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arXiv:1311.2773 · Europhysics Letters (EPL)

Mutually unbiased measurements for high-dimensional time-bin–based photonic states

Thomas Brougham, Stephen Mark Barnett

Published Nov 1, 2013• 9 citations
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Abstract

The task of measuring in two mutually unbiased bases is central to many quantum information protocols, as well as being of fundamental interest. Increasingly, there is an experimental focus on generating and controlling high-dimensional photonic states. One approach is to use the arrival time of a photon, which can be split into discrete time bins. An important problem associated with such states is the difficulty in experimentally realizing a measurement that is mutually unbiased with respect to the time of arrival. We propose a simple and compact scheme to measure in both the time-of-arrival basis and a basis that is approximately mutually unbiased with respect to the arrival time.

DOI
10.1209/0295-5075/104/30003

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The task of measuring in two mutually unbiased bases is central to many quantum information protocols, as well as being of fundamental interest. Increasingly, there is an experimental focus on generating and controlling high-dimensional photonic states. One approach is to use the arrival time of a photon, which can be split into discrete time bins. An important problem associated with such states is the difficulty in experimentally realizing a measurement that is mutually unbiased with respect to the time of arrival. We propose a simple and compact scheme to measure in both the time-of-arrival basis and a basis that is approximately mutually unbiased with respect to the arrival time.
High-dimensional quantum key distribution (QKD) offers the possibility of encoding multiple bits of key on a single entangled photon pair. An experimentally promising approach to realizing this is to use energy–time entanglement. Currently, however, the control of very high-dimensional entangled photons is challenging. We present a simple and experimentally compact approach, which is based on a cavity that allows one to measure two different bases: the time of arrival and another that is approximately mutually unbiased to the arrival time. We quantify the errors in the setup, due both to the approximate nature of the mutually unbiased measurement and as a result of experimental errors. It is shown that the protocol can be adapted using a cut-off so that it is robust against the considered errors, even within the regime of up to 10 bits per photon pair.
We propose a method for implementing mutually unbiased generation and measurement of time-bin qudits using a cascade of electro-optic phase modulator–coded fiber Bragg grating pairs. Our approach requires only a single spatial mode and can switch rapidly between basis choices. We obtain explicit solutions for dimensions d = 2, 3, and 4 that realize all d + 1 possible mutually unbiased bases and analyze the performance of our approach in quantum key distribution. Given its practicality and compatibility with current technology, our approach provides a promising springboard for scalable processing of high-dimensional time-bin states.
Measurements in mutually unbiased bases (MUBs) are integral to quantum science. An underlying principle of a set of MUBs is that a measurement in one such basis provides no information about a measurement in one of the others. This feature is especially relevant for quantum key distribution. We perform a complete set of mutually unbiased measurements for high-dimensional orbital angular momentum (OAM) states of light. For a two-photon entangled state we show that specific joint measurements provide the minimum number of experimental settings required to characterize the quantum state. From these measurements we reconstruct the density matrix describing the high-dimensional two-photon entangled state. The experimental procedure that we outline can be applied to many different multipartite and high-dimensional systems.
Quantum states encoded in the time-bin degree of freedom of photons represent a fundamental resource for quantum information protocols. Traditional methods for generating and measuring time-bin-encoded quantum states face severe challenges due to optical instabilities, complex setups, and timing resolution requirements. To circumvent these issues, we leverage an approach based on Hong-Ou-Mandel interference and we propose and demonstrate a robust and scalable protocol to generate and measure arbitrary high-dimensional time-bin quantum states. We experimentally implement the protocol in a photonic setup reaching high-fidelity quantum state tomographies of two- and three-dimensional quantum states encoded in time bins with short temporal separation. We also certify intrasystem polarization-time entanglement of single photons through a nonclassicality test. The demonstrated approach enables access to high-dimensional states and tasks that are practically inaccessible with standard schemes, thereby advancing fundamental quantum information science and opening applications in quantum communication.
Mutually unbiased bases (MUBs) play a key role in many protocols in quantum science, such as quantum key distribution. However, defining MUBs for arbitrary high-dimensional systems is theoretically difficult, and measurements in such bases can be hard to implement. We show experimentally that efficient quantum state reconstruction of a high-dimensional multipartite quantum system can be performed by considering only the MUBs of the individual parts. The state spaces of the individual subsystems are always smaller than the state space of the composite system. Thus, the benefit of this method is that MUBs need to be defined for the small Hilbert spaces of the subsystems rather than for the large space of the overall system. This becomes especially relevant where the definition or measurement of MUBs for the overall system is challenging. We illustrate this approach by implementing measurements for a high-dimensional system consisting of two photons entangled in the orbital angular momentum degree of freedom, and we reconstruct the state of this system for dimensions of the individual photons from d = 2 to 5.