe states that have extremal entanglement according to the negativity, concurrence, and relative entropy of entanglement for a given value of the CHSH violation. We explicitly find these extremal states by applying the generalized method of Lagrange multipliers based on the Karush-Kuhn-Tucker conditions. The found minimal and maximal states define the range of entanglement accessible for any two-qubit states that violate the CHSH inequality by the same amount. We also find extremal states for the concurrence versus negativity by considering only such states which do not violate the CHSH inequality. Furthermore, we describe an experimentally efficient linear-optical method to determine the highest Horodecki degree of the CHSH violation for arbitrary mixed states of two polarization qubits. By assuming to have access simultaneously to two copies of the states, our method requires only six discrete measurement settings instead of nine settings, which are usually considered.