Quantum state tomography with a single measurement setup

Oren, Dikla; Mutzafi, Maor; Eldar, Yonina C.; Segev, Mordechai

doi:10.1364/optica.4.000993

Cited by 32 publications

(40 citation statements)

References 57 publications

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“…The lower bound can be achieved if the same unitaries g α enable also the inversion Y for < N. The reconstruction algorithm is then Eq. (36).…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…The latter approach was recently extended to two-photon, two-mode states using a six-mode interferometer and it was conjectured this method would work for larger systems [34,35]. Related work has investigated how the number of additional modes required for high-fidelity state estimation depends on the purity of the input state [36]. Our results generalize these photonic studies that use a single measurement configuration by proving tomographic feasibility, deriving a bound on the minimum number of measurement modes, and providing an explicit reconstruction protocol.We numerically show that use of random interferometer configurations, in particular those corresponding to Haarrandom transformations, enable tomography using the minimum number of configurations.…”

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confidence: 99%

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Multiphoton Tomography with Linear Optics and Photon Counting

2018

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Determining an unknown quantum state from an ensemble of identical systems is a fundamental, yet experimentally demanding, task in quantum science. Here we study the number of measurement bases needed to fully characterize an arbitrary multi-mode state containing a definite number of photons, or an arbitrary mixture of such states. We show this task can be achieved using only linear optics and photon counting, which yield a practical though non-universal set of projective measurements. We derive the minimum number of measurement settings required and numerically show that this lower bound is saturated with random linear optics configurations, such as when the corresponding unitary transformation is Haar-random. Furthermore, we show that for N photons, any unitary 2N-design can be used to derive an analytical, though non-optimal, state reconstruction protocol.Introduction-An unknown quantum state can be determined by making a set of suitable measurements on identically prepared copies [1][2][3][4][5]. This procedure, known as quantum state tomography, is a fundamental concept in quantum science with wide ranging applications. For example, tomography allows one to assess quantum systems for use in quantum information processing by quantifying resources such as entanglement [6], quantum correlations [7], and coherence [8]. Indeed, since most measures of these resources require complete knowledge of the density matrix describing a system, full quantum tomography is often necessary. Similarly, tomography can be applied to quantum sensing [9, 10] to evaluate the capacity of a quantum probe state to yield enhanced measurement precision [11,12].A well-established framework for photonic quantum information uses a single photon and multiple modes to encode discrete-variable quantum states. A qubit may be encoded using a single-photon, two-mode state [13], and a qudit may be encoded by incorporating additional modes [14]. Multiqubit states of this form have been employed widely, including entanglement-based quantum-key distribution [15], quantum simulation [16], tests of quantum nonlocality [17], entanglement generation [18], and linear optical quantum computing [19]. For these states, optical tomography can be readily achieved using combinations of single-qudit measurements [2,20], which require only linear optics and single-photon detection. Exact reconstruction of N qubits can thus be achieved using 2 N + 1 measurement bases. Using this method, full tomography of up to six single-photon qubits has been demonstrated [21].However, this approach to optical tomography does not apply to more general states of multiple modes containing a definite total number of photons. In this case, a mode may contain multiple photons, which enables new applications including approaches to quantum sampling [22], imaging [23], and error-correction [24,25]. An alternate approach to state tomography for such states is to use balanced homodyne detection and well-developed continuous-variable algorithms to reconstruct the phase-space Wigner funct...

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“…The lower bound can be achieved if the same unitaries g α enable also the inversion Y for < N. The reconstruction algorithm is then Eq. (36).…”

Section: Proof Of Theoremmentioning

confidence: 99%

mentioning

confidence: 99%

Multiphoton Tomography with Linear Optics and Photon Counting

2018

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“…Since it is not necessary to reconfigure the experimental apparatus, the occurrence of errors is also reduced. Circuits that do not require reconfiguration are called static circuits [55].…”

Section: Introductionmentioning

confidence: 99%

“…It is necessary to mention that the proposed circuit design realizes full quantum tomography with the minimum number of measurements and no hypothesis about the initial system state are necessary. Others schemes are able to realize quantum tomography with a number of measurements smaller than the minimum for a particular class of systems states using prior knowledge of the input state [55].…”

Section: Introductionmentioning

confidence: 99%

Implementing positive-operator-valued-measurement elements in photonic circuits for performing minimum quantum state tomography of path qudits

Cardoso

Barros

et al. 2019

Phys. Rev. A

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Manipulation of qudits in optical tables is a difficult and non-scalable task. The use of integrated optical circuits opens new possibilities for the generation, manipulation and characterization of high dimensional states besides the ease of transmission of these states through an optical fiber. In this work, we propose photonic circuits to perform minimum quantum state tomography of path qudits and show how to determine all the constituents parameters of these circuits (beam splitters and phase shifters). Our strategies were based on the symmetries of the involved POVMs suggested for minimum tomography and allowed us to obtain interferometers smaller than those obtained by other already known methods. The calculations of the transmittances and reflectivities of the beam splitters were made using the definition of probability operators in an extended Hilbert spaces and the application of the Naimark's theorem. The employment of equidistant states for the definition of the POVM elements allowed us to develop a recipe applicable to the tomography of qudits of any dimension, generalizing our scheme.

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“…24 Here we demonstrate experimentally that, with the appropriate optical transformation, multiplexed detection can also allow full quantum state tomography, and we present a comprehensive theoretical analysis of multi-photon case building on our original proposal for two-photon tomography with static measurement setups 25 (after the initial submission of this work, a similar approach to quantum state tomography was investigated theoretically in ref. 26 ). When an N photon state is spread coherently across an increased number of outputs, the number of possible N-photon correlation measurement outcomes follows a similar scaling to the number of parameters in the density matrix.…”

Section: Introductionmentioning

confidence: 99%

Scalable on-chip quantum state tomography

et al. 2018

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Quantum information systems are on a path to vastly exceed the complexity of any classical device. The number of entangled qubits in quantum devices is rapidly increasing, and the information required to fully describe these systems scales exponentially with qubit number. This scaling is the key benefit of quantum systems, however it also presents a severe challenge. To characterize such systems typically requires an exponentially long sequence of different measurements, becoming highly resource demanding for large numbers of qubits. Here we propose and demonstrate a novel and scalable method for characterizing quantum systems based on expanding a multi-photon state to larger dimensionality. We establish that the complexity of this new measurement technique only scales linearly with the number of qubits, while providing a tomographically complete set of data without a need for reconfigurability. We experimentally demonstrate an integrated photonic chip capable of measuring two-and three-photon quantum states with statistical reconstruction fidelity of 99.71%.

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Quantum state tomography with a single measurement setup

Cited by 32 publications

References 57 publications

Multiphoton Tomography with Linear Optics and Photon Counting

Multiphoton Tomography with Linear Optics and Photon Counting

Implementing positive-operator-valued-measurement elements in photonic circuits for performing minimum quantum state tomography of path qudits

Scalable on-chip quantum state tomography

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