experimental examination of entanglement estimates

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Experimental Examination of Entanglement Estimates

Affiliations.

• 1 Center for Coherence and Quantum Optics, and Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA.
• 2 Peng Cheng Laboratory, Shenzhen 518052, China.
• 3 CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China.
• 4 CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China.
• 5 Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China.
• PMID: 37115884
• DOI: 10.1103/PhysRevLett.130.150801

Recently, a proper genuine multipartite entanglement measure has been found for three-qubit pure states [see Xie and Eberly, Phys. Rev. Lett. 127, 040403 (2021)PRLTAO0031-900710.1103/PhysRevLett.127.040403], but capturing useful entanglement measures for mixed states has remained an open challenge. So far, it requires not only a full tomography in experiments, but also huge calculational labor. A leading proposal was made by Gühne, Reimpell, and Werner [Phys. Rev. Lett. 98, 110502 (2007)PRLTAO0031-900710.1103/PhysRevLett.98.110502], who used expectation values of entanglement witnesses to describe a lower bound estimation of entanglement. We provide here an extension that also gives genuine upper bounds of entanglement. This advance requires only the expectation value of any Hermitian operator. Moreover, we identify a class of operators A_{1} that not only give good estimates, but also require a remarkably small number of experimental measurements. In this Letter, we define our approach and illustrate it by estimating entanglement measures for a number of pure and mixed states prepared in our recent experiments.

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• DOI: 10.1103/PhysRevLett.130.150801
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Experimental Examination of Entanglement Estimates.

• S. Xie , Yuan-Yuan Zhao , +4 authors J. Eberly
• Published in Physical Review Letters 15 July 2022

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15 Jul 2022  ·  Songbo Xie , Yuan-Yuan Zhao , Chao Zhang , Yun-Feng Huang , Chuan-Feng Li , Guang-Can Guo , Joseph H. Eberly · Edit social preview

Recently a proper genuine multipartite entanglement (GME) measure has been found for three-qubit pure states [see Xie and Eberly, Phys. Rev. Lett. 127, 040403 (2021)], but capturing useful entanglement measures for mixed states has remained an open challenge. So far, it requires not only a full tomography in experiments, but also huge calculational labor. A leading proposal was made by G\"uhne, Reimpell, and Werner [Phys. Rev. Lett. 98, 110502 (2007)], who used expectation values of entanglement witnesses to describe a lower bound estimation of entanglement. We provide here an extension that also gives genuine upper bounds of entanglement. This advance requires only the expectation value of {\em any} Hermitian operator. Moreover, we identify a class of operators $\A_1$ which not only give good estimates, but also require a remarkably small number of experimental measurements. In this note we define our approach and illustrate it by estimating entanglement measures for a number of pure and mixed states prepared in our recent experiments.

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• Published: 02 April 2015

Demonstration of entanglement-enhanced phase estimation in solid

• Gang-Qin Liu 1 ,
• Yu-Ran Zhang 1 ,
• Yan-Chun Chang 1 ,
• Jie-Dong Yue 1 ,
• Heng Fan 1 , 2 &
• Xin-Yu Pan 1 , 2  

Nature Communications volume  6 , Article number:  6726 ( 2015 ) Cite this article

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• Optical physics
• Quantum metrology

Precise parameter estimation plays a central role in science and technology. The statistical error in estimation can be decreased by repeating measurement, leading to that the resultant uncertainty of the estimated parameter is proportional to the square root of the number of repetitions in accordance with the central limit theorem. Quantum parameter estimation, an emerging field of quantum technology, aims to use quantum resources to yield higher statistical precision than classical approaches. Here we report the first room-temperature implementation of entanglement-enhanced phase estimation in a solid-state system: the nitrogen-vacancy centre in pure diamond. We demonstrate a super-resolving phase measurement with two entangled qubits of different physical realizations: an nitrogen-vacancy centre electron spin and a proximal 13 C nuclear spin. The experimental data shows clearly the uncertainty reduction when entanglement resource is used, confirming the theoretical expectation. Our results represent an elemental demonstration of enhancement of quantum metrology against classical procedure.

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Introduction, system description.

The phase estimation scheme is implemented by optically detected magnetic resonance (ODMR) 14 , 16 technique on a home-built confocal microscope system. The description of the system can be found in ref. 27 . The spin-1 electron spin of NV centre has triplet ground states with a zero-field splitting of Δ≈2.87 GHz between the states |0› and |±1›. As an external magnetic field of about 507 Gauss is applied along [111] direction of the diamond crystal, the degeneration of |±1› states can be well relieved, and the first qubit is encoded on the |0› and |−1› subspace. The electron spin state can be initialized to |0› state by a short 532 nm laser pulse (3  μ s) and manipulated by resonant microwave (MW) pulses of tunable duration and phase. The electron spin state is readout by collecting the spin-dependent fluorescence. To enhance the fluorescence collection efficiency, a solid immersion lens (SIL) 30 is etched above the selected NV center, typical count rate in this experiment is 250 k.p.s. with SIL, see Methods for details.

The second qubit is encoded on the |↑› and |↓› states of a nearby 13 C nuclear spin. See Fig. 1b for the energy levels of the two-qubit system. The coupling strength between the target nuclear spin and centre electron spin is 12.8 MHz, which indicates the 13 C atom sites on the third shell from the NV centre 31 . The polarization and readout procedure of the nuclear spin is more complicated than that of a electron spin. The 507 Gauss magnetic field causes excited-state level anti-crossing (ESLAC) of centre electron spin, in which the optical spin polarization of centre electron will transfer to nearby nuclear spins 32 , 33 . So the host 14 N nuclear spin, the nearby 13 C nuclear spin as well as the center electron spin are polarized by the same laser pulse under this magnetic field. To readout the nuclear spin state, a mapping gate, which transfers nuclear spin state to electron spin, and a following optical readout of electron spin state are employed 17 , 33 , see Methods for details.

( a ) Phase estimation schemes of the independent states and the electron–nuclear-entangled state. By harnessing entanglement, quantum metrology yields higher statistical precision than classical approaches. ( b ) Energy levels and physical encoding of the two-bit system. The electron spin and a nearby 13 C nuclear spin of an NV center are employed to demonstrate the metrology scheme. At excited-state level anti-crossing (ESLAC), both spins can be polarized, manipulated and readout with high fidelity. Two-bit conditional quantum gates are implemented by applying selective microwave (MW) or (RF) pulses.

The nearby nuclear spin couples to the centre electron spin through strong dipolar interaction, which provides excellent conditions to implement two-qubit controlled gate. On the one hand, the resonant frequency of |0↑›↔|1↑› transition and |0↓›↔|1↓› transition are separated by 12.8 MHz from each other, so we can selectively manipulation one branch of nuclear spin with high fidelity while keep the other branch untouched (using weak MW pulses, see black arrow in Fig. 1b ). On the other hand, the nuclear spin state evolution is strong affected by the state of electron spin: when electron spin is on the |0› state (or |−1› state), the dynamics of the nuclear spin is dominated by the external magnetic filed (or the dipolar interaction, respectively), its Zeeman splitting between the |↑› and |↓› states is about 500 kHz, which is far away from the dipolar interaction strength of 12.8 MHz. Therefore, we can selectively manipulate nuclear spin state in one branch of electron spin, as well (using RF pulses, see red arrow in Fig. 1b and Supplementary Note 1 ).

Phase preparation and measurement

To improve phase estimation accuracy, one can increase the repeat number, which means longer measurement time is needed. An equivalent way is to employ more qubits. As mentioned before, the state of the multiqubit system, independent or entanglement, determines the accuracy limit of phase estimation. For the investigated two-qubit system, the electron and nuclear spin can be prepared and measured independently. Fig. 3a plots the state tomography result of a nuclear spin superposition state. Using such independent state (either nuclear spin or electron spin) will get a phase relation as depicted in Fig. 3c , the amplitude of Rabi signal has cosine dependence on the phase of input state.

Entanglement-enhanced phase estimation

To demonstrate the merits of entangled state over independent state in phase estimation application, we compare their performances on different repeat number and different input phase, the measured results are summarized in Fig. 4 . The experimental procedure is: first, single-spin states (electron and nuclear) of the same input phase (30°) are prepared and measured independently. Then the output phases extracted from the same repeat number are counted together, no weight is added for either electron or nuclear spin states. For a fair comparison with entangled state, half of the statistic samples ( v ) are extracted from electron spin states, and the other half ( v ) are extracted from nuclear spin states. In the case of entangled state phase estimation, the entangled states are prepared and measured using the same repeat number ( v ). Note the same MW and RF channels are used to prepare the independent and entangled states.

Figure 4c,d show the phase estimation results of different input phases. The phase error of entangled state is smaller than independent state in all input phases, which indicates the enhancement of phase estimation accuracy by entanglement is phase independent.

As summarized in Fig. 4 by different figures of merit quantifying the uncertainty of phase estimation, the entanglement-enhanced precision is clearly shown by experimental data. This experiment demonstrates the advantage of the quantum metrology scheme. Practically by using quantum metrology, the measured physical quantity should have the same interaction on the probe system no matter it is prepared as a single-qubit or entangled state. In our special designed experiment, the measured phases are artificially encoded to the probe state such that the enhancement of precision can be shown by entangled probe state. However, in principle, the confirmation of theoretical expectation by experimental data provides a solid evidence that quantum phase estimation is applicable in this solid-state system.

In this experiment, we use repeating measurement to overcome the low photon collection efficiency of NV center. The phase estimation accuracy can be further improved by employing single-shot measurement technique, which is now available in NV system 19 , 20 , 34 . Although the photon collection efficiency is not perfect (<20%, not every measurement is stored and counted), the following two facts guarantee the reliability of the demonstration: (1) we use the same scheme to measure single and entangled states, that is, the phase information is finally converted to fluorescence signal of NV center and detected. (2) The detection efficiency of the system is stable (though not perfect as single-shot readout) for all the measurement, so we can directly compare the measured phase noise of single and entangled states.

As the phase estimation accuracy is determined by the total number of entangled qubits, a straightforward way to improve the phase accuracy is increasing the involved spin number. The large amount of weakly coupled 13 C nuclear spins around NV center are one of the best candidates. With the assistance of dynamical decoupling on center electron spin, up to six 13 C nuclear spins can be coherent manipulated 35 , 36 , 37 . Multiqubit application such as error correction has been demonstrated in this system 24 , 23 .

In conclusion, we report the first room-temperature implementation of entanglement-enhanced phase estimation in a solid-state system: the NV centre in pure diamond. We demonstrate a super-resolving phase measurement with two entangled qubits of different physical realizations: a NV centre electron spin and a proximal 13 C nuclear spin. Thus, our results represent a more generalized and elemental demonstration of enhancement of quantum metrology against classical procedure, which fully exploits the quantum nature of the system and probes.

Cramér-Rao bound and quantum Fisher information

Sample preparation.

High purity single crystal diamond (Element Six, N concentration <5 p.p.b.) is used for this experiment. There is almost no natural NV center in this diamond. NV centres are produced by electron implantation (7.5 Mev) and a following 2 h vacuum annealing (at 800 °C). Due to the random distribution of 13 C nuclear spins, the spin bath of individual NV center can be very different 31 . We choose NV centers with nearby 13 C nuclear spins, which can be identified by the extra splitting in ODMR signal, to implement the two-bit metrology scheme. Fig. 5a presents the physical structure of an NV center and a nearby 13 C nuclear spin. Fig. 5c is ODMR signal of this two-bit system. The coupling strength between electron spin and the selected 13 C nuclear is 12.8 MHz. Fig. 5b shows two dimensional fluorescence image of the FIB-etched SIL. The cross-cursor marked bright spot (blue) is the one used for this experiment.

( a ) NV centre with a nearby 13 C nuclear spin. ( b ) 2D fluorescence scan of the FIB-etched solid immersion lens (SIL, 12  μ m diameter). The bright spot is the investigated NV centre. ( c ) ODMR of NV electron spin under magnetic field of 690 Gauss. The splitting is caused by the nearby 13 C nuclear spin. ( d ) Rabi oscillation and free-induction decay (FID) of electron spin ( B =507 Gauss). Due to the thermal fluctuation of the spin bath, centre electron spin picks up random phase during free precession and loses coherence. With the help of resonant MW pulses, electron spin can be flipped in short time, and is less affected by the bath noise.

Coherence of electron spin and nearby nuclear spin

Coherent manipulation of electron spin and nuclear spin at eslac.

As mentioned in the main text, we work at the ESLAC point to achieve fast and high fidelity initialization of the electron–nuclear two-qubit system. Under an external magnetic field of 507 Gauss (along the quantization axis of the selected NV) and laser excitation (532 nm), the electron and nuclear spins are polarized simultaneously. Fig. 6a shows ODMR spectrum of centre electron spin at such magnetic field. From the contrast difference of two peaks, which correspond to |↑› and |↓› states of 13 C nuclear spin, we estimate the polarization rate of this nuclear spin is about 85% (in |↑› state). Furthermore, by measuring the pulse-ODMR spectrum of electron spin, we conclude that the host 14 N nuclear spin is completely polarized under this magnetic field.

( a ) ODMR spectrum of electron spin at B =507 Gauss. ( b ) Pulse sequence to manipulate electron and nuclear spins at ESLAC. ( c ) ODMR spectrum and ( d ) Rabi oscillation of the nearby 13 C nuclear spin. At ESLAC, electron spin and nearby nuclear spins (including host 14 N nuclear spin and nearby 13 C nuclear spins) can be polarized by a short laser pulse. Then spin states are manipulated by resonant MW or RF pulses. Electron spin states are readout by counting the fluorescence intensity of NV centre; nuclear spin states are mapped to electron spin and readout in the same way. The resonant frequency of this nuclear spin is 495 kHz, which is slightly modified by centre electron spin.

Figure 6b shows the pulse sequence of electron and nuclear spin manipulation. After polarization with high fidelity, both spin states can be manipulated with resonant MW (or RF) pulses. For electron spin, the final state is readout by counting the fluorescence intensity of NV center, since |0› state is brighter than |1› state. For nuclear spin state, we use a mapping gate, which is composed by a weak pulse of |0↑›↔|1↑› transition, to transfer the its state to electron spin and then readout optically. For example, an unknown nuclear spin state of |0› ⊗ ( α |↑›+ β |↓›) is transferred to α |1↑›+ β |0↓› after applying the mapping gate. We carefully tuned the MW power and pulse duration to maximum the flip efficiency while avoiding the unwanted non-resonant excitation. By comparing the Rabi amplitude of nuclear spin ( Fig. 6d , with mapping gate) and electron spin ( Fig. 5d , without mapping gate), we conclude that the mapping gate has transfer efficiency of more than 92%.

Figure 6c,d present the pulse-ODMR spectrum and Rabi oscillation of the nearby nuclear spin when electron spin is at |0› state. The resonant frequency of this nuclear spin is 495 kHz, which is smaller than the Larmor frequency of 13 C nuclear spin under this magnetic field (542 kHz). We attribute this modification to the ‘enhance effect’ of center electron spin. As the nuclear spin is close to the electron spin, the nonsecular terms of their dipole interaction contribute some electronic character to the nuclear-spin levels and modify its magnetic moment 16 . The Rabi frequency of nuclear spin is about 100 kHz, which reaches 20% of the Zeeman splitting, such fast manipulation also benefits from the electron enhance effect. We discuss the validity of rotating wave approximation in Supplementary Note 1.

Synchronization of pulse generators and phase calibration

Synchronization of the MW and RF generators is one of the main challenges in this experiment. We use the same clock reference for all the generators. For each cable connection and pulse sequence, we measure the phase of prepared state as we scan the phase of input MW pulses. This gives us a phase relation between MW and RF channels, which is used to compensate the difference between the two rotating frames. We check the phase relation before and after each data acquisition. The phase drift of our system is about 2° in 2 h measurement.

Data normalization and state tomography

Since the population information of electron spin is the only directly measurable signal in NV system, we normalize all the data to the fluorescence intensity of electron spin |0› state. Specifically, we apply two readout pulses (300 ns) at the end of each measurement. See pulse sequences in Figs 2b and 6b . The first readout pulse gets the instant population information of NV electron spin, and the second readout pulse (1 μs later) records a reference for the first one, as electron spin is polarized to |0› again after the 1-μs laser excitation. The ratio between the first signal and the reference signal is used for further data analysis, such as phase estimation or state tomography.

To carry out state tomography, we adopt the method detailed in refs 18 , 27 . Total three working transitions, |0↑›↔|0↑›↔|1↑› and |0↓›↔|1↓› are selected. The real and imaginary parts of the matrix elements in each working transition are measured by using RF (or MW) pulses of 0° and 90° phases, respectively. Other three transitions are measured in the same way, but extra transfer pulses are added before Rabi measurement in the working transitions. The full procedure of state tomography can be found in Supplementary information ( Supplementary Fig. 2 ).

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Acknowledgements

This work was supported by the National Basic Research Program of China (‘973’ Program under Grant Nos. 2014CB921402 and 2015CB921103), National Natural Science Foundation of China (under Grant No. 11175248), and the Strategic Priority Research Program of the Chinese Academy of Sciences (under Grant Nos. XDB07010300 and XDB01010000).

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Gang-Qin Liu, Yu-Ran Zhang, Yan-Chun Chang, Jie-Dong Yue, Heng Fan & Xin-Yu Pan

Collaborative Innovation Center of Quantum Matter, Beijing, 100190, China

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Contributions

X.-Y.P. and H.F. designed the experiment. X.-Y.P. is in charge of the experiment, H.F. is in charge of the theory. G.-Q.L., Y.-C. C. and X.-Y.P. performed the experiment. Y.-R. Z. and J.-D. Y carried out the theoretical study. G.-Q.L. and Y.-R. Z. wrote the paper. All authors analysed the data and commented on the manuscript.

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Correspondence to Heng Fan or Xin-Yu Pan .

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Supplementary information

Supplementary information.

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Liu, GQ., Zhang, YR., Chang, YC. et al. Demonstration of entanglement-enhanced phase estimation in solid. Nat Commun 6 , 6726 (2015). https://doi.org/10.1038/ncomms7726

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Experimental Examination of Entanglement Estimates

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• Experimental Examination of Entanglement Estimates APS
• Estimating Lower and Upper Bounds of Entanglement arXiv
• Experimental Examination of Entanglement Estimates arXiv
• Name: Xie, Songbo

Affiliations:

• U. Rochester
• Rochester U.

Affiliations (original):

• Center for Coherence and Quantum Optics, and Department of Physics and Astronomy, University of Rochester, Rochester, New York 14627, USA APS
• Center for Coherence and Quantum Optics, and Department of Physics and Astronomy, University of Rochester, Rochester NY 14627 USA.
• Phonetic signature: XYs
• UUID: 558c8ae9-9abf-4aab-bccd-c5bb75d6f1c9
• Name: Zhao, Yuan-Yuan
• PCL, Shenzhen
• Peng Cheng Laboratory, Shenzhen 518052, China APS
• Peng Cheng Laboratory, Shenzhen 518052 People’s Republic of China
• Phonetic signature: Zy
• UUID: 56e2a82d-54da-42aa-a593-ac39208dbfb5
• Name: Zhang, Chao
• USTC, Hefei
• Hefei, NSRL
• CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China APS
• Hefei National Laboratory, University of Science and Technology of China, Hefei 230088, China APS
• CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China APS
• CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People’s Republic of
• CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People’s Republic of China
• Phonetic signature: ZANGc
• UUID: 95452508-bdc9-4410-9a8f-e81f937d2e23
• Name: Huang, Yun-Feng
• Phonetic signature: HANGy
• UUID: 088ee15e-86cb-41e9-919a-6e2a7019a952
• Name: Li, Chuan-Feng
• Phonetic signature: Lc
• UUID: 09a44bf2-5d99-45da-b15f-a47883120af8
• Name: Guo, Guang-Can
• Phonetic signature: Gg
• UUID: 06b44159-5c4b-41b7-8548-c1ffa6fffd20
• Name: Eberly, Joseph H.
• Center for Coherence and Quantum Optics, and Department of Physics and Astronomy, University of Rochester, Rochester NY 14627 USA
• Phonetic signature: EBARLYj
• UUID: 07aba3b5-c3a4-4155-9345-1ca5fa31fb03

Recently, a proper genuine multipartite entanglement measure has been found for three-qubit pure states [see Xie and Eberly, Phys. Rev. Lett. 127, 040403 (2021)], but capturing useful entanglement measures for mixed states has remained an open challenge. So far, it requires not only a full tomography in experiments, but also huge calculational labor. A leading proposal was made by Gühne, Reimpell, and Werner [Phys. Rev. Lett. 98, 110502 (2007)], who used expectation values of entanglement witnesses to describe a lower bound estimation of entanglement. We provide here an extension that also gives genuine upper bounds of entanglement. This advance requires only the expectation value of any Hermitian operator. Moreover, we identify a class of operators <math display="inline"><msub><mi mathvariant="script">A</mi><mn>1</mn></msub></math> that not only give good estimates, but also require a remarkably small number of experimental measurements. In this Letter, we define our approach and illustrate it by estimating entanglement measures for a number of pure and mixed states prepared in our recent experiments.

Recently a proper genuine multipartite entanglement (GME) measure has been found for three-qubit pure states [see Xie and Eberly, Phys. Rev. Lett. 127, 040403 (2021)], but capturing useful entanglement measures for mixed states has remained an open challenge. So far, it requires not only a full tomography in experiments, but also huge calculational labor. A leading proposal was made by Gühne, Reimpell, and Werner [Phys. Rev. Lett. 98, 110502 (2007)], who used expectation values of entanglement witnesses to describe a lower bound estimation of entanglement. We provide here an extension that also gives genuine upper bounds of entanglement. This advance requires only the expectation value of {\em any} Hermitian operator. Moreover, we identify a class of operators $\A_1$ which not only give good estimates, but also require a remarkably small number of experimental measurements. In this note we define our approach and illustrate it by estimating entanglement measures for a number of pure and mixed states prepared in our recent experiments.

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• Volume: 130
• Article ID: 150801
• arXiv:2207.07584 ( PDF , PostScript , other formats ) [quant-ph]
• 10.1103/PhysRevLett.130.150801 [publication] APS
• 10.1103/PhysRevLett.130.150801 [publication] arXiv

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• Date: 2023-04-12
• Citations: 6
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Quantum Physics

Title: entanglement meter: estimation of entanglement with single copy in interferometer.

Abstract: 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.

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Physical Review A

Covering atomic, molecular, and optical physics and quantum science.

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Sample-efficient estimation of entanglement entropy through supervised learning

Maximilian rieger, moritz reh, and martin gärttner, phys. rev. a 109 , 012403 – published 2 january 2024.

• Citing Articles (1)
• INTRODUCTION
• PHYSICAL SYSTEM
• THEORETICAL CONCEPTS
• DATA GENERATION
• NETWORK ARCHITECTURE
• TRAINING AND UNCERTAINTY QUANTIFICATION
• ACKNOWLEDGMENTS

We explore a supervised machine-learning approach to estimate the entanglement entropy of multiqubit systems from few experimental samples. We put a particular focus on estimating both aleatoric and epistemic uncertainty of the network's estimate and benchmark against the best-known conventional estimation algorithms. For states that are contained in the training distribution, we observe convergence in a regime of sample sizes in which the baseline method fails to give correct estimates, while extrapolation only seems possible for regions close to the training regime. As a further application of our method, highly relevant for quantum simulation experiments, we estimate the quantum mutual information for nonunitary evolution by training our model on different noise strengths.

• Received 14 September 2023
• Accepted 4 December 2023

DOI: https://doi.org/10.1103/PhysRevA.109.012403

©2024 American Physical Society

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Authors & Affiliations

• 1 Kirchhoff-Institut für Physik, Universität Heidelberg, Im Neuenheimer Feld 227, 69120 Heidelberg, Germany
• 2 Physikalisches Institut, Universität Heidelberg, Im Neuenheimer Feld 226, 69120 Heidelberg, Germany
• 3 Institut für Theoretische Physik, Ruprecht-Karls-Universität Heidelberg, Philosophenweg 16, 69120 Heidelberg, Germany
• 4 Institute of Condensed Matter Theory and Optics, Friedrich-Schiller-University Jena, Max-Wien-Platz 1, 07743 Jena, Germany
• * [email protected]
• † [email protected]

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Vol. 109, Iss. 1 — January 2024

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Visual description of the proposed procedure. We classically simulate a system of interest described by ρ and store a quantity of interest, e.g., the entanglement entropy S = − ln ( tr ρ 2 ) , which is difficult to estimate from samples. Simultaneously, we generate synthetic measurement data S , which we use in a next step to learn a function that maps S to S . The function we propose for this is a deep neural network that, in a first step, embeds all samples in a latent space, before mapping the set of latent space embeddings into another latent space, in a permutation invariant fashion. From there on, we use a feed-forward net to predict both the mean and the standard deviation of the estimate. Once trained, the model can be employed on unlabeled experimental data to give estimates for quantities with otherwise infeasible sample complexity, such as entanglement entropies.

Training the network for N = 10 . Network performance on the training (green crosses) and validation (blue circles) dataset, compared to training labels (red line). After 4000 epochs of training, the parameters with minimal loss on the validation dataset are selected. Inset: Loss on training (green line, lower) and validation (blue line, upper) datasets.

Performance of the network (blue circles) trained only on the left-hand side of the time interval (dashed gray line). Violet crosses: Baseline method for N U = 2 and N M = 500 . Yellow pluses: Baseline method for N U = 300 and N M = 5000 .

(a) Training and validation data points in the ( γ z , γ − ) plane. Green diamonds: Training set. Blue circles: Validation set, consisting of 20 randomly distributed data points and 40 data points along a randomly chosen cross section (lower line). Red crosses: Test set, consisting of 30 randomly distributed data points and 30 validation data points along a randomly chosen cross section (lower line). (b) Performance of network (blue circles) on randomly scattered data points (red crosses). (c) Performance of the network (blue circles) on a cross section (red line) of 30 data points.

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