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Experiment configuration for compressive sensing of high-dimensional quantum states entangled in the orbital angular momentum degree of freedom.
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Accurately establishing the state of large-scale quantum systems is an
important tool in quantum information science; however, the large number of
unknown parameters hinders the rapid characterisation of such states, and
reconstruction procedures can become prohibitively time-consuming. Compressive
sensing, a procedure for solving inverse problems...
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Citations
... An efficient method for characterizing quantum states is of importance for quantum information applications as well as quantum fundamentals. Quantum state tomography (QST) [1][2][3][4][5][6][7][8][9][10][11][12] is a conventional method for reconstructing the density matrix of a state, which typically involves a large number of projection measurements on several basis states, followed by a complex reconstruction algorithm. [13] Alternatively, Lundeen et al. [14] proposed an efficient direct measurement scheme to determine a quantum wavefunction through weak measurements. ...
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.
... To circumvent the above-mentioned difficulties various methods have been proposed in recent years for states with high purity [27][28][29][30]. Other approaches include determination of Schmidt number [31,32], comparison of measurement results corresponding to two or more mutually unbiased bases [33][34][35][36][37], using the violation of the entropic Einstein-Podolsky-Rosen-steering inequality [38] or other suitably defined statistical correlators [39]. ...
Efficient certification and quantification of high dimensional entanglement of composite systems are challenging both theoretically as well as experimentally. Here, we demonstrate how to measure the linear entropy, negativity and the Schmidt number of bipartite systems from the visibility of Mach-Zehnder interferometer using single copies of the quantum state. Our result shows that for any two qubit pure bipartite state, the interference visibility is a direct measure of entanglement. We also propose how to measure the mutual predictability experimentally from the intensity patterns of the interferometric set-up without having to resort to local measurements of mutually unbiased bases. Furthermore, we show that the entanglement witness operator can be measured in a interference setup and the phase shift is sensitive to the separable or entangled nature of the state. Our proposal bring out the power of Interferometric set-up in entanglement detection of pure and several mixed states which paves the way towards design of entanglement meter.
... To circumvent the above-mentioned difficulties various methods have been proposed in recent years for states with high purity [27][28][29][30]. Other approaches include determination of Schmidt number [31,32], comparison of measurement results corresponding to two or more mutually unbiased bases [33,34], using the violation of the entropic EPR-Steering inequality [35] or other suitably defined statistical correlators [36]. ...
... Given a pure state ρ AB = |Ψ Ψ| and |Ψ AB = j λ j |j A |j B , the quantity tr(U A ⊗ U B ρ AB ) contains the following three terms Ψ| X A ⊗ X B |Ψ , Ψ| X A ⊗ I B |Ψ = Ψ| I A ⊗ X B |Ψ . Eqs. (28) and (27) shows that the first term is proportional to negativity, N (|Ψ ), second and third terms are zero. We then have the following ...
... (25) and (26), then the joint observable X A ⊗ X B , satisfies the relationships given by Eqs. (27) and (28). In particular, ...
Efficient certification and quantification of high dimensional entanglement of composite systems are challenging both theoretically as well as experimentally. Here, we demonstrate that several entanglement detection methods can be implemented efficiently in a Mach-Zehnder Interferometric set-up. In particular, we demonstrate how to measure the linear entropy and the negativity of bipartite systems from the visibility of Mach-Zehnder interferometer using single copy of the input state. Our result shows that for any two qubit pure bipartite state, the interference visibility is a direct measure of entanglement. We also propose how to measure the mutual predictability experimentally from the intensity patterns of the interferometric set-up without having to resort to local measurements of mutually unbiased bases. Furthermore, we show that the entanglement witness operator can be measured in a interference setup and the phase shift is sensitive to the separable or entangled nature of the state. Our proposals bring out the power of Interferometric set-up in entanglement detection of pure and several mixed states which paves the way towards design of entanglement meter.
... Formulating experimentally efficient methods for the characterization of higher-dimensional entangled states based on a limited number of measurements has become an active area of research [42][43][44][45][46]. There are recent studies to certify entanglement by determining a lower bound to EOF from a very few local measurements [30,47]. ...
Higher dimensional quantum systems (qudits) present a potentially more efficient means, compared to qubits, for implementing various information theoretic tasks. One of the ubiquitous resources in such explorations is entanglement. Entanglement monotones (EMs) are of key importance, particularly for assessing the efficacy of a given entangled state as a resource for information theoretic tasks. Till date, investigations towards determination of EMs have focused on providing their tighter lower bounds. There is yet no general scheme available for direct determination of the EMs. Consequently, an empirical determination of any EM has not yet been achieved for entangled qudit states. The present paper fills this gap, both theoretically as well as experimentally. First, we derive analytical relations between statistical correlation measures i.e. mutual predictability (MP), mutual information (MI) and Pearson correlation coefficient (PCC) and standard EMs i.e. negativity (N) and entanglement of formation (EOF) in arbitrary dimensions. As a proof of concept, we then experimentally measure MP, MI and PCC of two-qutrit pure states and determine their N and EOF using these derived relations. This is a useful addition to the experimenter’s toolkit wherein by using a limited number of measurements (in this case 1 set of measurements), one can directly measure the EMs in a bipartite arbitrary dimensional system. We obtain the value of N for our bipartite qutrit to be 0.907 ± 0.013 and the EOF to be 1.323 ± 0.022. Since the present scheme enables determination of more than one EM by the same limited number of measurements, we argue that it can serve as a unique experimental platform for quantitatively comparing and contrasting the operational implications of EMs as well as showing their non-monotonicity for a given bipartite pure qudit state.
... Moreover, we work in the regime of overexposure, for which P N (typically, N /P ∼ 10 4 -10 5 , depending on the system dimensionality). This setting is common in compressive sensing experiments, where shot noise in the outcome probability estimation should be diminished [6,20,21]. Preliminary tomography simulations show that feature prediction accuracy is limited by the finite P even though N = ∞ (the observed frequency f i is substituted for the exact outcome probability). In this sense, P is more important than N . ...
Full quantum tomography of high-dimensional quantum systems is experimentally infeasible due to the exponential scaling of the number of required measurements on the number of qubits in the system. However, several ideas have been proposed recently for predicting the limited number of features for these states, or estimating the expectation values of operators, without the need for full state reconstruction. These ideas go under the general name of shadow tomography. Here, we provide an experimental demonstration of property estimation based on classical shadows proposed in Huang et al. [Nat. Phys. 16, 1050 (2020)] and study its performance in a quantum-optical experiment with high-dimensional spatial states of photons. We show by means of experimental data how this procedure outperforms conventional state reconstruction in fidelity estimation from a limited number of measurements.
... Moreover, we worked in the regime of overexposure, for which P N (typically, N/P ∼ 10 4 -10 5 depending on the system dimensionality). This setting is common in compressive sensing experiments, where shot noise in the outcome probability estimation should be diminished [17][18][19]. Preliminary tomography simulations showed that feature prediction accuracy was limited by finite P even though N = ∞ (observed frequency f i was substituted with exact outcome probability). ...
Full quantum tomography of high-dimensional quantum systems is experimentally infeasible due to the exponential scaling of the number of required measurements on the number of qubits in the system. However, several ideas were proposed recently for predicting the limited number of features for these states, or estimating the expectation values of operators, without the need for full state reconstruction. These ideas go under the general name of shadow tomography. Here we provide an experimental demonstration of property estimation based on classical shadows proposed in [H.-Y. Huang, R. Kueng, J. Preskill. Nat. Phys. https://doi.org/10.1038/s41567-020-0932-7 (2020)] and study its performance in the quantum optical experiment with high-dimensional spatial states of photons. We show on experimental data how this procedure outperforms conventional state reconstruction in fidelity estimation from a limited number of measurements.
... They showed that their method can reconstruct a rank r unknown density matrix with only O(rN log 2 N ) measurements, in contrast to the O(N 2 ) measurements for the standard method. In another study [28], a high-dimensional entangled state was reconstructed from a significantly smaller number of measurements, using a related approach based on compressive sensing. While this method for the reconstruction of signals from an underdetermined system of equations is very popular in signal and image processing applications, it has now attracted interest and become topical in quantum information science related applications as well [27,[29][30][31][32][33]. ...
... This entire process leads to a photon detection which represents a measurement. The compressive sensing model used in this work is based on the singular value thresholding algorithm [37] that has been modified in a way similar to the one used in Ref. [28]. ...
... Here we briefly review the compressing sensing procedure that we used for our work. For this purpose, we follow the procedure of Tonolini et al. [28]. ...
Compressive sensing is used to perform high-dimensional quantum channel estimation with classical light. As an example, we perform a numerical simulation for the case of a three-dimensional classically non-separable state that is propagated through atmospheric turbulence. Using singular value thresholding algorithm based compressive sensing, we determine the channel matrix, which we subsequently use to correct for the atmospheric turbulence induced distortions. As a measure of the success of the procedure, we calculate the fidelity and the trace distance of the corrected density matrix with respect to the input state, and compare the results with those of the density matrix for the uncorrected state. Furthermore, we quantify the amount of classical non-separability in the density matrix of the corrected state by calculating its negativity. The results show that compressive sensing could contribute in the development and implementation of free-space quantum and optical communication systems.
... Next, inferring information on the state is usually done via the density matrix, which itself is a computationally intensive task to calculate from the QST data. There are approaches that overcome some of these limitations, for example, using compressive sensing, 198 which has been demonstrated up to d ¼ 17, or optimal bases, 199 which do hold promise for a speed up in state determination. Finally, even with a high dimensional density matrix on hand, it is not a trivial task to infer the degree of entanglement, a largely open challenge with some recent prospects for a faster measurement up to d ¼ 8. 200 It would be helpful to have entanglement witnesses for high dimensions that did not require a full QST nor a density matrix, an area of active research both theoretically and experimentally. ...
Quantum mechanics is now a mature topic dating back more than a century. During its scientific development, it fostered many technological advances that now are integrated into our everyday lives. More recently, over the past few decades, the authors have seen the emergence of a second quantum revolution, ushering in control of quantum states. Here, the spatial modes of light, “patterns of light,” hold tremendous potential: light is weakly interacting and so an attractive avenue for exploring entanglement preservation in open systems, while spatial modes of light offer a route to high dimensional Hilbert spaces for larger encoding alphabets, promising higher information capacity per photon, better security, and enhanced robustness to noise. Yet, progress in harnessing high dimensional spatial mode entanglement remains in its infancy. Here, the authors review the recent progress in this regard, outlining the core concepts in a tutorial manner before delving into the advances made in creation, manipulation, and detection of such quantum states. The authors cover advances in using orbital angular momentum as well as vectorial states that are hybrid entangled, combining spatial modes with polarization to form an infinite set of two-dimensional spaces: multidimensional entanglement. The authors highlight the exciting work in pushing the boundaries in both the dimension and the photon number, before finally summarizing the open challenges, and the questions that remain unanswered.
... They showed that their method can reconstruct a rank r unknown density matrix with only O(rN log 2 N ) measurements, in contrast to the O(N 2 ) measurements for the standard method. In another study [28], a high-dimensional entangled state was reconstructed from a significantly smaller number of measurements, using a related approach based on compressive sensing. While this method for the reconstruction of signals from an underdetermined system of equations is very popular in signal and image processing applications, it has now attracted interest and become topical in quantum information science related applications as well [27,[29][30][31][32][33]. ...
... This entire process leads to a photon detection which represents a measurement. The compressive sensing model used in this work is based on the singular value thresholding algorithm [37] that has been modified in a way similar to the one used in Ref. [28]. ...
... Here we briefly review the compressing sensing procedure that we used for our work. For this purpose, we follow the procedure of Tonolini et al. [28]. ...
Compressive sensing is used to perform high-dimensional quantum channel estimation. As an example, we perform a numerical simulation for the case of a qutrit photonic state that is propagated through atmospheric turbulence. Using singular value thresholding algorithm based compressive sensing, we determine the channel matrix, which we subsequently use to correct for the atmospheric turbulence induced distortions. As a measure of the success of the procedure we calculate the fidelity and the trace distance of the corrected density matrix against the input qutrit density matrix, and compare the results with those of the uncorrected quantum state density matrix. Furthermore, we quantify the amount of the entanglement in the corrected density matrix by calculating the negativity. The results show that compressive sensing could contribute in the development and implementation of free-space quantum and optical communication systems.
... One popular technique is compressed sensing 30 , which has massively disrupted conventional thinking about sampling. Applied to quantum systems, compressed sensing reduced measurement resources significantly for tasks, including tomography [31][32][33][34][35][36][37] and witnessing entanglement 38,39 . ...
Entanglement is the powerful and enigmatic resource central to quantum information processing, which promises capabilities in computing, simulation, secure communication, and metrology beyond what is possible for classical devices. Achieving a quantum advantage requires scaling quantum systems to sizes that can support a large amount of entanglement. Because real-world systems are varied and imperfect, the quantum resources they provide must be characterized before use. However, exactly quantifying the entanglement of an unknown system requires completely determining its quantum state, a task which requires an intractable number of measurements even for modestly sized systems. Here we demonstrate a new method for rigorously quantifying high-dimensional entanglement from extremely limited data. We improve an entropic, quantitative entanglement witness to operate directly on compressed experimental data acquired via an adaptive, multilevel sampling procedure. Only 6,456 measurements are needed to certify an entanglement-of-formation of $7.11 \pm .04$ ebits shared between spatially-entangled photon pairs. With a Hilbert space exceeding 68 billion dimensions, this is 10-million-fold fewer measurements than traditional approaches. The procedure does not computationally recover an underlying state and allows straightforward error analysis. Our technique offers a universal method for quantifying entanglement in any large quantum system shared by two parties.
