Searching for an absolutely maximally five-level entangled state

In this thesis we experimentally investigate quantum nonlocality: entangled states of spatially separated objects. Entanglement is one of the most striking consequences of the quantum formalism developed in the 1920's; the predicted outcomes of independent measurements on entangled objects reveal strong correlations that cannot be explained by classical physics. Early on, such predictions led physicists to doubt the validity and completeness of quantum theory. At the same time, entanglement is a key resource for applications in quantum information processing and a pre-requisite for many tasks in quantum communication and computation. This thesis attempts to answer two application driven questions: Firstly, can we generate useful entangled states of solid state spins for applications in quantum information processing? Secondly, can we use such entangled states as a resource to teleport an unknown quantum state? Finally, we ask a foundational question: Are our entangled states indeed inconsistent with the classical notions of free choice, locality and realism? Can we prove this experimentally, under the minimal assumptions of a loophole-free Bell test? To answer these questions we use single spins in ultra-pure diamonds. In particular, we use the electronic and nuclear spins associated with single nitrogen-vacancy (NV) defects. The NV centre is a point defect in diamond, consisting of a substitutional nitrogen (N) atom and a neighbouring missing carbon atom (vacancy, V). The NV centre possesses bound electronic states, whose energy levels lie well within the bandgap of the diamond host and whose spin degree of freedom can be used as a quantum bit (qubit). Because of the large diamond bandgap and the 99% spin free carbon-12 environment, the electronic spin qubit has exceptional coherence properties even at room temperature. Optical and microwave fields allow control of the electronic spin, which in turn allows control of nearby nuclear spins (the host nitrogen nuclear spin, and nearby carbon-13 spins). At liquid helium temperatures, spin-preserving optical transitions provide a powerful optical interface to the electronic spin, allowing, for example, projective readout of the spin state. By employing a protocol where entanglement is heralded by the detection of a single photon from each of two NV centres in diamonds separated by three metres, we find we can answer the first question in the affirmative. We show for the first time heralded entanglement between solid state quantum systems separated by a human-scale distance. Then, by combining the heralded entanglement with a deterministic local Bell state measurement and fast feed-forward, we show for the first time unconditional quantum teleportation over human-scale distances. We teleport an unknown quantum state from a nuclear spin in one diamond to an electronic spin in a diamond three meters away. Finally, by employing techniques from the previous experiments, we implement the first loophole-free Bell test. We separate two diamonds by 1.3 kilometres and optimize all operational fidelities, collection efficiencies and rates. This allows us to generate heralded entanglement between them approximately once an hour. The distance provides us with time to read out the electronic spin state in each diamond, faster than any lightlike signal could travel between them. The high-fidelity entangled state preparation and spin readout are sufficient to violate the Clauser-Horne-Shimony-Holt Bell-inequality. Combined with fast random number generators and a robust statistical analysis, we find a significant rejection of the local-realist hypothesis, without requiring additional experimental assumptions. The results in this thesis open the door to various applications in quantum information processing. In particular, a remote photonic entangling operation may enable future quantum networks. In such a network the nodes would be formed by the NV centre's combined electronic and nuclear spin register. The nodes would be linked by photonic entanglement operations. Such a network could be used for long distance secure communication, provide a connection between separated quantum computers, or form the basis of a fault tolerant quantum computer by itself. Furthermore, a loophole-free Bell test demonstrates the possibility to do device independent randomness generation and key distribution, that could form the basis for future secure communication channels.

We describe the normal ordering method (NOM) as a convinient way to obtaining explicit forms of the state vectors and density matrices of quantum states, usually generated in multiple coupled parametric optical processes. As a specific example, our method is demonstrated for the states, generated in three second-order parametric optical processes - one parametric down-conversion accompanied by two up-conversion processes. The application of the method allowed us to calculate the entropic characteristics of the multipartite quantum system.

The circuit model of quantum computation [1, 2, 3] has been a powerful tool for the development of quantum computation, acting both as a framework for theoretical investigations and as a guide for experiment. In the circuit model (also called the network model), unitary operations are represented by a network of elementary quantum gates such as the CNOT gate and single-qubit rotations. Many proposals for the implementation of quantum computation are designed around this model, including physical prescriptions for implementing the elementary gates. By formulating quantum computation in a different way, one can gain both a new framework for experiments and new theoretical insights. One-way quantum computation [4] has achieved both of these. Measurements on entangled states play a key role in many quantum information protocols, such as quantum teleportation and entanglement-based quantum key distribution. In these applications an entangled state is required, which must be generated beforehand. Then, during the protocol, measurements are made which convert the quantum correlations into, for example, a secret key. To repeat the protocol a fresh entangled state must be prepared. In this sense, the entangled state, or the quantum correlations embodied by the state, can be considered a resource which is “used up” in the protocol. In one-way quantum computation, the quantum correlations in an entangled state called a cluster state [6] or graph state [7] are exploited to allow universal quantum computation through single-qubit measurements alone. The quantum algorithm is specified in the choice of bases for these measurements and the “structure” of the entanglement (as explained below) of the resource state. The name “one-way” reflects the resource nature of the graph state. The state can be used only once, and (irreversible) projective measurements drive the computation forward, in contrast to the reversibility of every gate in the standard network model. In this chapter, we will provide an introduction to one-way quantum computation, and several of the techniques one can use to describe it. In this section we will introduce graph and cluster states and develop a notation for general single-qubit measurements. In section 2 we will introduce the key concepts of one-way quantum computation with some simple examples. After this, in section 3, we shall investigate how one-way quantum computation can

Quantum networks promise to be the future architecture for secure communication and distributed quantum computation. This thesis describes experiments on nitrogenvacancy (NV) centres that lead towards a versatile multi-node quantum network consisting ofmulti-qubit nodes. The NV centre in diamond is a spinful optically-active crystal defect. NVs are a prime network-node candidate due to demonstrated coherence times beyond 100ms and longitudinal relaxation times exceeding 1s and their spin-selective optical interface which facilitates the generation of spin-photon entanglement. Entangling links between nodes are therefore readily created by overlapping the emission of two NVs on a beam splitter. Besides NVs, we further address individual 13C nuclear spins in the vicinity and use these spins as a quantum resource. Our goal is to propel these nuclear spins to constitute robust quantummemories which store and manipulate quantum information in an NV-based quantum network. The experiments described in this thesis are thematically separated into three groups. First, we explore the NV-nuclear interplay. We demonstrate nuclear-spin control by observing the Zeno effect on up to two logical qubits within the state space of three nuclear spins (Chapter 3). We further realize that the always-on magnetic hyperfine interaction between NV and nuclear spins will limit the nuclear spin coherence when entangling distant NV centres (Chapter 4). A systematic experimental study probes our theoretical prediction and we additionally demonstrate improved robustness for logical states within decoherence-protected state spaces (Chapter 5) and finally for individual nuclear spins (Chapter 6). Second,we use remoteNV-NV entangled states to demonstrate experimental milestones in quantumnetworks. The realization of a high-fidelity entangled link over a distance of 1.3km permits the loophole-free violation of Bell’s inequality (Chapter 7). We further increase the entangling rate by three orders of magnitude such that it exceeds the decoherence rate of an entangled state on our network. This allows us to convert our probabilistic entanglement generation into a deterministic process which delivers entangled states at prespecifiedmoments in time (Chapter 8). Third, we finally combine the concepts of nuclear-spin quantum memories and remote entanglement generation to demonstrate entanglement distillation in a network setting (Chapter 9). We subsequently generate two raw entangled input states between two remote NV centres. The first state is stored on nuclear spins to liberate both NVs for the second round of state generation. Finally, a higher-fidelity entangled state is distilled via local operations. This constitutes the first quantum-network demonstration that relies on the control of multiple fully-coherent quantum systems per network node.

A common model of quantum computing is the gate model with binary basis states. Here, we consider the gate model of quantum computing with a non-binary radix resulting in more than two basis states to represent a quantum digit, or qudit. Quantum entanglement is an important phenomenon that is a critical component of quantum computation and communications algorithms. The generation and use of entanglement among radix-2 qubits is well-known and used often in quantum computing algorithms. Quantum entanglement exists in higher-radix systems as well although little is written regarding the generation of higher-radix entangled states. We provide background describing the feasibility of multiple-valued logic quantum systems and describe a new systematic method for generating maximally entangled states in quantum systems of dimension greater than two. This method is implemented in a synthesis algorithm that is described. Experimental results are included that demonstrate the transformations needed to create specific forms of maximally entangled quantum states.

Quantum entanglement is one of the primary features which distinguish quantum computers from classical computers. In gate-based quantum computing, the creation of entangled states or the distribution of entanglement across a quantum processor often requires circuit depths, which grow with the number of entangled qubits. However, in teleportation-based quantum computing, one can deterministically generate entangled states with a circuit depth that is constant in the number of qubits, provided that one has access to an entangled resource state, the ability to perform mid-circuit measurements, and can rapidly transmit classical information. In this work, aided by fast classical field programmable gate array-based control hardware with a feedback latency of only 150 ns, we explore the utility of teleportation-based protocols for generating non-local, multi-partite entanglement between superconducting qubits. First, we demonstrate well-known protocols for generating Greenberger–Horne–Zeilinger states and non-local controlled-NOT gates in constant depth. Next, we utilize both protocols for implementing a quantum fan-out gate in constant depth among three non-local qubits (i.e., controlled-NOT-NOT). Finally, we demonstrate deterministic state teleportation and entanglement swapping between qubits on opposite sides of our quantum processor. Throughout this work, we find that the fidelity of our teleportation-based protocols is limited by measurement-induced dephasing on idling spectator qubits. Therefore, our work serves as a useful study of the current benefits and limitations of teleportation-based protocols on contemporary superconducting quantum processors.

Building a working quantum computer that is able to perform useful calculations remains a challenge. With this thesis, we are trying to contribute a small piece to this puzzle by addressing three of the many fundamental questions one encounters along the way of reaching that goal. These questions are: (i) What is an easy way to create highly entangled states as a resource for quantum computation? (ii) What can we do to efficiently quantify states of noisy entanglement in systems coupled to the outside world? (iii) How can we protect and store fragile quantum states for arbitrary long times?
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\nThe first two questions are the subject of part one of this thesis, `Entanglement Measures & Highly Entangled States'. We devise a particular proposal for generating entanglement within a solid-state setup, starting first with the tripartite case and continuing with a generalization to four and more qubits. The main idea there is to realize systems with highly entangled ground states in order for entanglement to be created by merely cooling to low enough temperatures. We have addressed the issue of quantifying entanglement in these systems by numerically calculating mixed-state entanglement measures and maximizing the latter as a function of the external magnetic field strength. The research along these lines has led to the development of the numerical library 'libCreme'.
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\nThe second part of the thesis, 'Self-Correcting Quantum Memories', addresses the question how to reliably store quantum states long enough to perform useful calculations. Every computer, be it classical or quantum, needs the information it processes to be protected from corruption caused by faulty gates and perturbations from interactions with its environment. However, quantum states are much more susceptible to these adverse effects than classical states, making the manipulation and storage of quantum information a challenging task. Promising candidates for such 'quantum memories' are systems exhibiting topological order, because they are robust against local perturbations, and information encoded in their ground state can only be manipulated in a non-local fashion. We extend the so-called toric code by repulsive long-range interactions between anyons and show that this makes the code stable against thermal fluctuations. Furthermore, we investigate incoherent effects of quenched disorder in the toric code and similar systems.

How to concentrate non-maximally entangled states for quantum communication is a fundamental problem in quantum information. In this paper, we will apply generalized measurements to entanglement concentration of known non-maximally entangled pure states in arbitrary dimensional system. How to design the generalized measurements for the unambiguous discrimination of linearly independent non-orthogonal states is crucial for the concentration of the known non-maximally entangled states. The result shows that, any known non-maximally entangled pure state (for arbitrary dimensional system) can be transformed to the maximally entangled state only by introducing a qubit as ancilla and a joint unitary transformation operation on one of the entangled particles and the ancilla. In addition, because the less entangled state of each fail round will be re-concentrated too, the entanglement waste during the concentration process will be greatly reduced.

Questioning of quantum information Guolin Wu The Center of Philosophy and Technology, School of Marxism South China University of Technology, Guangzhou, Guangdong, P. R. China. E-Mails: ssglwu@scut.edu.cn * The Center of Philosophy and Technology, School of Marxism. South China University of Technology, Guangzhou, Guangdong, P. R. China,510640; Tel.:0086-13660190516; Fax: 0086-2087114979 Accepted: 19 February 2015

In recent years, quantum computing and quantum information science have become one of the most important and attractive research areas in a variety of disciplines, e. g., mathematics, information science, physics, chemistry, etc1. These new kinds of technologies are predicted to be much more advantageous compared with the classical computers and classical information science and the benefit obtained by these technologies is assumed to be beyond measure in our every-day life. For instance, quantum computers are predicted to be able to solve mathematical problems that today’s fastest computers could not solve in years. In particular, entanglement or entangled state plays a key role for quantum computing and quantum information processing. For example, arbitrary quantum states of two-level system can be teleported through classical communication with the help of maximally entangled Bell state from one place to other macroscopic distant places (quantum teleportation)2, which has no counterpart in classical mechanics. As opposed to the quantum teleportation, classical information can be teleported by using the maximally entangled Bell state (superdense coding)3. Needless to say, entanglement is also an essential ingredient in quantum computing1. At present, theoretical investigations of the mechanism of quantum computing and quantum information science have become mature although some of the important theoretical problems, e. g., definition of entanglement degree of multipartite systems, have not yet been solved and are still controversial. Yet, one can say that we are now reaching a stage of experimental realizations of quantum computing and quantum information processing proposed and investigated theoretically and numerically. To apply quantum computing and quantum information processing to realistic quantum systems, a number of microscopic quantum systems have been proposed. Just to mention a few, cavity quantum electrodynamics (cavity QED)4, trapped ions5 7, neutral atoms trapped in optical lattices8, nuclear magnetic resonance (NMR)9, 10, superconducting circuits11, silicon-based nuclear spin12, diamond-based quantum computer13, 14 are some of the promising candidates of quantum computing devices. However, investigation of utilization of molecular internal degrees of freedom for quantum computing and quantum information science, in particular, electronic, vibrational, and rotational degrees of freedom, is still in its infancy. Although molecules are also quantum systems, very few chemists have yet examined how to use molecular internal degrees of

Quantum coherence is a prime resource in quantum computing and quantum communication. Quantum coherence of an arbitrary qubit state can be created at a remote location using maximally entangled state, local operation and classical communication. However, if there is a noisy channel acting on one side of the shared resource, then, it is not possible to create perfect quantum coherence remotely. Here, we present a method for the creation of quantum coherence at a remote location via the use of entangled state and indefinite causal order. We show this specifically for the superposition of two completely depolarizing channels, two partially depolarizing channels and one completely depolarizing channel along with a unitary operator. We find that when the indefinite causal order of channels act on one-half of the entangled pair, then the shared state looses entanglement, but can retain non-zero quantum discord. This finding may have some interesting applications on its own where discord can be consumed as a resource. Our results suggest that the indefinite causal order along with a tiny amount of quantum discord can act as a resource in creating non-zero quantum coherence in the absence of entanglement.

Quantum teleportation enables the secure transfer of unknown quantum states between remote users and is a key technology in quantum information science. Networks based on continuous-variable entangled states can extend both the user capacity and the transmission distance of quantum teleportation. This paper analyzes quantum teleportation network schemes based on three types of continuous-variable entangled states (EPR entangled state, GHZ entangled state, and linear cluster entangled state). The results show that due to the correlation properties of different types of entangled states, different quantum teleportation networks have advantages in terms of fidelity, transmission distance, and quantum resource consumption of quantum teleportation. For low-error-rate applications such as quantum computing, EPR states provide the highest fidelity. When parallel teleportation of multiple states is required, networks based on EPR or cluster entangled states provide the necessary throughput performance. In scenarios where quantum resources are severely limited, the GHZ-based teleportation protocols minimize the number of entangled modes while preserving acceptable fidelity. For applications demanding controlled teleportation, both GHZ and cluster states supply the essential multi-party correlations. Notably, cluster states offer a practical trade-off between fidelity and resource overhead, rendering them attractive for certain implementations. This study provides a reference for the design of multi-user metropolitan quantum teleportation networks.

Descrying all aspects of the quantum entanglement of multipartite quantum systems is an essential part when researching the quantum entanglement. Maximally multi-qubit entangled state is one of the research objects. Recently, attracted by a criterion for maximally entangled nine-qubit state and construct a nine-qubit maximally entangled state, we construct a new genuine maximally nine-qubit entangled state via the recurrence relation. Further, nine-qubit entangled state is found which can be seen as a new form maximally nine-qubit entangled state. There are 108 of purity equal to 1/16, 18 of purity equal to 1/8 in observation. We believe the result could provide a new idea for constructing more new maximally multi-qubit states.

The advent of a new kind of entangled state known as hybrid entangled state, i.e., entanglement between different degrees of freedom, makes it possible to perform various quantum computational and communication tasks with lesser amount of resources. Here, we aim to exploit the advantage of these entangled states in communication over quantum networks. Unfortunately, the entanglement shared over the network deteriorates due to its unavoidable interaction with surroundings. Thus, an entanglement concentration protocol is proposed to obtain a maximally entangled hybrid Omega-type state from the corresponding non-maximally entangled states. The advantage of the proposed entanglement concentration protocol is that it is feasible to implement this protocol with linear optical components and present technology. The corresponding linear optical quantum circuit is provided for experimental realizations, while the success probability of the concentration protocol is also reported. Thereafter, we propose an application of maximally entangled hybrid state in the hierarchical quantum teleportation network by performing information splitting using Omega-type state, which is also the first hierarchical quantum communication scheme in the hybrid domain so far. The present hybrid entangled state has advantage in circumventing Pauli operations on the coherent state by polarization rotation of single qubit, which can be performed with lesser errors.

The quantification of quantum entangled states is essential for developing entanglement resources, and this process has profound implications for advancing quantum computing and quantum communication. To address the challenge of quantifying such systems beyond 10 qubits, we present a method for constructing high-dimensional k-uniform states using a 9-qubit entangled state and recurrence relations. By incrementally increasing the number of qubits, we construct 10-, 11-, and 12-qubit entangled states and verify their uniformity using reduced density matrices. We demonstrate for the first time that the 12-qubit state is 4-uniform as all 4-qubit subsystems exhibit maximal mixedness. Compared to conventional designs, our recurrence-relation-based approach emphasizes intrinsic multiqubit correlations, offering a direct method for quantifying high-dimensional entanglement. The 4-uniform state offers distinct advantages for quantum error correction and fault-tolerant computation, laying a foundation for large-scale quantum information systems.