Stabilizer states are a prime resource for a number of applications in quantum information science, such as secret-sharing and measurement-based quantum computation. This motivates us to study the entanglement of noisy stabilizer states across a bipartition. We show that the spectra of the corresponding reduced states can be expressed in terms of properties of an associated stabilizer code. In particular, this allows us to show that the coherent information is related to the so-called syndrome entropy of the underlying code. We use this viewpoint to find stabilizer states that are resilient against noise, allowing for more robust entanglement distribution in near-term quantum networks. We specialize our results to the case of graph states, where the found connections with stabilizer codes reduces back to classical linear codes for dephasing noise. On our way we provide an alternative proof of the fact that every qubit stabilizer code is equivalent up to single-qubit Clifford gates to a graph code.
Stabilizer states are a prime resource for a number of applications in quantum information science, such as secret-sharing and measurement-based quantum computation. This motivates us to study the entanglement of noisy stabilizer states across a bipartition. We show that the spectra of the corresponding reduced states can be expressed in terms of properties of an associated stabilizer code. In particular, this allows us to show that the coherent information is related to the so-called syndrome entropy of the underlying code. We use this viewpoint to find stabilizer states that are resilient against noise, allowing for more robust entanglement distribution in near-term quantum networks. We specialize our results to the case of graph states, where the found connections with stabilizer codes reduces back to classical linear codes for dephasing noise. On our way we provide an alternative proof of the fact that every qubit stabilizer code is equivalent up to single-qubit Clifford gates to a graph code.
Characterizing large noisy multiparty quantum states using genuine multipartite entanglement is a challenging task. In this paper, we calculate lower bounds of genuine multipartite entanglement localized over a chosen multiparty subsystem of multiqubit stabilizer states in the noiseless and noisy scenarios. In the absence of noise, adopting a graph-based technique, we perform the calculation for arbitrary graph states as representatives of the stabilizer states and show that the graph operations required for the calculation have a polynomial scaling with the system size. As demonstrations, we compute the localized genuine multipartite entanglement over subsystems of large graphs having linear, ladder, and square structures. We also extend the calculation for graph states subjected to single-qubit Markovian or non-Markovian Pauli noise on all qubits and demonstrate, for a specific lower bound of the localizable genuine multipartite entanglement corresponding to a specific Pauli measurement setup, the existence of a critical noise strength beyond which all of the postmeasured states are biseparable. The calculation is also useful for arbitrary large stabilizer states under noise due to the local unitary connection between stabilizer states and graph states. We demonstrate this by considering a toric code defined on a square lattice and computing a lower bound of localizable genuine multipartite entanglement over a nontrivial loop of the code. Similar to the graph states, we show the existence of the critical noise strength in this case also and discuss its interesting features.
Stabilizer states and graph states find application in quantum error correction, measurement-based quantum computation and various other concepts in quantum information theory. In this work, we study party-local Clifford (PLC) transformations among stabilizer states. These transformations arise as a physically motivated extension of local operations in quantum networks with access to bipartite entanglement between some of the nodes of the network. First, we show that PLC transformations among graph states are equivalent to a generalization of the well-known local complementation, which describes local Clifford transformations among graph states. Then, we introduce a mathematical framework to study PLC equivalence of stabilizer states, relating it to the classification of tuples of bilinear forms. This framework allows us to study decompositions of stabilizer states into tensor products of indecomposable ones, that is, decompositions into states from the entanglement generating set (EGS). While the EGS is finite up to 3 parties [Bravyi et al., J. Math. Phys. 47 , 062106 (2006)], we show that for 4 and more parties it is an infinite set, even when considering party-local unitary transformations. Moreover, we explicitly compute the EGS for 4 parties up to 10 qubits. Finally, we generalize the framework to qudit stabilizer states in prime dimensions not equal to 2 , which allows us to show that the decomposition of qudit stabilizer states into states from the EGS is unique.
Graph states are a key resource for a number of applications in quantum information theory. Due to the inherent noise in noisy intermediate-scale quantum (NISQ) era devices, it is important to understand the effects noise has on the usefulness of graph states. We consider a noise model where the initial qubits, prepared in | + ⟩ states, undergo depolarizing noise before the application of the CZ operations that generate edges between qubits situated at the nodes of the resulting graph state. For this model we develop a method for calculating the coherent information – a lower bound on the rate at which entanglement can be distilled, across a bipartition of the graph state. We also identify some patterns on how adding more nodes or edges affects the bipartite distillable entanglement. As an application, we find a family of graph states that maintain a strictly positive coherent information for any amount of (non-maximal) depolarizing noise.
Cluster states and graph states in general offer a useful model of the stabilizer formalism and a path toward the development of measurement-based quantum computation. Their defining structure – the stabilizer group – encodes all possible correlations that can be observed during measurement. The measurement outcomes which are consistent with the stabilizer structure make error correction possible. Here, we leverage both properties to design feasible families of states that can be used as robust building blocks of quantum computation. This procedure reduces the effect of experimentally relevant noise models on the extraction of smaller entangled states from the larger noisy graph state. In particular, we study the extraction of Bell pairs from linearly extended graph states – this has the immediate consequence for state teleportation across the graph. We show that robust entanglement can be extracted by proper design of the linear graph with only a minimal overhead of the physical qubits. This scenario is relevant to systems in which the entanglement can be created between neighboring sites. The results shown in this work provide a mathematical framework for noise reduction in measurement-based quantum computation. With proper connectivity structures, the effect of noise can be minimized for a large class of realistic noise processes.
We show how entanglement-assisted codes can be constructed from arbitrary quantum codes by associating them with quantum codes for erasure channels. If a subset of physical qubits is correctable for an erasure error, then it naturally forms the receiver's share of a bipartite state that can be used for entanglement-assisted communications, both in the noiseless and noisy ebit error models. In the case of degenerate codes, we show that the receiver's share of the bipartite state can sometimes be compressed, at the cost of potentially reduced error-correction ability in the noisy ebit error model. We also give examples of permutation-invariant and XP-stabilizer entanglement-assisted codes, the first outside of the stabilizer and codeword-stabilized frameworks.
Quantum information science is an interdisciplinary field that investigates the principles and applications of quantum mechanics in information processing. Upcoming quantum technologies promise a range of valuable applications, including powerful simulation of complex systems, enhanced computational capabilities, high-precision sensing and metrology, and secure communication via quantum key distribution. A central concept is entanglement, a unique type of correlation between quantum systems that has no classical counterpart. Entanglement plays a pivotal role in many quantum protocols, particularly as a key resource in quantum communication, where protocols such as distributed quantum computation and secret sharing are executed in a distributed setting. Hence, distributing entanglement over long distances and manipulating it locally is essential for enabling quantum communication applications. However, the effectiveness of such applications is limited by the challenges of transmitting quantum information and maintaining precise control over quantum systems. For example, quantum information carriers, such as photons traveling through optical fibers, may be lost, or the operations performed may not be perfect. Consequently, it is crucial to develop robust tools for distributing and manipulating entanglement in distributed settings in order to fully unlock the capabilities of quantum technologies. In this thesis, we explore the influence of noise on the manipulation of entanglement in entanglement-based quantum networks. First, we examine the noise resilience of various resource states, i.e., pre-established entangled states, used to achieve bipartite or tripartite communication. This is followed by an analysis of a fully asymmetric network scenario, incorporating the noise introduced by the delays from classical messaging within the quantum network. We then propose classical communication protocols to achieve bipartite communication and evaluate their noise impact under the optimization of the arrangement of the pre-established entanglement. In addition, we propose an alternative approach to quantum repeaters, a key device for entanglement distribution, based on growing multipartite states, allowing entanglement to be recycled and patched. All investigations have been carried out employing a novel formalism, also developed in this thesis, for efficiently and accurately simulating noisy quantum processes on classical hardware, with broader applicability beyond quantum communication.