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Open Systems & Information Dynamics

Direct Measurement of Nonlinear Properties of Bipartite Quantum States

Fabio Antonio Bovino, Giuseppe Castagnoli, ARTUR K. EKERT, Carolina Alves, Paweł Horodecki, Alexander V. Sergienko

Published Aug 31, 2006• 4 citations
Publisher

Abstract

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.

DOI
10.1007/s11080-006-9008-0

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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.
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.
Abstract Using the type-I SPDC process in BBO nonlinear crystal (NLC), we generate a polarization-entangled state near to the maximally-entangled Bell-state with high-visibility (high-brightness) 98.50 ±1.33 % (87.71 ± 4.45 %) for HV (DA) basis. We calculate the CHSH version of the Bell inequality, as a nonlocal realism test, and find a strong violation from the classical physics or any hidden variable theory (HVT), S= 2.71±0.10. Via measuring the coincidence count (CC) rate in the SPDC process, we obtain the quantum efficiency of single-photon detectors (SPDs) around (25.5 ±3.4)% , which is in good agreement to their manufacturer company. As expected, we verify the linear dependency of the CC rate vs. pump power of input CW-laser, which may yield to find the effective second-order susceptibility crystal. Using the theory of the measurement of qubits, includes a tomographic reconstruction of quantum states due to the linear set of 16 polarization-measurement, together with a maximum-likelihood-technique (MLT), which is based on the numerical optimization, we calculate the physical non-negative definite density matrices, which implies on the non-separability and entanglement of prepared state. By having the maximum likelihood density operator, we calculate precisely the entanglement measures such as Concurrence, entanglement of formation, tangle, logarithmic negativity, and different entanglement entropies such as linear entropy, Von-Neumann entropy, and Renyi 2-entropy. Finally, this high-brightness and low-rate entangled photons source can be used for short-range quantum measurements in the Lab.
Quantum coherence is the most distinguished feature of quantum mechanics, which characterizes the superposition properties of quantum states. It plays a critical role in various fields, ranging from quantum information technology to quantum biology. Although various coherence quantifiers have been proposed since the resource theory of coherence was established, there are a lack of experimental methods to estimate them efficiently, which restricts the applications of coherence. Relative entropy of coherence is one of the main quantifiers of coherence, and is frequently used in quantum information science. Such nonlinear properties of quantum states are usually calculated from full descriptions of the quantum state, although they are not related to all parameters that specify the state. Here, we experimentally measure the relative entropy of coherence for the arbitrary qubit states directly in the photonic system without using standard state tomography. In the experiment, we directly measure the von Neumann entropy of the quantum states through interference, thus obtaining the relative entropy of coherence, and finding that the experimental results are in good agreement with the theory. Our work provides a nice alternative experimental scheme for measuring the relative entropy of coherence.
The relation between the violation of the Bell-CHSH inequalities and entanglement properties of quantum states is not clear so one may consider the mixedness of the system to understand the entanglement properties better than the Bell-CHSH inequality. In this respect, we prove that if the mixedness of the state measured by the linear entropy is less than 2/3 but strictly greater than zero then the two qubit states are entangled. But if the linear entropy is greater or equal to 2/3 then the state may or may not be entangled. Further we show that our entanglement criterion detects larger set of entangled state than Bell-CHSH inequality and Santos's entropic criterion [Phys. Rev. A 69, 022305 (2004)]. Lastly we illustrate our result by citing few examples.
In recent years two fundamental aspects of quantum mechanics have attracted a great deal of interest, namely the investigation on the irreducible nonlocal properties of Nature implied by quantum entanglement and the physical realization of the 'Schroedinger Cat'. In the present work a macro - state consisting of about 3.5 x 10^4 photons in a quantum superposition and entangled with a far apart single - photon state is generated. Then, the non-separability of the overall micro-macro system is demonstrated and the corresponding Bell's inequalities are found to be violated. Precisely, an entangled photon pair is created by a nonlinear optical process, then one photon of the pair is injected into an optical parametric amplifier operating for any input polarization state, i.e. into a phase-covariant cloning machine. Such transformation establishes a connection between the single photon and the multi particle fields. We then demonstrate the non-separability of the bipartite system by adopting a local filtering technique within a positive operator valued measurement.