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arXiv:1506.03659 · Physical Review A

Generalized Hofmann quantum process fidelity bounds for quantum filters

Michal Sedlák, Jaromı́r Fiurášek

Published Apr 11, 2016• 3 citations
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Abstract

We propose and investigate bounds on the quantum process fidelity of quantum filters, i.e., probabilistic quantum operations represented by a single Kraus operator $K$. These bounds generalize the Hofmann bounds on the quantum process fidelity of unitary operations [H. F. Hofmann, Phys. Rev. Lett. 94, 160504 (2005)] and are based on probing the quantum filter with pure states forming two mutually unbiased bases. Determination of these bounds therefore requires far fewer measurements than full quantum process tomography. We find that it is particularly suitable to construct one of the probe bases from the right eigenstates of $K$, because in this case the bounds are tight in the sense that if the actual filter coincides with the ideal one, then both the lower and the upper bounds are equal to 1. We theoretically investigate the application of these bounds to a two-qubit optical quantum filter formed by the interference of two photons on a partially polarizing beam splitter. For an experimentally convenient choice of factorized input states and measurements we study the tightness of the bounds. We show that more stringent bounds can be obtained by more sophisticated processing of the data using convex optimization and we compare our methods for different choices of the input probe states.

DOI
10.1103/physreva.93.042316

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We propose and investigate bounds on the quantum process fidelity of quantum filters, i.e., probabilistic quantum operations represented by a single Kraus operator KKK. These bounds generalize the Hofmann bounds on the quantum process fidelity of unitary operations [H. F. Hofmann, Phys. Rev. Lett. 94, 160504 (2005)] and are based on probing the quantum filter with pure states forming two mutually unbiased bases. Determination of these bounds therefore requires far fewer measurements than full quantum process tomography. We find that it is particularly suitable to construct one of the probe bases from the right eigenstates of KKK, because in this case the bounds are tight in the sense that if the actual filter coincides with the ideal one, then both the lower and the upper bounds are equal to 1. We theoretically investigate the application of these bounds to a two-qubit optical quantum filter formed by the interference of two photons on a partially polarizing beam splitter. For an experimentally convenient choice of factorized input states and measurements we study the tightness of the bounds. We show that more stringent bounds can be obtained by more sophisticated processing of the data using convex optimization and we compare our methods for different choices of the input probe states.
To certify that an experimentally implemented quantum transformation is a certain unitary operation UUU on a ddd-dimensional Hilbert space, it suffices to determine fidelities of output states for d+1d+1d+1 suitably chosen pure input states [Reich et al., Phys. Rev. A 88, 042309 (2013)]. The set of these d+1d+1d+1 probe states can consist of ddd orthogonal states that form a basis and one additional state which is a balanced superposition of all ddd basis states. Here we provide an analytical lower bound on quantum process fidelity for two-qubit quantum gates which results from the knowledge of average state fidelity for the basis states and the fidelity of the superposition state. We compare this bound with the Hofmann bound that is based on knowledge of average state fidelities for two mutually unbiased bases. We also discuss the possible extension of our findings to NNN-qubit operations.
We present a systematic comparison of different methods of fidelity estimation of a linear optical quantum-controlled-ZZZ gate implemented by two-photon interference on a partially polarizing beam splitter. We utilize a linear fidelity estimator based on the Monte Carlo sampling technique as well as a nonlinear estimator based on maximum likelihood reconstruction of a full quantum-process matrix. In addition, we evaluate the lower bound on quantum-gate fidelity determined by average quantum-state fidelities for two mutually unbiased bases. In order to probe various regimes of operation of the gate we introduce a tunable time delay between the two photons. This allows us to move from high-fidelity operation to a regime where the photons become distinguishable and the success probability of the scheme significantly depends on the input state. We discuss in detail possible systematic effects that could influence the gate-fidelity estimation.
We investigate the problem of bounding the quantum process fidelity given bounds on the fidelities between target states and the action of a process on a set of pure input states. We formulate the problem as a semidefinite program and prove convexity of the minimum process fidelity as a function of the errors on the output states. We characterize the conditions required to uniquely determine a process in the case of no errors and derive a lower bound on its fidelity in the limit of small errors for any set of input states satisfying these conditions. We then consider sets of input states whose one-dimensional projectors form a symmetric positive operator-valued measure (POVM). We prove that for such sets the minimum fidelity is bounded by a linear function of the average output state error. A symmetric POVM with minimal number of elements contains d+1d+1d+1 states, where ddd is the Hilbert space dimension. Our bounds applied to such states provide an efficient method for estimating the process fidelity without the use of full process tomography.
Conditional quantum optical processes enable a wide range of technologies from generation of highly non-classical states to implementation of quantum logic operations. The process fidelity that can be achieved in a realistic implementation depends on a number of system parameters. Here we experimentally examine Fock state filtration, a canonical example of a broad class of conditional quantum operations acting on a single optical field mode. This operation is based upon interference of the mode to be manipulated with an auxiliary single-photon state at a beam splitter, resulting in the entanglement of the two output modes. A conditional projective measurement onto a single photon state at one output mode heralds the success of the process. This operation, which implements a measurement-induced nonlinearity, is capable of suppressing particular photon-number probability amplitudes of an arbitrary quantum state. We employ coherent-state process tomography to determine the precise operation realized in our experiment, which is mathematically represented by a process tensor. To identify the key sources of experimental imperfection, we develop a realistic model of the process and identify three main contributions that significantly hamper its efficacy. The experimentally reconstructed process tensor is compared with the model, yielding a fidelity better than 0.95. This enables us to identify three key challenges to overcome in realizing a filter with optimal performance—namely the single-photon nature of the auxiliary state, high mode overlap of the optical fields involved, and the need for photon-number-resolving detection when heralding. The results show that the filter does indeed exhibit a non-linear response as a function of input photon number and preserves the phase relation between Fock layers of the output state, providing promise for future applications.
Abstract We theoretically and experimentally investigate conditional enhancement of overall coherence of quantum states by probabilistic quantum operations that apply to the input state a quantum filter diagonal in the basis of incoherent states. We identify the optimal filters that for a given probability of successful filtering maximize the output coherence. We verify the performance of the studied quantum filters in a proof-of-principle experiment with linear optics, where a pair of two-level quantum systems is represented by polarization states of two photons. We comprehensively characterize the implemented two-qubit linear optical quantum filters by full quantum process tomography and we experimentally observe the optimal quantum coherence enhancement by quantum filtering.
We propose an efficiently measurable lower bound on quantum process fidelity of N-qubit controlled-Z gates. This bound is determined by average output state fidelities for N partially conjugate product bases. A distinct advantage of our approach is that only fidelities with product states need to be measured while keeping the total number of measurements much smaller than what is necessary for full quantum process tomography. As an application, we use this method to experimentally estimate quantum process fidelity F of a three-qubit linear optical quantum Toffoli gate and we find that F≥0.83. We also demonstrate the entangling capability of the gate by preparing Greenberger-Horne-Zeilinger-type three-qubit entangled states from input product states.