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Chinese Physics B

Direct measurement of nonlocal quantum states without approximation

Gang 冈 Yang 杨, Ran 然 Yang 杨, Yan-Xiao Gong, Shi-Ning 世宁 Zhu 祝

Published Sep 1, 2023• 3 citations
Publisher

Abstract

Efficient acquiring information from a quantum state is important for research in fundamental quantum physics and quantum information applications. Instead of using standard quantum state tomography method with reconstruction algorithm, weak values were proposed to directly measure density matrix elements of quantum state. Recently, similar to the concept of weak value, modular values were introduced to extend the direct measurement scheme to nonlocal quantum wavefunction. However, this method still involves approximations, which leads to inherent low precision. Here, we propose a new scheme which enables direct measurement for ideal value of the nonlocal density matrix element without taking approximations. Our scheme allows more accurate characterization of nonlocal quantum states, and therefore has greater advantages in practical measurement scenarios.

DOI
10.1088/1674-1056/acf5d7

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Efficient acquiring information from a quantum state is important for research in fundamental quantum physics and quantum information applications. Instead of using standard quantum state tomography method with reconstruction algorithm, weak values were proposed to directly measure density matrix elements of quantum state. Recently, similar to the concept of weak value, modular values were introduced to extend the direct measurement scheme to nonlocal quantum wavefunction. However, this method still involves approximations, which leads to inherent low precision. Here, we propose a new scheme which enables direct measurement for ideal value of the nonlocal density matrix element without taking approximations. Our scheme allows more accurate characterization of nonlocal quantum states, and therefore has greater advantages in practical measurement scenarios.
New techniques based on weak measurements have recently been introduced to the field of quantum state reconstruction. Some of them allow the direct measurement of each matrix element of an unknown density operator and need only O(d) different operations, compared to d^{2} linearly independent projectors in the case of standard quantum state tomography, for the reconstruction of an arbitrary mixed state. However, due to the weakness of these couplings, these protocols are approximated and prone to large statistical errors. We propose a method which is similar to the weak measurement protocols but works regardless of the coupling strength: our protocol is not approximated and thus improves the accuracy and precision of the results with respect to weak measurement schemes. We experimentally apply it to the polarization state of single photons and compare the results to those of preexisting methods for different values of the coupling strength. Our results show that our method outperforms previous proposals in terms of accuracy and statistical errors.
Abstract In the field of quantum information, the acquisition of information for unknown quantum states is very important. When we only need to obtain specific elements of a state density matrix, the traditional quantum state tomography will become very complicated, because it requires a global quantum state reconstruction. Direct measurement of the quantum state allows us to obtain arbitrary specific matrix elements of the quantum state without state reconstruction, so direct measurement schemes have obtained extensive attention. Recently, some direct measurement schemes based on weak values have been proposed, but extra auxiliary states in these schemes are necessary and it will increase the complexity of the practical experiment. Meanwhile, the post-selection process in the scheme will reduce the utilization of resources. In order to avoid these disadvantages, a direct measurement scheme without auxiliary states is proposed in this paper. In this scheme, we achieve the direct measurement of quantum states by using quantum circuits, then we extend it to the measurement of general multi-particle states and complete the error analysis. Finally, when we take into account the dephasing of the quantum states, we modify the circuits and the modified circuits still work for the dephasing case.
Efficient, reliable determination of a system's state is at the heart of quantum science. Directly measuring any desired density-matrix elements is a unique challenge for both theorists and experimentalists, as standard methods have always suffered from the rapid increase in complexity of measurements and reconstruction algorithms. This study presents an efficient scheme for directly measuring arbitrary density matrices with only one strong measurement, or a weak measurement, of each qudit. This method is important for characterizing large-scale quantum systems, and the technology that will be derived from it will be easy to expand and integrate, for example, on a quantum chip.
The time-symmetric formalism endows the weak measurement and its outcome, the weak value, with many unique features. In particular, it allows a direct tomography of quantum states without resorting to complicated reconstruction algorithms and provides an operational meaning to wave functions and density matrices. Here, we propose and experimentally demonstrate the direct tomography of a measurement apparatus by taking the backward direction of weak measurement formalism. Our protocol works rigorously with the arbitrary measurement strength, which offers improved accuracy and precision. The precision can be further improved by taking into account the completeness condition of the measurement operators, which also ensures the feasibility of our protocol for the characterization of the arbitrary quantum measurement. Our work provides new insight on the symmetry between quantum states and measurements, as well as an efficient method to characterize a measurement apparatus.
Recent work has revealed that the wave function of a pure state can be measured directly and that complementary knowledge of a quantum system can be obtained simultaneously by weak measurements. However, the original scheme applies only to pure states, and it is not efficient because most of the data are discarded by post-selection. Here, we propose tomography schemes for pure states and for mixed states via weak measurements, and our schemes are more efficient because we do not discard any data. Furthermore, we demonstrate that any matrix element of a general state can be directly read from an appropriate weak measurement. The density matrix (with all of its elements) represents all that is directly accessible from a general measurement.
A direct measurement protocol allows reconstructing specific elements of the density matrix of a quantum state without using quantum state tomography. However, the direct measurement protocols to date are primarily based on weak or strong measurements with an ancillary pointer, which interacts with the investigated system to extract information about the specified elements. Here, we present a direct measurement scheme based on phase-shifting operations which do not need ancillary pointers. In this method, estimates of at most six expectation values of projective observables suffice to determine any specific element of an unknown quantum density matrix. A concrete quantum circuit to implement this direct measurement protocol for multiqubit states is provided, which is composed of just single-qubit gates and two multiqubit controlled-phase gates. This scheme is also extended for the direct measurement of the density matrix of continuous-variable quantum states. Our method can be used in quantum information applications where only partial information about the quantum state needs to be extracted, for example, problems such as entanglement witnessing, fidelity estimation of quantum systems, and quantum coherence estimation.