## Profile
Entanglement, Superselection Rules and Quantum Resources
Quantum Communication and Computation
Foundations of Quantum Mechanics ## Outputs
Title | Category | Date | Authors |
Constructing monotones for quantum phase references in totally dephasing channels University of Calgary | Publication | 2011-01-01 | B. Toloui Semnani, G. Gour, B. C. Sanders | Superselection rule-resource theory in the presence of partial prior knowledgeThe study of new resource theories that arise from restrictions on possible quantum operations is turning into a very exciting field of research in quantum information theory. An interesting and remarkably fruitful approach has been to view the restrictions on quantum operations as coming from lack of access to classical reference frames. Superselection rules (SSR) are usually regarded to be axiomatic in nature. Surprisingly it turns out that the absence of reference frames can also give rise to superselection rules in a given situation. From a resource theory point of view this second outlook is much more productive. Each reference frame can be characterized by the group of its transformations. The corresponding superselection rule comes about by asking the allowed states and operations to remain invariant under the actions of this group. This is because such states and operations are all that can be prepared and implemented without the reference frame in question. Other states thus become resources in the presenceof the superselection rule [1].
Gour and Spekkens [2] have extensively studied the resources that arise from three types of superselection rules for pure unipartite quantum states: (1) Chirality or Z2-SSR, (2) phase reference or U(1)-SSR, and (3) (special cases of) Cartesian frames for spatial orientation or SU(2)-SSR. They have identified the form of allowed operations and the corresponding resources when no prior knowledge of the reference frame is assumed. They have found various relevant resource measures, the so called frameness measures, and developed their corresponding resource theories.
A very interesting generalization of the above results is to consider the more practical case where the parties already have some knowledge of the reference frame, and the way this partial knowledge modifies the resource theory in question. This is introduced in the formalism by using a non-uniform measure overall possible transformations of the reference frame.
Another challenging task is to broaden the scope ofthe theory to include mixed states. This is specially important since in real situations it is quite hard to prepare and work with pure states, and mixed states play a crucial role in almost all implementations of quantum information theory.
Finally investigating the combined situation of mixed state resources in presence of prior partial knowledge and the interplay between the two can lead to many stimulating results.
Our research shows that by restricting ourselves to pure states only in the cases studied, prior partial knowledge gives no new theory and leads to the identical set of resources as the case of completely unknown reference frame.
To include mixed states, we extend the notion of frameness of formation in an analogous way to the entanglement of formation, as follows: The frameness measure of pure states in a given decomposition of the state in question are averaged with their relative weights. Minimizing this average over all possible decompositions then gives the frameness of formation for that mixed state.
We have shown that a similar technique to Wootters’ regarding the entanglement of formation for bipartite states can be used to calculate the frameness of formation for a set of unipartite states in the Z2-SSR. This is a very interesting result since it shows that Wootters’ method also works in other resource theories, and is therefore more general than previously known.
[References:
[1] Reference frames, superselection rules, and quantum information, S. D. Bartlett, T. Rudolph,and R. W. Spekkens, Rev. Mod. Phys. 79, 555(2007)
[2] The resource theory of quantum reference frames: manipulations and monotones, G. Gourand R. W. Spekkens, New Journal of Physics 10(2008) 033023 ] University of Calgary | Presentation | 2008-08-20 | B. Toloui Semnani, G. Gour | Mixed state quantum reference frame resourcesSituations where the operations of a noisy channel used for the transmission and retrieval of quantum states belong to a specific group of transformations give rise to resources beside entanglement that allow us to overcome the ensuing constraints, such as when shared reference frames (RF) associated with symmetry groups are lacking between the nodes of a quantum channel. So far, most work on this new kind of resource, dubbed "frameness", has been focused on pure state transformations even though almost all states and operations in the lab involve some degree of mixedness. Here we address the problem of quantifying the frameness of mixed states. We introduce a new family of pure state frameness measures associated with Abelian Lie groups in a Hilbert space of arbitrary but finite dimensions, whose convex roof extensions remain monotonic. In particular, we show that this family of frameness monotones are closely related to generalized concurrence functions of the reduced density operators of entangled states. This highlights interesting and deep links between frameness and entanglement resource theories, and provides a new way of classifying all frameness monotones as functions of the "twirled" state that results from tracing out the RF, where the state plus the RF are treated as a joint entangled system. Finally, we use a member of this family of frameness monotones to determine the explicit analytical form of a qubit's frameness of formation. The frameness of formation denotes the minimum average cost of preparing the ensemble of pure states that realize a given mixed state, and can be used to quantify the frameness of that state under certain conditions. Our results thus extends Wootter's formula for the entanglement of formation of bipartite qubit states to a whole new and different class of resources. University of Calgary | Presentation | 2010-02-19 | B. Toloui Semnani, G. Gour, B. C. Sanders | Measures For Quantum Reference Frame Resources And Their Link To Entanglement Monotones University of Calgary | Presentation | 2011-07-06 | B. Toloui Semnani, G. Gour, B. C. Sanders | Simulating covariant transformations with local operationsWe show how quantum evolutions that are characterized by covariant transformations and restricted by superselection rules can be mapped to LOCC operations. We further show how measures of entanglement can be used to quantify the asymmetry, or frameness, of any state, pure or mixed.
Our results make it possible for the first time to construct a wide range of asymmetry monotones for general symmetry groups associated with different superselection rules, and highlights the deep links that exist between entanglement theory and the resource theories of asymmetry. University of Calgary | Presentation | 2012-06-08 | B. Toloui Semnani, G. Gour | Measuring asymmetry with entanglementWe demonstrate how G-covariant transformations can be simulated by LOCC operations. This technique allows for the asymmetry of states to be quantified via bipartite entanglement measures. University of Calgary | Presentation | 2012-05-02 | B. Toloui Semnani, G. Gour | Linking asymmetry of quantum states to entanglementQuantum evolutions that preserve a certain symmetry are expressed as covariant transformations. We show how covariant transformations can be simulated by local operations by embedding the system's Hilbert space in the tensor product of two Hilbert spaces. The embedding maps symmetric states to separable bipartite states in the larger Hilbert space and some asymmetric states to entangled states. We show how entanglement of the bipartite image state can be used to quantify the asymmetry of the original state. Our results make it possible for the first time to construct a wide range of asymmetry monotones for general symmetry groups associated with different superselection rules, and highlights the deep links that exist between entanglement theory and the resource theories of asymmetry. University of Calgary | Presentation | 2012-07-25 | B. Toloui Semnani, G. Gour | Constructing asymmetry monotones from entanglement monotonesWe show that any entanglement monotone for bipartite states can be turned into an `asymmetry' monotone, or a quantity that changes monotonically under dynamical time evolutions that preserve the system's phase symmetries. Asymmetry monotones hold information about how a system evolves under symmetry preserving transformations, and are important tools for the detailed study of a system's symmetry properties. Asymmetry monotones also quantify the ability of bounded-size quantum states to substitute for ideal external reference frames and are known as `frameness' monotones in this context. We introduce various new classes of asymmetry monotones both for pure and mixed states and investigate how their properties compare with known bipartite entanglement measures. University of Calgary | Presentation | 2012-06-12 | B. Toloui Semnani, G. Gour | Frameness of formation for a qubitAlmost all states and operations in the lab involve some degree of mixedness, so it is necessary to extend the results of the newly developed reference frame resource theories to include mixed states. We produce, for the first time, explicit results for a qubit's frameness of formation. The frameness of formation denotes the average resource cost of generating a mixed state. This cost is measured in terms of standard resource states, called refbits, that are chosen as units of frameness. In order to determine the exact value of this frameness measure, we develop a novel technique that generalizes Wootter's idea for entanglement of formation to a wide class of reference frame resource theories. We introduce the "concurrence of frameness" as a generalization of the concurrence measure to the case of reference frames. The concurrence of a resource state is explicitly determined, and the cost of preparing a resource is expressed as a simple function of this concurrence. This approach is applicable to resource measures of any given group of transformations associated with a superselection rule, as long as the related resource cost can be written as an explicit function of the concurrence of frameness. Finally, we demonstrate the application of our result to the resource theories of the groups Z_2 and U(1) that are associated with chiral and phase reference frames respectively. University of Calgary | Presentation | 2009-08-23 | B. Toloui Semnani, G. Gour, B. C. Sanders | Quantum frameness for CPT inversion symmetry University of Calgary | Publication | 2013-01-01 | M. Skotiniotis, B. Toloui Semnani, I. T. Durham, B. C. Sanders | Quantum frameness for charge-parity-time inversion symmetry University of Calgary | Publication | 2013-01-01 | M. Skotiniotis, B. Toloui Semnani, I. Durham, B. C. Sanders |
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