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arXiv:quant-ph/0409083 · Physical Review Letters

Complementary Classical Fidelities as an Efficient Criterion for the Evaluation of Experimentally Realized Quantum Operations

Holger Friedrich Hofmann

Published Apr 29, 2005• 98 citations
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Abstract

It is shown that a good estimate of the fidelity of an experimentally realized quantum process can be obtained by measuring the outputs for only two complementary sets of input states. The number of measurements required to test a quantum network operation is therefore only twice as high as the number of measurements required to test a corresponding classical system.

DOI
10.1103/physrevlett.94.160504

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It is shown that a good estimate of the fidelity of an experimentally realized quantum process can be obtained by measuring the outputs for only two complementary sets of input states. The number of measurements required to test a quantum network operation is therefore only twice as high as the number of measurements required to test a corresponding classical system.
Quantum information processes utilize the potential of quantum coherence to achieve improvements in communication and computation protocols. In order to develop appropriate technologies, it is therefore necessary to test the successful implementation of quantum coherent operations in experimental devices. In this presentation, it is shown how the quantum coherent performance of a device can be evaluated from complementary test measurements. Despite the limitation of test measurements to only two orthogonal basis sets of states, this method provides a surprisingly detailed and intuitively accessible picture of errors in quantum operations, making it possible to assess the quantum parallelism of non-classical operations in terms of the directly observable "classical" device properties.
The information obtained from the operation of a quantum gate on only two complementary sets of input states is sufficient to estimate the quantum process fidelity of the gate. In the case of entangling gates, these conditions can be used to predict the multi qubit entanglement capability from the fidelities of two non-entangling local operations. It is then possible to predict highly non-classical features of the gate such as violations of local realism from the fidelities of two completely classical input-output relations, without generating any actual entanglement.
The information obtained from the operation of a quantum gate on only two complementary sets of input states is sufficient to estimate the quantum process fidelity of the gate. In the case of entangling gates, these conditions can be used to predict the multi qubit entanglement capability from the fidelities of two non-entangling local operations. It is then possible to predict highly non-classical features of the gate such as violations of local realism from the fidelities of two completely classical input-output relations, without generating any actual entanglement.
It is shown that a quantum controlled-NOT gate simultaneously performs the logical functions of three distinct conditional local operations. Each of these local operations can be verified by measuring a corresponding truth table of four local inputs and four local outputs. The quantum parallelism of the gate can then be observed directly in a set of three simple experimental tests, each of which has a clear intuitive interpretation in terms of classical logical operations. Specifically, quantum parallelism is achieved if the average fidelity of the three classical operations exceeds 2∕32∕32∕3. It is thus possible to evaluate the essential quantum parallelism of an experimental controlled-NOT gate by testing only three characteristic classical operations performed by the gate.
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 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.