Distillation Protocol Research Articles

Resource theory of imaginarity in distributed scenarios

The resource theory of imaginarity studies the operational value of imaginary parts in quantum states, operations, and measurements. Here we introduce and study the distillation and conversion of imaginarity in distributed scenario. This arises naturally in bipartite systems where both parties work together to generate the maximum possible imaginarity on one of the subsystems. We give exact solutions to this problem for general qubit states and pure states of arbitrary dimension. We present a scenario that demonstrates the operational advantage of imaginarity: the discrimination of quantum channels without the aid of an ancillary system. We then link this scenario to local operations and classical communications(LOCC) discrimination of bipartite states. We experimentally demonstrate the relevant assisted distillation protocol, and show the usefulness of imaginarity in the aforementioned two tasks.

Experimental Virtual Distillation of Entanglement and Coherence.

Noise is, in general, inevitable and detrimental to practical and useful quantum communication and computation. Under the resource theory framework, resource distillation serves as a generic tool to overcome the effect of noise. Yet, conventional resource distillation protocols generally require operations on multiple copies of resource states, and strong limitations exist that restrict their practical utilities. Recently, by relaxing the setting of resource distillation to only approximating the measurement statistics instead of the quantum state, a resource-frugal protocol, "virtual resource distillation," is proposed, which allows more effective distillation of noisy resources. Here, we report its experimental implementation on a photonic quantum system for the distillation of quantum coherence (up to dimension four) and bipartite entanglement. We show the virtual distillation of the maximal superposed state of dimension four from the state of dimension two, an impossible task in conventional coherence distillation. Furthermore, we demonstrate the virtual distillation of entanglement with operations acting only on a single copy of the noisy Einstein-Podolsky-Rosen (EPR) pair and showcase the quantum teleportation task using the virtually distilled EPR pair with a significantly improved fidelity of the teleported state. These results illustrate the feasibility of the virtual resource distillation method and pave the way for accurate manipulation of quantum resources with noisy quantum hardware.

Partially Fault-Tolerant Quantum Computing Architecture with Error-Corrected Clifford Gates and Space-Time Efficient Analog Rotations

Quantum computers are expected to drastically accelerate certain computing tasks versus classical computers. Noisy intermediate-scale quantum (NISQ) devices, which have tens to hundreds of noisy physical qubits, are gradually becoming available, but it is still challenging to achieve useful quantum advantages in meaningful tasks. On the other hand, full fault-tolerant quantum computing (FTQC) based on quantum error correction code remains far beyond realization due to its extremely large requirement of high-precision physical qubits. In this study, we propose a quantum computing architecture to close the gap between NISQ and FTQC architectures. Our architecture is based on erroneous arbitrary rotation gates and error-corrected Clifford gates implemented by lattice surgery. We omit the typical distillation protocol to achieve direct analog rotations and small qubit requirements, and minimize the remnant errors of the rotations by a carefully designed state injection protocol. Our estimation based on numerical simulations shows that for early-FTQC devices that consist of 104 physical qubits with physical error probability p=10−4, we can perform roughly 1.72×107 Clifford operations and 3.75×104 arbitrary rotations on 64 logical qubits. Such computations cannot be realized by the existing NISQ and FTQC architectures on the same device, as well as classical computers. We hope that our proposal and the corresponding development of quantum algorithms based on it will bring new insights into the realization of practical quantum computers in the future. Published by the American Physical Society 2024

GHZ distillation protocols in the presence of decoherence

In this paper, we introduce a novel heuristic approach designed to optimize the performance of Greenberger-Horne- Zeilinger (GHZ) creation and distillation protocols under decoherence. Our methodology converts these protocols into a practical set of instructions, demonstrating, through simulations, the production of higher-quality GHZ states than previously known protocols. This advancement contributes to the field of distributed quantum computing by addressing the need for high-quality entanglement required for operations between different quantum computers.

The Round Complexity of Local Operations and Classical Communication (LOCC) in Random-Party Entanglement Distillation

A powerful operational paradigm for distributed quantum information processing involves manipulating pre-shared entanglement by local operations and classical communication (LOCC). The LOCC round complexity of a given task describes how many rounds of classical communication are needed to complete the task. Despite some results separating one-round versus two-round protocols, very little is known about higher round complexities. In this paper, we revisit the task of one-shot random-party entanglement distillation as a way to highlight some interesting features of LOCC round complexity. We first show that for random-party distillation in three qubits, the number of communication rounds needed in an optimal protocol depends on the entanglement measure used; for the same fixed state some entanglement measures need only two rounds to maximize whereas others need an unbounded number of rounds. In doing so, we construct a family of LOCC instruments that require an unbounded number of rounds to implement. We then prove explicit tight lower bounds on the LOCC round number as a function of distillation success probability. Our calculations show that the original W-state random distillation protocol by Fortescue and Lo is essentially optimal in terms of round complexity.

Seven definitions of bipartite bound entanglement

An entangled state is bound entangled if one cannot combine any number of copies of the state to a maximally entangled state by using only local operations and classical communication. If one formalizes this notion of bound entanglement, one arrives immediately at four different definitions. In addition, at least three more definitions are commonly used in the literature, in particular in the very first paper on bound entanglement. Here we review critical distillation protocols and we examine how different results from quantum information theory interact in order to prove that all seven definitions are eventually equivalent. Our self-contained analysis unifies and extends previous results scattered in the literature and reveals details of the structure of bound entanglement.

Distilling Nonlocality in Quantum Correlations.

Nonlocality, as established by the seminal Bell's theorem, is considered to be the most striking feature of correlations present in spacelike separated events. Its practical application in device independent protocols, such as secure key distribution, randomness certification, etc., demands identification and amplification of such correlations observed in the quantum world. In this Letter we study the prospect of nonlocality distillation, wherein, by applying a natural set of free operations (called wirings) on many copies of weakly nonlocal systems, one aims to generate correlations of higher nonlocal strength. In the simplest Bell scenario, we identify a protocol, namely, logical OR-AND wiring, that can distill nonlocality to a significantly high degree starting from arbitrarily weak quantum nonlocal correlations. As it turns out, our protocol has several interesting facets: (i)it demonstrates that a set of distillable quantum correlations has nonzero measure in the full eight-dimensional correlation space, (ii)it can distill quantum Hardy correlations by preserving its structure, (iii)it shows that (nonlocal) quantum correlations sufficiently close to the local deterministic points can be distilled by a significant amount. Finally, we also demonstrate efficacy of the considered distillation protocol in detecting postquantum correlations.

Bipartite entanglement distillation by unilateral and bilateral local filters using polarizing Mach–Zehnder interferometers

The extraordinary correlation seen in entangled states in space-like separated regions is one of the most intriguing aspects of quantum states. Practical utility of entanglement as a resource for quantum key distribution, dense coding, or teleportation generally requires maximally entangled states. In practice, entanglement quality degrades substantially due to channel noise. The problem may be mitigated by entanglement distillation. The simplest distillation protocol is enforced by local filtering operations and classical communication. The local filtering operations are merely generalized positive operator valued measures utilizing additional degrees of freedom (DoFs) as ancillary qubits. In this work, we experimentally show that filtering on a single channel (unilateral) is equally effective as filtering on both channels (bilateral) for distillation of pure non-maximally entangled bipartite states. This result holds for a non-maximally entangled multi-qubit Greenberger-Horne-Zeilinger (GHZ) like states as well, as they show a straightforward extension of the Bell state structure. Further, we provide a theoretical comparison of the efficacy of unilateral and bilateral filtering for the case of mixed states resulting from local depolarizing noise introduced either in one or both of the non-maximally entangled pairs. Surprisingly, when noise is introduced in either one of the channels, we find that unilateral filtering on the noise-free channel outperforms the filtering on the noisy channel and bilateral filtering on both channels. We also find that bilateral filtering is more effective when both channels are noisy. A reduction in the number of local operations, in general, has the advantage of reducing the complexity of the experimental apparatus and the reduction of measurement related errors.

Overcoming entropic limitations on asymptotic state transformations through probabilistic protocols

The quantum relative entropy is known to play a key role in determining the asymptotic convertibility of quantum states in general resource-theoretic settings, often constituting the unique monotone that is relevant in the asymptotic regime. We show that this is no longer the case when one allows stochastic protocols that may only succeed with some probability, in which case the quantum relative entropy is insufficient to characterize the rates of asymptotic state transformations, and a new entropic quantity based on a regularization of the Hilbert projective metric comes into play. Such a scenario is motivated by a setting where the cost associated with transformations of quantum states, typically taken to be the number of copies of a given state, is instead identified with the size of the quantum memory needed to realize the protocol. Our approach allows for constructing transformation protocols that achieve strictly higher rates than those imposed by the relative entropy. Focusing on the task of resource distillation, we give broadly applicable strong converse bounds on the asymptotic rates of probabilistic distillation protocols, and show them to be tight in relevant settings such as entanglement distillation with nonentangling operations. This generalizes and extends previously known limitations that are only applicable to deterministic protocols. Our methods are based on recent results for probabilistic one-shot transformations as well as a new asymptotic equipartition property for the projective relative entropy.

Measurement‐Based Hyperentanglement Distillation for Lossy and Distortion Photon State

AbstractThe information content of a photon system can be extended by hyperentanglement, but the quality of hyperentanglement will be decreased by the complicated transmission loss and channel noise in quantum information processing. Here, an efficient measurement‐based hyperentanglement distillation protocol (MB‐HDP) is presented for depressing the effects of complicated transmission loss and channel noise on hyperentanglement. In the MB‐HDP, the nonlocal lossy and distortion photon states are coupled to local hyperentangled Greenberger–Horne–Zeilinger (GHZ) states using parity measurement and qubit amplification device, and the decoherence caused by bit‐flip (phase‐flip) error, diverse transmission coefficients and transmission loss can be depressed by the successful measurement results, which can increase the quality of nonlocal hyperentangled photon state. This MB‐HDP broadens the application scope of hyperentanglement distillation to nonlocal lossy and distortion photon state with a lower degree of entanglement. In addition, the MB‐HDP can further improve the quality of nonlocal hyperentangled photon state by coupling multiple copies of lossy and distortion hyperentangled photon state with local hyperentangled GHZ states. This work demonstrates the ability of measurement‐based method for ensuring the quality of nonlocal hyperentanglement, which can improve the integrity and capacity of long‐distance quantum information processing.

Advantages of Multicopy Nonlocality Distillation and Its Application to Minimizing Communication Complexity.

Nonlocal correlations are a central feature of quantum theory, and understanding why quantum theory has a limited amount of nonlocality is a fundamental problem. Since nonlocality also has technological applications, e.g., for device-independent cryptography, it is useful to understand it as a resource and, in particular, whether and how different types of nonlocality can be interconverted. Here we focus on nonlocality distillation which involves using several copies of a nonlocal resource to generate one with more nonlocality. We introduce several distillation schemes which distill an extended part of the set of nonlocal correlations including quantum correlations. Our schemes are based on a natural set of operational procedures known as wirings that can be applied regardless of the underlying theory. Some are sequential algorithms that repeatedly use a two-copy protocol, while others are genuine three-copy distillation protocols. In some regions we prove that genuine three-copy protocols are strictly better than two-copy protocols. By applying our new protocols we also increase the region in which nonlocal correlations are known to give rise to trivial communication complexity. This brings us closer to an understanding of the sets of nonlocal correlations that can be recovered from information-theoretic principles, which, in turn, enhances our understanding of what is special about quantum theory.

Catalytic transformations in coherence theory

In 2016, Winter and Yang [A. Winter and D. Yang, Phys. Rev. Lett. 116, 120404 (2016)] provided an operational meaning to relative entropy of coherence and coherence of formation by introducing coherence distillation and dilation protocols in an asymptotic setup. Here we have provided relative entropy of coherence [via incoherent operations (IO)] and coherence of formation (via IO) and quantum incoherent relative entropy [via local quantum incoherent operations with classical communications (LQICC)] as a measure using the concept of a catalyst. We have proved an existential correspondence between asymptotic and catalytic state transformation using IO, local incoherent operations with classical communications, and LQICC. We have also discussed two very important protocols, assisted distillation and quantum incoherent state merging using a catalyst. Monotone properties of relative entropy of coherence, coherence of formation, and quantum incoherent relative entropy under the catalytic transformation are also discussed here.

Annihilating Entanglement Between Cones

Every multipartite entangled quantum state becomes fully separable after an entanglement breaking quantum channel acted locally on each of its subsystems. Whether there are other quantum channels with this property has been an open problem with important implications for entanglement theory (e.g., for the distillation problem and the PPT squared conjecture). We cast this problem in the general setting of proper convex cones in finite-dimensional vector spaces. The max-entanglement annihilating maps transform the k-fold maximal tensor product of a cone {textsf{C}}_1 into the k-fold minimal tensor product of a cone {textsf{C}}_2, and the pair ({textsf{C}}_1,{textsf{C}}_2) is called resilient if all max-entanglement annihilating maps are entanglement breaking. Our main result is that ({textsf{C}}_1,{textsf{C}}_2) is resilient if either {textsf{C}}_1 or {textsf{C}}_2 is a Lorentz cone. Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation: As a warm-up, we use the multiplication tensors of real composition algebras to construct a finite family of generalized distillation protocols for Lorentz cones, containing the distillation protocol for entangled qubit states by Bennett et al. (Phys Rev Lett 76(5):722, 1996) as a special case. Then, we construct an infinite family of protocols using solutions to the Hurwitz matrix equations. After proving these results, we focus on maps between cones of positive semidefinite matrices, where we derive necessary conditions for max-entanglement annihilation similar to the reduction criterion in entanglement distillation. Finally, we apply results from the theory of Banach space tensor norms to show that the Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied.

Achieving the ultimate end-to-end rates of lossy quantum communication networks

The field of quantum communications promises the faithful distribution of quantum information, quantum entanglement, and absolutely secret keys, however, the highest rates of these tasks are fundamentally limited by the transmission distance between quantum repeaters. The ultimate end-to-end rates of quantum communication networks are known to be achievable by an optimal entanglement distillation protocol followed by teleportation. In this work, we give a practical design for this achievability. Our ultimate design is an iterative approach, where each purification step operates on shared entangled states and detects loss errors at the highest rates allowed by physics. As a simpler design, we show that the first round of iterations can purify completely at high rates. We propose an experimental implementation using linear optics and photon-number measurements which is robust to inefficient operations and measurements, showcasing its near-term potential for real-world practical applications.

Open-Air Microwave Entanglement Distribution for Quantum Teleportation

Microwave technology plays a central role in current wireless communications, standing among them mobile communication and local area networks (LANs). The microwave range shows relevant advantages with respect to other frequencies in open-air transmission, such as low absorption losses and low energy consumption, and it is additionally the natural working frequency in superconducting quantum technologies. Entanglement distribution between separate parties is at the core of secure quantum communications. Therefore, understanding its limitations in realistic open-air settings, specially in the rather unexplored microwave regime, is crucial for transforming microwave quantum communications into a mainstream technology. Here, we investigate the feasibility of an open-air entanglement distribution scheme with microwave two-mode squeezed states. First, we study the reach of direct entanglement transmission in open-air, obtaining a maximum distance of approximately 500 meters in a realistic setting with state-of-the-art experimental parameters. Afterwards, we adapt entanglement distillation and entanglement swapping protocols to microwave technology in order to reduce environmental entanglement degradation. While entanglement distillation helps to increase quantum correlations in the short-distance low-squeezing regime by up to $46\%$, entanglement swapping increases the reach by $14\%$. Then, we compute the fidelity of a continuous-variable quantum teleportation protocol using open-air-distributed entanglement as a resource. Finally, we adapt the machinery to explore the limitations of quantum communication between satellites, where the thermal noise impact is substantially reduced and diffraction losses are dominant.

Privacy Amplification Strategies in Sequential Secret Key Distillation Protocols Based on Machine Learning

It is well known that Renyi’s entropy of order 2 determines the maximum possible length of the distilled secret keys in sequential secret key distillation protocols so that no information is leaked to the eavesdropper. There have been no attempts to estimate this key quantity based on information available to the legitimate parties to this protocol in the literature. We propose a new machine learning system, which estimates the lower bound of conditional Renyi entropy with high accuracy, based on 13 characteristics locally measured on the side of legitimate participants. The system is based on a prediction intervals deep neural network, trained for a given source of common randomness. We experimentally evaluated this result for two different sources, namely 14 and 6-dimensional EEG signals, of 50 participants, with varying advantage distillation and information reconciliation strategies with and without additional lossless compression block. Across all proposed systems and analyzed sources on average, the best machine learning strategy, called the hybrid strategy, increases the quantity of generated keys 2.77 times compared to the classical strategy. By introducing the Huffman lossless coder before the PA block, the loss of potential source randomness was reduced from 68.48% to a negligible 0.75%, while the leakage rate per one bit remains in the order of magnitude 10−4.

Tight constraints on probabilistic convertibility of quantum states

We develop two general approaches to characterising the manipulation of quantum states by means of probabilistic protocols constrained by the limitations of some quantum resource theory. First, we give a general necessary condition for the existence of a physical transformation between quantum states, obtained using a recently introduced resource monotone based on the Hilbert projective metric. In all affine quantum resource theories (e.g. coherence, asymmetry, imaginarity) as well as in entanglement distillation, we show that the monotone provides a necessary and sufficient condition for one-shot resource convertibility under resource-non-generating operations, and hence no better restrictions on all probabilistic protocols are possible. We use the monotone to establish improved bounds on the performance of both one-shot and many-copy probabilistic resource distillation protocols. Complementing this approach, we introduce a general method for bounding achievable probabilities in resource transformations under resource-non-generating maps through a family of convex optimisation problems. We show it to tightly characterise single-shot probabilistic distillation in broad types of resource theories, allowing an exact analysis of the trade-offs between the probabilities and errors in distilling maximally resourceful states. We demonstrate the usefulness of both of our approaches in the study of quantum entanglement distillation.

Quantum Pin Codes

We introduce quantum pin codes: a class of quantum CSS codes. Quantum pin codes are a generalization of quantum color codes and Reed-Muller codes and share a lot of their structure and properties. Pin codes have gauge operators, an unfolding procedure and their stabilizers form so-called <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell $ </tex-math></inline-formula> -orthogonal spaces meaning that the joint overlap between any <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell $ </tex-math></inline-formula> stabilizer elements is always even. This last feature makes them interesting for devising magic-state distillation protocols, for instance by using puncturing techniques. We study examples of these codes and their properties.

Classification of small triorthogonal codes

Triorthogonal codes are a class of quantum error correcting codes used in magic state distillation protocols. We classify all triorthogonal codes with $n+k \le 38$, where $n$ is the number of physical qubits and $k$ is the number of logical qubits of the code. We find $38$ distinguished triorthogonal subspaces and show that every triorthogonal code with $n+k\le 38$ descends from one of these subspaces through elementary operations such as puncturing and deleting qubits. Specifically, we associate each triorthogonal code with a Reed-Muller polynomial of weight $n+k$, and classify the Reed-Muller polynomials of low weight using the results of Kasami, Tokura, and Azumi and an extensive computerized search. In an appendix independent of the main text, we improve a magic state distillation protocol by reducing the time variance due to stochastic Clifford corrections.

Enumerating all bilocal Clifford distillation protocols through symmetry reduction

Entanglement distillation is an essential building block in quantum communication protocols. Here, we study the class of near-term implementable distillation protocols that use bilocal Clifford operations followed by a single round of communication. We introduce tools to enumerate and optimise over all protocols for up to n=5 (not necessarily equal) Bell-diagonal states using a commodity desktop computer. Furthermore, by exploiting the symmetries of the input states, we find all protocols for up to n=8 copies of a Werner state. For the latter case, we present circuits that achieve the highest fidelity with perfect operations and no decoherence. These circuits have modest depth and number of two-qubit gates. Our results are based on a correspondence between distillation protocols and double cosets of the symplectic group, and improve on previously known protocols.