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Physical Review A

Direct measurement of density-matrix elements with a phase-shifting technique on a quantum photonic chip

Congxiu Li, Yaxin Wang, Tianfeng Feng, Zhihao Li, Changliang Ren, Xiaoqi Zhou

Published Jun 7, 2022• 5 citations
Publisher

Abstract

A direct measurement protocol allows reconstructing specific elements of the density matrix of a quantum state without using quantum state tomography. Until now, most direct measurement protocols have relied on ancillary pointers, which add complexity and are not always easy to implement experimentally. In this paper, we experimentally validate a direct measurement protocol requiring no ancillary pointers that calculates the value of a specific element of a quantum state's density matrix using only six projection measurements. We prepare path-encoded arbitrary four-dimensional quantum states on a silicon-based quantum photonic chip and measure the density-matrix elements of the four-dimensional quantum states using this protocol, with experimental results deviating from ideal values by an average of only $0.010\ifmmode\pm\else\textpm\fi{}0.004$. This approach has the potential to be applied to 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.

DOI
10.1103/physreva.105.062414

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A direct measurement protocol allows reconstructing specific elements of the density matrix of a quantum state without using quantum state tomography. Until now, most direct measurement protocols have relied on ancillary pointers, which add complexity and are not always easy to implement experimentally. In this paper, we experimentally validate a direct measurement protocol requiring no ancillary pointers that calculates the value of a specific element of a quantum state's density matrix using only six projection measurements. We prepare path-encoded arbitrary four-dimensional quantum states on a silicon-based quantum photonic chip and measure the density-matrix elements of the four-dimensional quantum states using this protocol, with experimental results deviating from ideal values by an average of only 0.010\ifmmode±\else\textpm\fi0.0040.010\ifmmode\pm\else\textpm\fi{}0.0040.010\ifmmode±\else\textpm\fi0.004. This approach has the potential to be applied to 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.
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.
Density matrix, which characterizes a quantum state, plays an important role in quantum mechanics. Recently, a method which can directly measure the elements of a density matrix was proposed. Compared with the conventional quantum state tomography which is widely used to reconstruct the density matrix, this measurement method has the advantages of directness and simplicity. However, this direct measurement method relies on an extra pointer space. The addition of this extra pointer can increase the complexity of an experiment. In this paper, we first review previous work on direct measurement, then we propose a scheme to directly measure the density matrix based on δ-quench, which is also a direct measurement method but needs no additional pointer. This proposal reduces the complexity of the measuring system and further simplifies the measurement. We propose two schemes to realize this δ-quench measurement, then analyse their superiorities in different situations of measurement. An experiment to measure photon's density matrix is also designed.
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.
We introduce a direct estimation framework for reconstructing multiple density matrix elements of an unknown quantum state using classical shadow tomography. Traditional direct measurement protocols (DMPs), while effective for individual elements, suffer from poor scalability due to post-selection losses and the need for element-specific measurement configurations. In contrast, our method, DMP-ST, leverages random Clifford or biased mutually unbiased basis measurements to enable global estimation: a single dataset suffices to estimate arbitrary off-diagonal entries with high accuracy. We prove that estimating KKK off-diagonal matrix elements up to additive error εεε requires only O(log⁡K/ε2)\mathcal{O}(\log K/ε^2)O(logK/ε2) samples, achieving exponential improvement over conventional DMPs. The number of required measurement configurations can also be exponentially reduced for large K. When extended to full state tomography, DMP-ST attains trace distance error ≤ε\le ε≤ε with sample complexity O(d3log⁡d/ε2)\mathcal{O}(d^3 \log d/ε^2)O(d3logd/ε2), which is closed to the optimal scaling for single-copy measurements. Moreover, biased MUB measurements reduce sample complexity by a constant factor than random Clifford measurements. This work provides both theoretical guarantees and explicit protocols for efficient, entrywise quantum state reconstruction. It significantly advances the practicality of direct tomography, especially for high-dimensional systems and near-term quantum platforms.
One drawback of conventional quantum state tomography is that it does not readily provide access to single density matrix elements since it requires a global reconstruction. Here, we experimentally demonstrate a scheme that can be used to directly measure individual density matrix elements of general quantum states. The scheme relies on measuring a sequence of three observables, each complementary to the last. The first two measurements are made weak to minimize the disturbance they cause to the state, while the final measurement is strong. We perform this joint measurement on polarized photons in pure and mixed states to directly measure their density matrix. The weak measurements are achieved using two walk-off crystals, each inducing a polarization-dependent spatial shift that couples the spatial and polarization degrees of freedom of the photons. This direct measurement method provides an operational meaning to the density matrix and promises to be especially useful for large dimensional states.
Quantum information protocols often rely on tomographic techniques to determine the state of the system. A popular method of encoding information is on the different paths a photon may take, for example, parallel waveguides in integrated optics. However, reconstruction of states encoded onto a large number of paths is often prohibitively resource intensive and requires complicated experimental setups. Addressing this, we present a simple method for determining the state of a photon in a superposition of d paths using a rotating one-dimensional optical Fourier Transform. We establish the theory and experimentally demonstrate the technique by measuring a wide variety of six-dimensional density matrices. The average fidelity of these with the expected state is as high as 0.9852 +/- 0.0008. This performance is comparable or exceeds established tomographic methods for other types of systems.