10 research outputs found
Equivalence classes of non-local unitary operations
We study when a multipartite non--local unitary operation can
deterministically or probabilistically simulate another one when local
operations of a certain kind -in some cases including also classical
communication- are allowed. In the case of probabilistic simulation and
allowing for arbitrary local operations, we provide necessary and sufficient
conditions for the simulation to be possible. Deterministic and probabilistic
interconversion under certain kinds of local operations are used to define
equivalence relations between gates. In the probabilistic, bipartite case this
induces a finite number of classes. In multiqubit systems, however, two unitary
operations typically cannot simulate each other with non-zero probability of
success. We also show which kind of entanglement can be created by a given
non--local unitary operation and generalize our results to arbitrary operators.Comment: (1) 9 pages, no figures, submitted to QIC; (2) reference added, minor
change
Optimal conversion of non--local unitary operations
We study when a non--local unitary operation acting on two --level systems
can probabilistically simulate another one when arbitrary local operations and
classical communication are allowed. We provide necessary and sufficient
conditions for the simulation to be possible. Probabilistic interconvertability
is used to define an equivalence relation between gates. We show that this
relation induces a finite number of classes, that we identify. In the case of
two qubits, two classes of non--local operations exist. We choose the CNOT and
SWAP as representatives of these classes. We show how the CNOT [SWAP] can be
deterministically converted into any operation of its class. We also calculate
the optimal probability of obtaining the CNOT [SWAP] from any operation of the
corresponding class and provide a protocol to achieve this task.Comment: 4 pages, no figure
Optimal Entanglement Generation from Quantum Operations
We consider how much entanglement can be produced by a non-local two-qubit
unitary operation, - the entangling capacity of . For a single
application of , with no ancillas, we find the entangling capacity and
show that it generally helps to act with on an entangled state.
Allowing ancillas, we present numerical results from which we can conclude,
quite generally, that allowing initial entanglement typically increases the
optimal capacity in this case as well. Next, we show that allowing collective
processing does not increase the entangling capacity if initial entanglement is
allowed.Comment: v1.0 15 pages, 3 figures, written in revtex4. v2.0 References
updated. Submitted to Phys. Rev. A v3.0 16 pages, 4 figures. Expanded
explanation in section 3A, figures corrected and made clearer. Definition of
entangling capacity in section 4 made explicit. Other minor typos correcte
GHZ extraction yield for multipartite stabilizer states
Let be an arbitrary stabilizer state distributed between three
remote parties, such that each party holds several qubits. Let be a
stabilizer group of . We show that can be converted by local
unitaries into a collection of singlets, GHZ states, and local one-qubit
states. The numbers of singlets and GHZs are determined by dimensions of
certain subgroups of . For an arbitrary number of parties we find a
formula for the maximal number of -partite GHZ states that can be extracted
from by local unitaries. A connection with earlier introduced measures
of multipartite correlations is made. An example of an undecomposable
four-party stabilizer state with more than one qubit per party is given. These
results are derived from a general theoretical framework that allows one to
study interconversion of multipartite stabilizer states by local Clifford group
operators. As a simple application, we study three-party entanglement in
two-dimensional lattice models that can be exactly solved by the stabilizer
formalism.Comment: 12 pages, 1 figur
Entrelazamiento como recurso para tareas de información
[ES] En una carta datada en 1947 Albert Einstein definió el entrelazamiento como ”acción fantasmal a distancia''. A lo largo de los años esta propiedad insólita y aterradora para algunos, se ha convertido en una gran oportunidad para otros. Gracias al carácter del entrelazamiento, el paradigma de los procesos de tratamiento de información puede llegar a cambiar de forma radical, llegando incluso a producir una revolución en la información y la comunicación. El presente trabajo pretende identificar algunas de las propiedades que describe el entrelazamiento y su posible uso como recurso en tareas de información
Entrelazamiento como recurso para tareas de información
[ES] En una carta datada en 1947 Albert Einstein definió el entrelazamiento como ”acción fantasmal a distancia''. A lo largo de los años esta propiedad insólita y aterradora para algunos, se ha convertido en una gran oportunidad para otros. Gracias al carácter del entrelazamiento, el paradigma de los procesos de tratamiento de información puede llegar a cambiar de forma radical, llegando incluso a producir una revolución en la información y la comunicación. El presente trabajo pretende identificar algunas de las propiedades que describe el entrelazamiento y su posible uso como recurso en tareas de información
Quantum multipartite entangled states, classical and quantum error correction
Studying entanglement is essential for our understanding of such diverse areas as high-energy physics, condensed matter physics, and quantum optics. Moreover, entanglement allows us to surpass classical physics and technologies enabling better information processing, computation, and improved metrology. Recently, entanglement also played a prominent role in characterizing and simulating quantum many-body states and in this way deepened our understanding of quantum matter. While bipartite entanglement is well understood, multipartite entanglement is much richer and leads to stronger contradictions with classical physics. Among all possible entangled states, a special class of states has attracted attention for a wide range of tasks. These states are called k-uniform states and are pure multipartite quantum states of n parties and local dimension q with the property that all of their reductions to k parties are maximally mixed. Operationally, in a k-uniform state any subset of at most k parties is maximally entangled with the rest. The k = bn/2c-uniform states are called absolutely maximally entangled because they are maximally entangled along any splitting of the n parties into two groups. These states find applications in several protocols and, in particular, are the building blocks of quantum error correcting codes with a holographic geometry, which has provided valuable insight into the connections between quantum information theory and conformal field theory. Their properties and the applications are however intriguing, as we know little about them: when they exist, how to construct them, how they relate to other multipartite entangled states, such as graph states, or how they connect under local operations and classical communication. With this motivation in mind, in this thesis we first study the properties of k-uniform states and then present systematic methods to construct closed-form expressions of them. The structure of our methods proves to be particularly fruitful in understanding the structure of these quantum states, their graph-state representation and classification under local operations and classical communication. We also construct several examples of absolutely maximally entangled states whose existence was open so far. Finally, we explore a new family of quantum error correcting codes that generalize and improve the link between classical error correcting codes, multipartite entangled states, and the stabilizer formalism. The results of this thesis can have a role in characterizing and studying the following three topics: multipartite entanglement, classical error correcting codes and quantum error correcting codes. The multipartite entangled states can provide a link to find different resources for quantum information processing tasks and quantify entanglement. Constructing two sets of highly entangled multipartite states, it is important to know if they are equivalent under local operations and classical communication. By understanding which states belong to the same class of quantum resource, one may discuss the role they play in some certain quantum information tasks like quantum key distribution, teleportation and constructing optimum quantum error correcting codes. They can also be used to explore the connection between the Antide Sitter/Conformal Field Theory holographic correspondence and quantum error correction, which will then allow us to construct better quantum error correcting codes. At the same time, their roles in the characterization of quantum networks will be essential to design functional networks, robust against losses and local noise.El estudio del entrelazamiento cuántico es esencial para la comprensión de diversas áreas como la óptica cuántica, la materia condensada e incluso la física de altas energías. Además, el entrelazamiento nos permite superar la física y tecnologías clásicas llevando a una mejora en el procesado de la información, la computación y la metrología. Recientemente se ha descubierto que el entrelazamiento desarrolla un papel central en la caracterización y simulación de sistemas cuánticos de muchos cuerpos, de esta manera facilitando nuestra comprensión de la materia cuántica. Mientras que se tiene un buen conocimiento del entrelazamiento en estados puros bipartitos, nuestra comprensión del caso de muchas partes es mucho más limitada, a pesar de que sea un escenario más rico y que presenta un contraste más fuerte con la física clásica. De entre todos los posibles estados entrelazados, una clase especial ha llamado la atención por su amplia gama de aplicaciones. Estos estados se llaman k-uniformes y son los estados multipartitos de n cuerpos con dimensión local q con la propiedad de que todas las reducciones a k cuerpos son máximamente desordenadas. Operacionalmente, en un estado k-uniforme cualquier subconjunto de hasta k cuerpos está máximamente entrelazado con el resto. Los estados k = n/2 -uniformes se llaman estados absolutamente máximamente entrelazados porque son máximamente entrelazados respecto a cualquier partición de los n cuerpos en dos grupos. Estos estados encuentran aplicaciones en varios protocolos y, en particular, forman los elementos de base para la construcción de los códigos de corrección de errores cuánticos con geometría holográfica, los cuales han aportado intuición importante sobre la conexión entre la teoría de la información cuántica y la teoría conforme de campos. Las propiedades y aplicaciones de estos estados son intrigantes porque conocemos poco sobre las mismas: cuándo existen, cómo construirlos, cómo se relacionan con otros estados con entrelazamiento multipartito, cómo los estados grafo, o como se relacionan mediante operaciones locales y comunicación clásica.
Con esta motivación en mente, en esta tesis primero estudiamos las propiedades de los estados k-uniformes y luego presentamos métodos sistemáticos para construir expresiones cerradas de los mismos. La naturaleza de nuestros métodos resulta ser muy útil para entender la estructura de estos estados cuánticos, su representación como estados grafo y su clasificación bajo operaciones locales y comunicación clásica. También construimos varios ejemplos de estados absolutamente máximamente entrelazados, cuya existencia era desconocida. Finalmente, exploramos una nueva familia de códigos de corrección de errores cuánticos que generalizan y mejoran la conexión entre los códigos de corrección de errores clásicos, los estados entrelazados multipartitos y el formalismo de estabilizadores.
Los resultados de esta tesis pueden desarrollar un papel importante en la caracterización y el estudio de las tres siguientes áreas: entrelazamiento multipartito, códigos de corrección de errores clásicos y códigos de corrección de errores cuánticos. Los estados de entrelazamiento multipartito pueden aportar una conexión para encontrar diferentes recursos para tareas de procesamiento de la información cuántica y cuantificación del entrelazamiento. Al construir dos conjuntos de estados multipartitos altamente entrelazados, es importante saber si son equivalentes entre operaciones locales y comunicación clásica. Entendiendo qué estados pertenecen a la misma clase de recurso cuántico, se puede discutir qué papel desempeñan en ciertas tareas de información cuántica, como la distribución de claves criptográficas cuánticas, la teleportación y la construcción de códigos de corrección de errores cuánticos óptimos.
También se pueden usar para explorar la conexión entre la correspondencia holográfica Anti-de Sitter/Conformal Field Theory y códigos de corrección de errores cuánticos, que nos permitiría construir mejores códigos de corrección de errores. A la vez, su papel en la caracterización de redes cuánticas será esencial en el diseño de redes funcionales, robustas ante pérdidas y ruidos locales
A study of entanglement in quantum information theory
Although the concept of quantum entanglement has been known for about seventy years, it only recently quit the realms of meta-theoretical discussions when it was discovered how entanglement can be exploited to compute and communicate with an unprecedented power. The primary motivation of the work presented in this thesis has been to contribute to the big effort that has been done during the last decade to understand and quantify quantum en- tanglement. We have developed advanced techniques of linear and multilinear algebra to investigate and classify entangled pure and mixed quantum states, and discussed some novel applications in the field of quantum information theory.
The results presented in this thesis are mainly of interest from a fundamental point a view: entanglement is the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought [186]. It is however a real privilege that fundamental research in quantum information theory bears the tools of tomorrow’s electrical engineers: the ongoing minia- turization of electronic components will soon reach a scale where quantum mechanical effects play a major role.
The first part of this thesis is devoted to the study of entanglement. Local equivalence classes of multipartite pure and mixed quantum systems are dis- cussed, and different entanglement measures are introduced and compared. The second part is mainly concerned with the problem of transmission and extraction of classical and quantum information through quantum channels. Optimal detection strategies for continuously monitored systems are derived, and we exploit a duality between quantum maps and entangled quantum states to present a unified description of quantum channels