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arXiv:2002.07446 · Physical Review Letters

Quantum State Interferography

Surya Narayan Sahoo, Sanchari Chakraborti, Arun Kumar Pati, Urbasi Sinha

Published Sep 16, 2020• 17 citations
arXivHTMLPDF

Abstract

Quantum state tomography (QST) has been the traditional method for characterization of an unknown state. Recently, many direct measurement methods have been implemented to reconstruct the state in a resource efficient way. In this Letter, we present an interferometric method, in which any qubit state, whether mixed or pure, can be inferred from the visibility, phase shift, and average intensity of an interference pattern using a single-shot measurement-hence, we call it quantum state interferography. This provides us with a "black box" approach to quantum state estimation, wherein, between the incidence of the photon and extraction of state information, we are not changing any conditions within the setup, thus giving us a true single shot estimation of the quantum state. In contrast, standard QST requires at least two measurements for pure state qubit and at least three measurements for mixed state qubit reconstruction. We then go on to show that QSI is more resource efficient than QST for quantification of entanglement in pure bipartite qubits. We experimentally implement our method with high fidelity using the polarization degree of freedom of light. An extension of the scheme to pure states involving d-1 interferograms for d-dimensional systems is also presented. Thus, the scaling gain is even more dramatic in the qudit scenario for our method, where, in contrast, standard QST, without any assumptions, scales roughly as d^{2}.

DOI
10.1103/physrevlett.125.123601

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Quantum state tomography (QST) has been the traditional method for characterization of an unknown state. Recently, many direct measurement methods have been implemented to reconstruct the state in a resource efficient way. In this Letter, we present an interferometric method, in which any qubit state, whether mixed or pure, can be inferred from the visibility, phase shift, and average intensity of an interference pattern using a single-shot measurement-hence, we call it quantum state interferography. This provides us with a "black box" approach to quantum state estimation, wherein, between the incidence of the photon and extraction of state information, we are not changing any conditions within the setup, thus giving us a true single shot estimation of the quantum state. In contrast, standard QST requires at least two measurements for pure state qubit and at least three measurements for mixed state qubit reconstruction. We then go on to show that QSI is more resource efficient than QST for quantification of entanglement in pure bipartite qubits. We experimentally implement our method with high fidelity using the polarization degree of freedom of light. An extension of the scheme to pure states involving d-1 interferograms for d-dimensional systems is also presented. Thus, the scaling gain is even more dramatic in the qudit scenario for our method, where, in contrast, standard QST, without any assumptions, scales roughly as d^{2}.
Quantum state interferography has recently been proposed as an interference-based technique for the measurement of the general qubit and pure qudit states [Phys. Rev. Lett. 125, 123601 (2020)]. This interferometric method exploits the relationship between the off-diagonal elements of the density matrix and the expectation values of non-Hermitian spin ladder operators. The extension of this method to a composite system is however nontrivial, as it involves the measurement of both single-photon and nonlocal multiphoton interferences. In this work, we present the theoretical analysis of the two-photon quantum state interferography of an arbitrary quantum state. We show that complete state information could in principle be obtained with two measurement settings and by recording six interferograms, using a single-photon sensitive imaging device. We then experimentally demonstrate two-photon nonlocal quantum state interferography of polarization-entangled photon pairs generated with type-I spontaneous parametric down conversion. Quantum state interferography of all four Bell states is performed. An upper bound on the separability of the bipartite states is established in terms of the visibility, phase average intensity, and phase shift of the interference patterns related to the antidiagonal elements of the matrix. Each of the four Bell states can be distinguished by the violation of these bounds uniquely.
Quantum state tomography (QST) estimates an unknown quantum state from a measurement process typically performed on identical copies of the target state and plays a core role in characterizing quantum information systems. However, because quantum measurement outcomes are probabilistic, QST methods often require many shots per basis to achieve high-fidelity estimates, which hinders scalability to many-qubit applications and slows the transmission of quantum information. A more efficient procedure is therefore required for modern-day quantum operations in quantum computing, communications, and sensing. We address this need with an adaptive QST approach that measures a single-qubit state in a wide set of bases. We include bases beyond the Pauli triad {X,Y,Z} by leveraging the infinite-dimensional Hilbert space of quantum mechanics, increasing the likelihood of producing highly polarized measurement outcomes and finding an optimally informative basis. With this basis, the state can be identified with fewer measurements by focusing on directions that most reduce uncertainty of the state. In addition to theoretical innovation in [9], we experimentally verify in a simplified photonic realization using 13 bases that having many more available measurement bases than the Pauli triad benefits QST. This engineering implementation demonstrates that adaptive QST is both efficient in performance and practical to implement.
Quantum computers solve ever more complex tasks using steadily growing system sizes.Characterizing these quantum systems is vital, yet becoming increasingly challenging.The gold-standard method for this task is quantum state tomography (QST), capable of fully reconstructing a quantum state without prior knowledge.The measurement and classical computing costs, however, increase exponentially with the number of constituents (e.g., qubits)-a daunting bottleneck given the scale of existing and nearterm quantum devices.Here, we demonstrate a scalable and practical QST approach that only uses a single measurement setting, namely symmetric informationally complete (SIC) positive operator-valued measures (POVMs).We implement these nonorthogonal measurements on an ion trap quantum processor by utilizing additional energy levels within each ion-without requiring ancillary ions to assist in measurements.More precisely, we locally map the SIC POVM to orthogonal states embedded in a higher-dimensional system, which we read out using repeated in-sequence detections, thereby providing full tomographic information in every shot.Combining this SIC tomography with the recently developed randomized measurement toolbox ("classical shadows") proves to be a powerful combination.SIC tomography alleviates the need for choosing measurement settings at random ("derandomization"), while classical shadows enable the estimation of arbitrary polynomial functions of the density matrix orders of magnitudes faster than standard methods.The latter enables in-depth entanglement characterization, which we experimentally showcase on a five-qubit absolutely maximally entangled state.Moreover, the fact that the full tomography information is available in every shot enables online QST in real time (i.e., while the experiment is running).We demonstrate this on an eight-qubit entangled state (which has 2 8 • 2 8 -1 = 65 535 degrees of freedom), as well as for fast state identification.All in all, these features single out SIC-based classical shadow estimation as a highly scalable and convenient tool for quantum state characterization.
We propose a method to perform the quantum state tomography (QST) of an nnn-partite qudit state embedded in single photons based on the Hong-Ou-Mandel interference between the target photon and ancillary probe light. Our method requires only passive beam splitters in the target modes by shifting all of the active devices used in conventional QST methods to the probe preparation system. This feature is useful when the active optical devices for QST have large optical losses. Moreover, even when the probe is prepared by a laser light having a mode mismatch with the target photon, the method offers an accurate estimation of the target state. As a proof-of-principle, we perform the experimental demonstration using a polarization qubit. Regardless of the photon statistics of the probe light, the estimated results of state reconstruction are as accurate as those verified by the conventional QST.
Quantum state tomography (QST) aims at estimating a quantum state from averaged quantum measurements made on copies of that state. Most quantum algorithms rely on QST at some point and it is a well-explored topic in the literature, mostly for mixed states. In this paper we focus on the QST of a pure quantum state using parallel unentangled measurements. Pure states are a small but useful subset of all quantum states, and their tomography requires fewer measurements and is essentially a phase recovery problem. Parallel unentangled measurements are easy to implement in practice because they allow the user to measure each qubit individually, e.g., using one-qubit Pauli measurements. We propose two sets of quantum measurements that one can make on a pure state as well as the algorithms that use the measurement outcomes in order to identify the state. We also discuss how those estimates can be fine tuned by finding the state that maximizes the likelihood of the measurements with different variants of the likelihood. The performances of the proposed three types of QST methods are validated by means of detailed numerical tests, including for mixed states that are close to being pure.
Quantum state tomography (QST) is a fundamental technique for characterizing quantum states, crucial for validating quantum technologies. However, traditional QST methods often suffer from limitations in measurement efficiency and accuracy, particularly when dealing with complex quantum systems. This work presents an advanced approach to QST utilizing entangled photon sources and a meticulously designed measurement strategy. The core of this method lies in exploiting the correlations inherent in entangled photon pairs to significantly enhance the reconstruction process. We detail a novel scheme that minimizes measurement errors and maximizes the information extracted from each measurement, leading to a substantial improvement in both the accuracy and resolution of the reconstructed quantum state. The theoretical framework outlines the key equations and algorithms used for state reconstruction, focusing on a practical implementation. The results presented suggest a pathway towards more robust and reliable QST, paving the way for more accurate characterization of quantum systems and the development of advanced quantum applications.