Bound entangled singlet-like states for quantum metrology
Károly F. Pál, Géza Tóth, Erika Bene, Tamás Vértesi
备注：
17 pages including 3 figures, revtex4.2; v2: presentation improved, some further results added, links to MATLAB programs generating the bound entangled states added; v3: published version; v4: published version, typo fixed in the first author's name in Ref. [8] and elsewhere
Bipartite entangled quantum states with a positive partial transpose (PPT), i.e., PPT entangled states, are usually considered very weakly entangled. Since no pure entanglement can be distilled from them, they are also called bound entangled. In this paper we present two classes of ($2d\times 2d$)-dimensional PPT entangled states for any $d\ge 2$ which outperform all separable states in metrology significantly. We present strong evidence that our states provide the maximal metrological gain achievable by PPT states for a given system size. When the dimension $d$ goes to infinity, the metrological gain of these states becomes maximal and equals the metrological gain of a pair of maximally entangled qubits. Thus, we argue that our states could be called "PPT singlets."
We present an improved phase estimation scheme employing entangled coherent states and demon- strate that the states give the smallest variance in the phase parameter in comparison to NOON, BAT and "optimal" states under perfect and lossy conditions. As these advantages emerge for very modest particle numbers, the optical version of entangled coherent state metrology is achievable with current technology.
To acquire the best path-entangled photon Fock states for robust quantum optical metrology with parity detection, we calculate phase information from a lossy interferometer by using twin entangled Fock states. We show that (a) when loss is less than 50% twin entangled Fock states with large photon number difference give higher visibility while when loss is higher than 50% the ones with less photon number difference give higher visibility; (b) twin entangled Fock states with large photon number difference give sub-shot-noise limit sensitivity for phase detection in a lossy environment. This res…
This Letter verifies the potential of several classes of entangled coherent state in well known quantum metrology which includes detection of classical external force, and shows that there is a class of entangled coherent state for the external force detection system without the quantum limit in the detection of the light. In the case of the precision measurement of continuos parameter like phase measurement, the entangled coherent state with perfect entanglement does not provide remarkable benefit, but we provide a concrete example that certain class of entangled coherent state gives a remark…
Entangled photons can be used to make measurements with an accuracy beyond that possible with classical light. While most implementations of quantum metrology have used states made up of a single colour of photons, we show that entangled states of two colours can show supersensitivity to optical phase and path-length by using a photonic crystal fibre source of photon pairs inside an interferometer. This setup is relatively simple and robust to experimental imperfections. We demonstrate sensitivity beyond the standard quantum limit and show super-resolved interference fringes using entangled st…
Weak-value amplification (WVA) provides a way for amplified detection of a tiny physical signal at the expense of a lower detection probability. Despite this trade-off, due to its robustness against certain types of noise, WVA has advantages over conventional measurements in precision metrology. Moreover, it has been shown that WVA-based metrology can reach the Heisenberg-limit using entangled resources, but preparing macroscopic entangled resources remains challenging. Here we demonstrate a novel WVA scheme based on iterative interactions, achieving the Heisenberg-limited precision scaling wi…
Quantum mechanics allows entanglement enhanced measurements to be performed, but loss remains an obstacle in constructing realistic quantum metrology schemes. However, recent work has revealed that entangled coherent states (ECSs) have the potential to perform robust sub-classical measurements [J. Joo et. al., Phys. Rev. Lett. 107, 83601 (2011)]. Up to now no read out scheme has been devised which exploits this robust nature of ECSs, but we present here an experimentally accessible method of achieving precision close to the theoretical bound, even with loss. We show substantial improvements ov…
We propose a class of path-entangled photon Fock states for robust quantum optical metrology, imaging, and sensing in the presence of loss. We model propagation loss with beam-splitters and derive a reduced density matrix formalism from which we examine how photon loss affects coherence. It is shown that particular entangled number states, which contain a special superposition of photons in both arms of a Mach-Zehnder interferometer, are resilient to environmental decoherence. We demonstrate an order of magnitude greater visibility with loss, than possible with N00N states. We also show that t…
We propose a generalized form of entangled coherent states (ECS) and apply them in a multi-arm optical interferometer to estimate multiple phase shifts. We obtain the quantum Cramer-Rao bounds for both the linear and nonlinear parameterization protocols. Through the analysis, we find that, utilizing the simultaneous estimation, this generalized form of ECS gives a better precision than the generalized NOON states [Phys. Rev. Lett. \textbf{111}, 070403 (2013)]. Moreover, comparing with the independent estimation, both the linear and nonlinear protocols have the same advantage in the relation to…
By considering matter wave bright solitons from weakly coupled Bose-Einstein condensates trapped in a double-well potential, we study the formation of macroscopic non-classical states, including Schrödinger-cat superposition states and maximally path entangled $N00N$-states. With these macroscopic states, we examine Mach-Zehnder interferometer in the context of parity measurements, in order to obtain Heisenberg limit accuracy for linear phase shift measurement. We reveal that the ratio between two-body scattering length and intra-well hopping parameter can be measured with the scaling beyond t…
We investigate the phase enhancement of quantum states subject to non-linear phase shifts. The optimal phase estimation of even entangled coherent states (ECSs) is shown to be better than that of NOON states and of odd ECS states with the same average particle number <n> and non-linearity exponent k. We investigate the creation of approximate ECSs (AECSs) from a squeezed vacuum with current optical technology methods and find that a pure AECS is even slightly better than an even ECS for large <n>. We also examine simple but physically relevant cases of loss in the non-linear interferometer arm…
We investigate an optimal distance of two components in an entangled coherent state for quantum phase estimation in lossy interferometry. The optimal distance is obtained by an economical point, representing the quantum Fisher information that we can extract per input energy. Maximizing the formula of the quantum Fisher information over an input mean photon number, we show that, as the losses of the interferometer increase, it can be more beneficial to prepare an initially entangled coherent state that is less entangled. This represents that the optimal distance of the two-mode components decr…
Quantum states of light, such as squeezed states or entangled states, can be used to make measurements (metrology), produce images, and sense objects with a precision that far exceeds what is possible classically, and also exceeds what was once thought to be possible quantum mechanically. The primary idea is to exploit quantum effects to beat the shot-noise limit in metrology and the Rayleigh diffraction limit in imaging and sensing. Quantum optical metrology has received a boost in recent years with an influx of ideas from the rapidly evolving field of optical quantum information processing. …
Quantum metrology with entangled states in realistic noisy environments always suffers from decoherence. Therefore, the measurement precision is greatly reduced. Here we applied the dynamical decoupling method to protect the $N$-qubit quantum metrology protocol and successfully revived the scaling of the measurement precision as ${N^{-k}}$ with $k\in \left[{5/6,11/12}\right]$. The degree of the precision revival, as determined by the noise spectrum distribution, indicates that the performance of the protected protocol can be further improved by controlling the noise spectrum. Such a protected …
We show that for some noisy channels, the optimal entanglement-assisted strategy depends on the noise level. We note that there is a non-trivial crossover between the parallel-entangled strategy and the ancilla-assisted strategy - in the former the probes are all entangled, the latter the probes are entangled with a noiseless ancilla but not amongst themselves. The transition can be explained by the fact that, separable states are more robust against noise and therefore are optimal in the high noise limit, but they are in turn outperformed by ancilla-assisted ones.
We investigate the role of coarsened measurement reference in quantum metrology. Coarsened measurement reference comes from the coarsened reference time and basis. When the measurement based on one common reference basis, the disadvantage can be removed by symmetry. Due to the coarsened reference basis, entangled states can not perform better than product states for large number of probe particles. Given a finite uncertainty of the coarsened reference basis, the optimal number of probe particle is obtained in estimating phase. Finally, we prove that the maximally entangled state always achieve…
Quantum metrology allows for a huge boost in the precision of parameters estimation. However, it seems to be extremely sensitive on the noise. Bound entangled states are states with large amount of noise what makes them unusable for almost all quantum informational tasks. Here we provide a counterintuitive example of a family of bound entangled states which can be used in quantum enhanced metrology. We show that these states give advantage as big as maximally entangled states and asymptotically reach the Heisenberg limit. Moreover, entanglement of the applied states is very weak which is refle…
Besides many interesting application to the study of foundations of quantum mechanics, entangled state are now assuming a large relevance for some practical application. In particular, we discuss most recent results obtained in our laboratory on the use of two photons entangled states produced in parametric down conversion for absolute quantum efficiency calibration of photodetectors, in photon counting regime.
We analyze the role of entanglement among probes and with external ancillas in quantum metrology. In the absence of noise, it is known that unentangled sequential strategies can achieve the same Heisenberg scaling of entangled strategies and that external ancillas are useless. This changes in the presence of noise: here we prove that entangled strategies can have higher precision than unentangled ones and that the addition of passive external ancillas can also increase the precision. We analyze some specific noise models and use the results to conjecture a general hierarchy for quantum metrolo…
We propose and analyze a probabilistic but heralded scheme to generate pure, entangled, non-Gaussian states of collective spin in large atomic ensembles by means of single-photon detection. One photon announces the preparation of a Dicke state, while two or more photons announce Schrödinger cat states. The method produces pure states even for finite photon detection efficiency and weak atom-photon coupling. The entanglement generation can be made quasi-deterministic by means of repeated trial and feedback, enabling metrology beyond the standard quantum limit.
In this thesis, we study about three subjects 1- Classical simulation of two spin-$s$ singlet correlations for all $s$ involving spin measurements, 2- Quantum correlations in successive single spin measurements, 3- Non locality without inequality for almost all two-qubit entangled states based on Cabello's non locality argument.
We introduce a new class of quantum many-particle entangled states, called the Dicke squeezed (or DS) states, which can be used to improve the precision in quantum metrology beyond the standard quantum limit. We show that the enhancement in measurement precision is characterized by a single experimentally detectable parameter, called the Dicke squeezing $ξ_{D}$, which also bounds the entanglement depth for this class of states. The measurement precision approaches the ultimate Heisenberg limit as $ξ_{D}$ attains the minimum in an ideal Dicke state. Compared with other entangled states, we show…
The usual notion of separability has to be reconsidered when applied to states describing identical particles. A definition of separability not related to any a priori Hilbert space tensor product structure is needed: this can be given in terms of commuting subalgebras of observables. Accordingly, the results concerning the use of the quantum Fisher information in quantum metrology are generalized and physically reinterpreted.
We consider the problem of establishing entangled states of optimal singlet fraction and negativity between two remote parties for every use of a noisy quantum channel and trace-preserving LOCC under the assumption that the parties do not share prior correlations. We show that for a family of quantum channels in every finite dimension $d\geq3$, one-shot optimal singlet fraction and entanglement negativity are attained only with appropriate nonmaximally entangled states. We further show that the generalization of the formula that exactly computes one-shot optimal singlet fraction for qubit chan…
There has been much interest in developing phase estimation schemes which beat the so-called Heisenberg limit, i.e., for which the phase resolution scales better than 1/n, where n is a measure of resources such as the average photon number or the number of atomic qubits. In particular, a number of nonlinear schemes have been proposed for which the resolution appears to scale as 1/n^k or even exp(-n), based on optimising the quantum Cramer-Rao bound. Such schemes include the use of entangled coherent states. However, it may be shown that the average root mean square errors of the proposed schem…
In quantum metrology schemes, one generally needs to prepare $m$ copies of $N$ entangled particles, such as entangled photon states, and then they are detected in a destructive process to estimate an unknown parameter. Here, we present a novel experimental scheme for estimating this parameter by using repeated indirect quantum nondemolition measurements in the setup called "photon box". This interaction-based scheme is able to achieve the phase sensitivity scaling as $1/N$ with a Fock state of $N$ photons. Moreover, we only need to prepare one initial $N$-photon state and it can be used repeti…
The study of optical parametric amplifiers (OPAs) has been successful in describing and creating nonclassical light for use in fields such as quantum metrology and quantum lithography [Agarwal, et al., J. Opt. Soc. Am. B, 24, 2 (2007)]. In this paper we present the theory of an OPA scheme utilizing an entangled state input. The scheme involves two identical OPAs seeded with the maximally path-entangled N00N state (|2,0>+|0,2>)/sqrt{2}. The stimulated amplification results in output state probability amplitudes that have a dependence on the number of photons in each mode, which differs greatly …
Environmental noise can hinder the metrological capabilities of entangled states. While the use of entanglement allows for Heisenberg-limited resolution, the largest permitted by quantum mechanics, deviations from strictly unitary dynamics quickly restore the standard scaling dictated by the central limit theorem. Product and maximally entangled states become asymptotically equivalent when the noisy evolution is both local and strictly Markovian. However, temporal correlations in the noise have been shown to lift this equivalence while fully (spatially) correlated noise allows for the identifi…
There are a number of different strategies to measure the phase shift between two pathways of light more efficiently than suggested by the standard quantum limit. One way is to use highly entangled photons. Another way is to expose photons to a non-linear or interacting Hamiltonian. This paper emphasises that the conditional dynamics of open quantum systems provides an interesting additional tool for quantum-enhanced metrology. As a concrete example, we review a recent scheme which exploits the conditional dynamics of a laser-driven optical cavity with spontaneous photon emission inside a quan…
There is growing belief that the next decade will see the emergence of sensing devices based on the laws of quantum physics that outperform some of our current sensing devices. For example, in frequency estimation, using a probe prepared in an entangled state can, in principle, lead to a precision gain compared to a probe prepared in a separable state. Even in the presence of some forms of decoherence, it has been shown that the precision gain can increase with the number of probe particles $N$. Usually, however, the entangled and separable state preparation and readout times are assumed to be…
A promising practical application of entanglement is metrology, where quantum states can be used to make measurements beyond the shot noise limit. Here we consider how metrology schemes could be realised using atomic Bose-Einstein condensates (BECs) trapped in different potentials. In particular, we show that if a trapped BEC is rotated at just the right frequency, it can undergo a quantum phase transition characterised by large-scale entanglement spreading across the system. This simple process of stirring can generate interesting quantum states such as macroscopic superpositions of all the a…
We analytically obtain the precision bounds of frequency measurements in correlated Markovian and non-Markovian environments by using a variational approach. It is verified that in standard Ramsey spectroscopy setup, the metrological equivalence of product and maximally entangled states persists in maximally correlated Markovian and non-Markovian environments. We find that the optimal measurement can achieve a much higher resolution than standard Ramsey spectroscopy in the correlated environments. When the number of particles in the maximally entangled states is even, the precision bound decre…
Quantum technologies exploit entanglement to enhance various tasks beyond their classical limits including computation, communication and measurements. Quantum metrology aims to increase the precision of a measured quantity that is estimated in the presence of statistical errors using entangled quantum states. We present a novel approach for finding (near) optimal states for metrology in the presence of noise, using variational techniques as a tool for efficiently searching the classically intractable high-dimensional space of quantum states. We comprehensively explore systems consisting of up…
The radical pair (RP) mechanism, which describes the quantum dynamics of a spatially separated electron pair, is considered as one of the principal models of avian magnetoreception. Different from the conventional phenomenological approach where the sensitivity of avian magnetoreception is characterized by the singlet yield $Φ_{S}$, we introduce the quantum Fisher information (QFI), which represents the maximum information about the magnetic field's direction extracted from the RP state, to give a precise measure of sensitivity of avian compass essentially. The consistency between our results …
An entangled state prepared in the decoherence free sub-space together with a Ramsey type measurement can probe parity violation in heavy alkali ions like Ba+ or Ra+. Here we propose an experiment with Ba+ ions as an example to measure the small parity violating effect in this system.
In quantum metrology, entanglement represents a valuable resource that can be used to overcome the Standard Quantum Limit (SQL) that bounds the precision of sensors that operate with independent particles. Measurements beyond the SQL are typically enabled by relatively simple entangled states (squeezed states with Gaussian probability distributions), where quantum noise is redistributed between different quadratures. However, due to both fundamental limitations and the finite measurement resolution achieved in practice, sensors based on squeezed states typically operate far from the true funda…
In this paper we explore the possibility of performing Heisenberg limited quantum metrology of a phase, without any prior, by employing only maximally entangled states. Starting from the estimator introduced by Higgins et al. in New J. Phys. 11, 073023 (2009), the main result of this paper is to produce an analytical upper bound on the associated Mean Squared Error which is monotonically decreasing as a function of the square of the number of quantum probes used in the process. The analysed protocol is non-adaptive and requires in principle (for distinguishable probes) only separable measureme…
Entangled states of rotating, trapped ultracold bosons form a very promising scenario for quantum metrology. In order to employ such states for metrology, it is vital to understand their detailed form and the enhanced accuracy with which they could measure phase, in this case generated through rotation. In this work we study the rotation of ultracold bosons in an asymmetric trapping potential beyond the lowest Landau level (LLL) approximation. We demonstrate that whilst the LLL can identify reasonably the critical frequency for a quantum phase transition and entangled state generation, it is v…
Quantum approaches relying on entangled photons have been recently proposed to increase the efficiency of optical measurements. We demonstrate here that, surprisingly, the use of classical light with entangled degrees of freedom can also bring outstanding advantages over conventional measurements in polarization metrology. Specifically, we show that radially polarized beams of light allow to perform real-time single-shot Mueller matrix polarimetry. Our results also indicate that quantum optical procedures requiring entanglement without nonlocality can be actually achieved in classical optics r…
Entanglement is recognized as a key resource for quantum computation and quantum cryptography. For quantum metrology, the use of entangled states has been discussed and demonstrated as a means of improving the signal-to-noise ratio. In addition, entangled states have been used in experiments for efficient quantum state detection and for the measurement of scattering lengths. In quantum information processing, manipulation of individual quantum bits allows for the tailored design of specific states that are insensitive to the detrimental influences of an environment. Such 'decoherence-free subs…
Quantum metrology exploits quantum correlations in specially prepared entangled or other non-classical states to perform measurements that exceed the standard quantum limit. Typically though, such states are hard to engineer, particularly when larger numbers of resources are desired. As an alternative, this paper aims to establish quantum jump metrology which is based on generalised sequential measurements as a general design principle for quantum metrology and discusses how to exploit open quantum systems to obtain a quantum enhancement. By analysing a simple toy model, we illustrate that par…
Quantum technologies exploit entanglement to revolutionize computing, measurements, and communications. This has stimulated the research in different areas of physics to engineer and manipulate fragile many-particle entangled states. Progress has been particularly rapid for atoms. Thanks to the large and tunable nonlinearities and the well developed techniques for trapping, controlling and counting, many groundbreaking experiments have demonstrated the generation of entangled states of trapped ions, cold and ultracold gases of neutral atoms. Moreover, atoms can couple strongly to external forc…
Preparing highly entangled quantum states between remote parties is a major challenge for quantum communications [1-8]. Particularly promising in this context are the N00N states, which are entangled N-photon wavepackets delocalized between two different locations, providing measurement sensitivity limited only by the uncertainty principle [1, 10-15]. However, these states are notoriously vulnerable to losses, making it difficult both to share them between remote locations, and to recombine them to exploit interference effects. Here we address this challenge by utilizing the reverse version of…
Quantum metrology exploits quantum resources and strategies to improve measurement precision of unknown parameters. One crucial issue is how to prepare a quantum entangled state suitable for high-precision measurement beyond the standard quantum limit. Here, we propose a scheme to find optimal pulse sequence to accelerate the one-axis twisting dynamics for entanglement generation with the aid of deep reinforcement learning (DRL). We consider the pulse train as a sequence of $π/2$ pulses along one axis or two orthogonal axes, and the operation is determined by maximizing the quantum Fisher info…
Invariant entangled states remain unchanged under simultaneous identical unitary transformations of all their subsystems. We experimentally generate and characterize such invariant two-, four-, and six-photon polarization entangled states. This is done only with a suitable filtering procedure of multiple emissions of entangled photon pairs from a single source, without any interferometric overlaps. We get the desired states utilizing bosonic emission enhancement due to indistinguishability. The setup is very stable, and gives high interference contrasts. Thus, the process is a very likely cand…
Maximally entangled states--a resource for quantum information processing--can only be shared through noiseless quantum channels, whereas in practice channels are noisy. Here we ask: Given a noisy quantum channel, what is the maximum attainable purity (measured by singlet fraction) of shared entanglement for single channel use and local trace preserving operations? We find an exact formula of the maximum singlet fraction attainable for a qubit channel and give an explicit protocol to achieve the optimal value. The protocol distinguishes between unital and nonunital channels and requires no loc…
We propose an interferometric scheme for the estimation of a linear combination with non-negative weights of an arbitrary number $M>1$ of unknown phase delays, distributed across an $M$-channel linear optical network, with Heisenberg-limited sensitivity. This is achieved without the need of any sources of photon-number or entangled states, photon-number resolving detectors or auxiliary interferometric channels. Indeed, the proposed protocol remarkably relys upon a single squeezed state source, an antisqueezing operation at the interferometer output, and on-off photodetectors.
