Remote Preparation and Distribution of Bipartite Entangled States University of California, San Diego, University of Calgary | Publication | 2004-12-01 | G. Gour, B. C. Sanders |

Reexamination of entanglement of superpositions University of Calgary | Publication | 2007-11-01 | G. Gour |

Deterministic entanglement of assistance and monogamy constraints University of California, San Diego, University of Calgary | Publication | 2005-10-01 | G. Gour, D. A. Meyer, B. C. Sanders |

Entanglement of assistance is not a bipartite measure nor a tripartite monotone University of California, San Diego | Publication | 2006-06-01 | G. Gour, R. W. Spekkens |

Questions of stability near black hole critical points University of Alberta | Publication | 2003-09-01 | G. Gour, A. J. Medved |

Algebraic approach to quantum black holes: Logarithmic corrections to black hole entropy University of Alberta | Publication | 2002-11-01 | G. Gour |

Why is the black hole entropy (almost) linear in the horizon area? University of Calgary | Publication | 2001-02-01 | G. Gour, A. E. Mayo |

Schwarzschild black hole as a grand canonical ensemble University of Calgary | Publication | 1999-12-01 | G. Gour |

Quantum mechanics of a black hole University of Calgary | Publication | 2000-05-01 | G. Gour |

The resource theory of quantum reference frames: manipulations and monotones University of Calgary | Publication | 2008-03-01 | G. Gour, R. W. Spekkens |

Evolution and Symmetry of Multipartite Entanglement University of Calgary | Publication | 2010-11-01 | G. Gour |

Necessary and sufficient conditions for local manipulation of multipartite pure quantum states University of Calgary | Publication | 2011-07-01 | G. Gour, N. R. Wallach |

The Minimum Entropy Output of a Quantum Channel Is Locally Additive University of California, San Diego, University of Calgary | Publication | 2013-01-01 | G. Gour, S. Friedland |

Construction of all general symmetric informationally complete measurements University of Calgary | Publication | 2014-08-01 | G. Gour, A. Kalev |

Classification of multipartite entanglement of all finite dimensionality University of California, San Diego, University of Calgary | Publication | 2013-01-01 | G. Gour, N. R. Wallach |

Entanglement of subspaces in terms of entanglement of superpositions University of Calgary | Publication | 2008-01-01 | G. Gour, A. Roy |

Duality for monogamy of entanglement University of Calgary | Publication | 2007-01-01 | G. Gour, S. Bandyopadhyay, B. C. Sanders |

Entanglement of collaboration University of California, San Diego | Publication | 2006-01-01 | G. Gour |

Entanglement in quantum informationQuantum information science is concerned with the manipulation, computation and communication of information, where the information is encoded in two (or more) level quantum systems called "qubits", unlike classical information, which is encoded in Boolean "bits". The devices used in this science are governed by the principles of quantum mechanics, which opens the possibility for a large range of applications. In this colloquium talk I will give a gentle review of this exciting field with the focus on the fascinating role entanglement plays in quantum information. In particular, I will start with a brief history of the discovery of non-intuitive quantum correlations (i.e. entanglement) and then show that quantum entanglement, besides being of interest from a fundamental point of view, is a valuable resource for many quantum information tasks such as quantum teleportation and superdense coding. University of Calgary | Presentation | 2007-03-22 | G. Gour |

From heisenberg uncertainty principle to the theory of majorization University of California, San Diego, University of Calgary | Presentation | 2013-06-04 | G. Gour |

From Heisenberg Uncertainty Principle to the theory of majorization University of Calgary | Presentation | 2013-12-22 | G. Gour |

From the Heisenberg uncertainty principle to the theory of majorization and quantum information University of Calgary | Presentation | 2014-03-27 | G. Gour |

Quantum resource theories and super selection rulesIn quantum information theory entanglement arises due to the restriction to local operations and classical communication (LOCC). In particular, entanglement can be considered as a quantum resource with which spatially separated parties can overcome or at least partly overcome the limitation of LOCC. Clearly, different types of restrictions corresponds to different kinds of quantum resource theories (QRTs). In this talk I will discuss the QRTs that emanate from various natural constraints. I will focus on QRTs that follow from the presence of super-selection rules or the absence of shared reference frames. In particular, I will discuss the analogies and distinctions between and among the different QRTs and show that, in general, QRTs in many aspects are very similar to entanglement theory. Such comparisons provide a much broader perspective on all of these resource theories and allow us to use the insights gained from one QRT to solve the problems that arise in the context of another QRT.
University of Calgary | Presentation | 2007-06-02 | G. Gour |

Polygamy of entanglement of assistance: duality for monogamy of entanglementIn contrast to classical multi-partite systems, which can enjoy arbitrary correlations between components, shared entanglement is restricted in a multipartite system. In this talk I will introduce a duality for monogamy of entanglement: whereas monogamy of entanglement inequalities provide an upper bound for bipartite sharability of entanglement in a multipartite system, I will show that the same quantity provides a lower bound for distribution of bipartite entanglement in a multipartite system. I will then show that our results for monogamy of entanglement can be used to establish relations between bipartite entanglement that separate one qubit from the rest vs separating two qubits from the rest. University of Calgary | Presentation | 2008-02-17 | G. Gour |

A few open problems in quantum informationQuantum information science, an interface area of mathematics,physics and computing science, is concerned with the manipulation, computation and communication of information, where the information is encoded in two (or more) level quantum systems called "qubits", unlike classical information, which is encoded in Boolean "bits". The devices used in this science are governed by the principles of quantum mechanics, which opens the possibility for a large range of applications. In this talk I will discuss several open problems in the field, focusing on the long standing additivity conjecture that the minimum entropy output of a completely positive trace preserving linear map, as measured using the von Neumann entropy, is additive under taking tensor products. Despite the enormous effort by the most experts in the field during the last 12 years, this problem remained unsolved. Here I will present some recent progress and future directions. This talk is based on a joint work with N. R. Wallach and A. Roy. University of Calgary | Presentation | 2008-05-02 | G. Gour |

Resource theories, SU(2) super selection rule, and time inversion University of Calgary | Presentation | 2008-08-24 | G. Gour, W. R. Spekkens, B. C. Sanders, P. Turner |

Quantum resource theoriesQuantum information theory can be viewed as a theory of inter-conversions among different resources. These resources are diversely classified as quantum or classical, static or dynamic, noisy or noiseless, and therefore enable plethora of quantum information processing tasks. In this talk I will discuss the characterization, manipulation and quantification of quantum resources. I will focus mostly on the resource theories that follow from the presence of super-selection rules or the absence of shared reference frames. In particular, I will compare and discuss measures of quantum resources, such as the relative entropy of a resource, and show that these measures get different operational interpretations in different resource theories. Such comparisons provide a much broader perspective on all of these resource theories and allow us to use the insights gained from one theory to solve the problems that arise in the context of another resource theory. University of Calgary | Presentation | 2010-01-12 | G. Gour |

Entanglement quantification and manipulation University of Calgary | Presentation | 2010-01-11 | G. Gour |

Basic quantum information protocols University of Calgary | Presentation | 2010-01-11 | G. Gour |

Quantum channels University of Calgary | Presentation | 2010-01-12 | G. Gour |

Multipartite Entanglement: Classification, Quantification, Manipulation, Evolution and Applications Exotic multipartite entangled states plays an important role in a variety of quantum information processing tasks such as conventional and measurement-based quantum computation, quantum error correction schemes, quantum secret sharing, quantum simulations, and in principle in the description of every composite system consisting of more than one subsystem. The amount of information needed to describe N-party quantum system grows exponentially with N, which makes it very difficult and almost impossible to classify multipartite entangled states. In this talk I will show that a new formalism based on the stabilizer group of a given multipartite state, not only makes it possible to classify and quantify the amount of entanglement in multipartite states, but also describes fully the manipulation of multipartite entanglement under separable operations. In particular, I will introduce necessary and sufficient conditions to transform one pure multipartite state to another multipartite state via separable operations. In addition, I will discuss the evolution of multipartite entanglement under noise and decoherence, and its quantification in terms of SL-invariant polynomials. I will end with few applications to quantum secret sharing. Some of the work presented here is based on a joint work with Nolan Wallach. University of Calgary | Presentation | 2011-04-29 | G. Gour |

Quantum multipartite entanglementExotic multipartite entangled states plays an important role in a variety of quantum information processing tasks such as conventional and measurement-based quantum computation, quantum error correction schemes, quantum secret sharing, quantum simulations, and in principle in the description of every composite system consisting of more than one subsystem. The amount of information needed to describe N-party quantum system grows exponentially with N, which makes it very difficult and almost impossible to classify multipartite entangled states. In this talk I will discuss my recent work with Nolan Wallach on this subject. I will show that a new formalism based on the stabilizer group of a given multipartite state, not only makes it possible to classify and quantify the amount of entanglement in multipartite states, but also describes fully the manipulation of multipartite entanglement under separable operations. In particular, I will introduce necessary and sufficient conditions to transform one pure multipartite state to another multipartite state via separable operations. In addition, I will discuss the evolution of multipartite entanglement under noise and decoherence, and its quantification in terms of SL-invariant polynomials. I will end with few applications to quantum secret sharing. University of Calgary | Presentation | 2011-03-20 | G. Gour |

Local additivity of the minimum entropy output of a quantum channelIn this talk I will show that the minimum von-Neumann entropy output of a quantum channel is locally additive. Hasting's counterexample for the global additivity conjecture, makes this result somewhat surprising. In particular, it indicates that the non-additivity of the minimum entropy output is related to a global effect of quantum channels. I will end with few related open problems. University of Calgary | Presentation | 2012-02-17 | G. Gour |

Local additivity of the minimum entropy output of a quantum channelOne of the major open problems in quantum information concerns
with the question whether entanglement between signal states can help to
send classical information on quantum channels. Recently, Hasting proved
that entanglement does help by finding a counter-example for the long
standing additivity conjecture that the minimum von-Neumann entropy
output of a quantum channel is additive under taking tensor products.
In this talk I will show that the minimum von-Neumann entropy output of
a quantum channel is locally additive. Hasting's counterexample for the
global additivity conjecture, makes this result somewhat surprising. In
particular, it indicates that the non-additivity of the minimum entropy
output is related to a global effect of quantum channels. I will end
with few related open problems. University of Calgary | Presentation | 2012-04-27 | G. Gour |

Closed formula for the relative entropy of entanglementA quantum state (positive semi-definite matrix) acting on a tensor product of two Hilbert spaces, is called a product state if it can be written as a tensor product of two quantum states. A separable state is a convex combination of product states, that describes a composite physical system with no quantum entanglement. Entanglement of a non-separable (i.e. entangled) quantum state is measured by the relative entropy "distance" of the state to the convex set of separable states. This distance is therefore called the relative entropy of entanglement (REE). Since it is NP hard to determine whether a quantum state is separable or not, the convex optimization problem posed by the REE can not be solved analytically. However, in this talk, I will show that a closed formula exists for the inverse problem. That is, for a quantum state on the boundary of the set of separable states, there is a closed formula for all the entangled state for which this state is the closest separable state (CSS). In addition I will show that if an entangled state is full rank, then its CSS is unique. My talk is based on a joint work with Shmuel Friedland. University of Calgary | Presentation | 2012-06-03 | G. Gour, S. Friedland |

Towards a complete classification of multipartite entanglementMulti-particle entanglement is an essential resource for a variety of quantum information processing tasks. Yet, despite an enormous amount of literature dedicated to its study, our current understanding of it is still in its infancy. In this talk I will introduce a
systematic classification of multi particle entanglement in terms of equivalence classes of states under stochastic local operations and classical communication (SLOCC). I will show that such an SLOCC equivalency class of states is characterized by ratios of homogenous polynomials that are invariant under local action of the special linear group. I will then introduce a complete construction of the set of all such SL-invariant polynomials (SLIPs).
The construction is based on Schur-Weyl duality and applies to any number of qudits in all (finite) dimensions. In addition, I will introduce an elegant formula for the dimension of the
homogenous SLIPs space of a fixed degree as a function of the number of qudits. The expressions for the SLIPs involve in general many terms, but for the case of qubits can be
written in a much simpler form. University of Calgary | Presentation | 2013-08-27 | G. Gour |

Universal uncertainty relationsUncertainty relations lie at the core of quantum mechanics and are a direct manifestation of the non-commutative structure of the theory. They impose intrinsic limitations on the precision with which physical properties can be simultaneously determined.
The modern work on uncertainty relations employs entropic measures to quantify the lack of knowledge associated with measuring non-commuting observables. However there is no
fundamental reason for using entropies as quantifiers; any functional relation that characterizes the uncertainty of the measurement outcomes defines an uncertainty relation.Starting from a simple requirement any reasonable measure of uncertainty has to satisfy, we show that Schur-concave functions are the most general uncertainty quantifiers. We then discover a novel fine-grained uncertainty relation written in terms of a majorization relation, which generates an infinite family of distinct scalar uncertainty relations via the application of uncertainty quantifiers. Our relation is universally valid and captures the essence of
uncertainty in quantum mechanics. University of Calgary | Presentation | 2013-07-02 | G. Gour |

Universal uncertainty relationsUncertainty relations lie at the core of quantum mechanics and are a direct manifestation of the non-commutative structure of the theory. They impose intrinsic limitations on the precision with which physical properties can be simultaneously determined.
The modern work on uncertainty relations employs entropic measures to quantify the lack of knowledge associated with measuring non-commuting observables. However there is no
fundamental reason for using entropies as quantifiers; any functional relation that characterizes the uncertainty of the measurement outcomes defines an uncertainty relation.
Starting from a simple requirement any reasonable measure of uncertainty has to satisfy, we show that Schur-concave functions are the most general uncertainty quantifiers. We then
discover a novel fine-grained uncertainty relation written in terms of a majorization relation, which generates an infinite family of distinct scalar uncertainty relations via the application of
uncertainty quantifiers. Our relation is universally valid and captures the essence of uncertainty in quantum mechanics. University of Calgary | Presentation | 2013-08-09 | G. Gour |

Towards a complete classification of multipartite entanglementMulti-particle entanglement is an essential resource for a variety of quantum information processing tasks. Yet, despite an enormous amount of literature dedicated to its study, our current understanding of it is still in its infancy. In this talk I will introduce a systematic classification of multiparticle entanglement in terms of equivalence classes of states under stochastic local operations and classical communication (SLOCC). I will show that such an SLOCC equivalency class of states is characterized by ratios of homogenous polynomials that are invariant under local action of the special linear group. I will then introduce a complete construction of the set of all such SL-invariant polynomials (SLIPs). The construction is based on Schur-Weyl duality and applies to any number of qudits in all (finite) dimensions. In addition, I will introduce an elegant formula for the dimension of the homogenous SLIPs space of a fixed degree as a function of the number of qudits. The expressions for the SLIPs involve in general many terms, but for the case of qubits can be written in a much simpler form. University of Calgary | Presentation | 2013-12-08 | G. Gour |

Universal uncertainty relations Uncertainty relations are a distinctive characteristic of quantum theory that imposes intrinsic limitations on the precision with which physical properties can be simultaneously determined. The modern work on uncertainty relations employs entropic measures to quantify the lack of knowledge associated with measuring non-commuting observables. However, I will show here that there is no fundamental reason for using entropies as quantifiers; in fact, any functional relation that characterizes the uncertainty of the measurement outcomes can be used to define an uncertainty relation. Starting from a simple assumption that any measure of uncertainty is non-decreasing under mere relabeling of the measurement outcomes, I will show that Schur-concave functions are the most general uncertainty quantifiers. I will then introduce a novel fine-grained uncertainty relation written in terms of a majorization relation, which generates an infinite family of distinct scalar uncertainty relations via the application of arbitrary measures of uncertainty. This infinite family of uncertainty relations includes all the known entropic uncertainty relations, but is not limited to them. In this sense, the relation is universally valid and captures the essence of the uncertainty principle in quantum theory. This talk is based on a joint work with Shmuel Friedland and Vlad Gheorghiu. University of Calgary | Presentation | 2014-03-03 | G. Gour |

Hiding entanglement in classical bitsEntanglement and in particular bipartite entanglement is the key
resource for many
quantum information processing tasks. Thus, it is extremely important to
find secure
ways to distribute entanglement among distant parties. In this talk I
will discuss several
methods to hide entanglement in classical bits and focus on the hidden
entanglement,
the entanglement of assistance, the localizable entanglement and the
entanglement of
collaboration. In particular, I will show that one classical bit can
unlock at most one ebit.
University of Calgary | Presentation | 2006-10-27 | G. Gour |

Quantum resource theories and super selection rulesIn quantum information theory entanglement arises due to the restriction to local operations and classical communication (LOCC). In particular, entanglement can be considered as a quantum resource with which spatially separated parties can overcome or at least partly overcome the limitation of LOCC. Clearly, different types of restrictions corresponds to different kinds of quantum resource theories (QRTs). In this talk I will discuss the QRTs that emanate from various natural constraints. I will focus on QRTs that follow from the presence of super-selection rules or the absence of shared reference frames. In particular, I will discuss the analogies and distinctions between and among the different QRTs and show that, in general, QRTs in many aspects are very similar to entanglement theory. Such comparisons provide a much broader perspective on all of these resource theories and allow us to use the insights gained from one QRT to solve the problems that arise in t he context of another QRT. University of Calgary | Presentation | 2007-02-18 | G. Gour |

Entanglement in Quantum Information Quantum information science is concerned with the manipulation, computation and communication of information, where the information is encoded in two (or more) level quantum systems called "qubits", unlike classical information, which is encoded in Boolean "bits". The devices used in this science are governed by the principles of quantum mechanics, which opens the possibility for a large range of applications. In this colloquium talk I will give a gentle review of this exciting field with the focus on the fascinating role entanglement plays in quantum information. In particular, I will start with a brief history of the discovery of non-intuitive quantum correlations (i.e. entanglement) and then show that quantum entanglement, besides being of interest from a fundamental point of view, is a valuable resource for many quantum information tasks such as quantum teleportation and superdense coding. University of Calgary | Presentation | 2007-05-04 | G. Gour |

On Symmetric SL-Invariant Polynomials in Four Qubits University of Calgary | Publication | 2014-05-01 | G. Gour, N. Wallach |

The resource theory of informational nonequilibrium in thermodynamics University of Calgary | Publication | 2013-12-01 | G. Gour, M. Müller, V. Narasimhachar, R. Spekkens, N. Halpern |

All maximally entangled four qubit states University of Calgary | Publication | 2010-11-01 | G. Gour, N. Wallach |

Measuring the quality of a quantum reference frame: the relative entropy of frameness University of Calgary | Publication | 2009-01-01 | G. Gour, I. Marvian, R. Spekkens |

Polygamy of distributed entanglement University of Calgary | Publication | 2009-07-01 | F. Buscemi, G. Gour, J. Kim |

Building blocks of a black hole University of Calgary | Publication | 2002-06-01 | J. D. Bekenstein, G. Gour |

Necessary conditions for entanglement catalysts University of Calgary | Publication | 2009-05-01 | Y. R. Sanders, G. Gour |

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 |

An explicit expression for the relative entropy of entanglement in all dimensions University of Calgary | Publication | 2011-01-01 | S. Friedland, G. Gour |

Reducing the Quantum Communication Cost of Quantum Secret Sharing University of Calgary, University of California, San Diego | Publication | 2012-10-01 | B. Fortescue, G. Gour |

Limitations to sharing entanglement University of Calgary | Publication | 2012-01-01 | J. Kim, G. Gour, B. C. Sanders |

Multipartite entanglement evolution under separable operations University of Calgary, University of California, San Diego | Publication | 2012-11-01 | V. Gheorghiu, G. Gour |

Alignment of reference frames and an operational interpretation for the G -asymmetry University of Calgary | Publication | 2012-07-01 | M. Skotiniotis, G. Gour |

Simulating symmetric time evolution with local operations University of California, San Diego, University of Calgary | Publication | 2012-12-01 | B. Toloui, G. Gour |

Mutually unbiased measurements in finite dimensions University of Calgary | Publication | 2014-05-01 | A. Kalev, G. Gour |

Phase-asymmetry resource interconversion via estimation University of Calgary | Publication | 2014-03-01 | V. Narasimhachar, G. Gour |

Alignment of reference frames and an operational interpretation for the G-asymmetry University of Calgary | Publication | 2012-01-01 | M. Skotiniotis, G. Gour |

Closed formula for the relative entropy of entanglement in all dimensions University of Calgary | Publication | 2011-05-01 | S. Friedland, G. Gour |

Polygamy of distributed entanglement University of Calgary | Publication | 2009-01-01 | F. Buscemi, G. Gour, J. Kim |

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 |

A contextual toy model University of Calgary | Presentation | 2008-08-21 | M. Skotiniotis, G. Gour, A. Roy, B. C. Sanders |

Concurrence monotones as conditions for entanglement catalysis University of Calgary | Presentation | 2008-08-21 | Y. Sanders, 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 |

Going beyond the share size bound in quantum secret sharingQuantum secret sharing (QSS) is an important cryptographic protocol which allows a quantum secret to be split between multiple "players", such that only certain authorised player subsets may recover the secret. It is, however, costly in terms of quantum communication and storage; perfect QSS using quantum states requires every player's share to be at least as large as the original secret.
I will discuss some of our recent results in which we improve upon this bound through the use of imperfect "ramp" secret sharing, which allows for smaller shares at the cost of weaker security. We find a specific class of "entanglement sharing" ramp protocols, which allow for smaller shares while still broadly restricting the protocols' information leakage.
Finally I will demonstrate how, by incorporating classical encryption into "hybrid" QSS protocols, quantum share size can be reduced (sometimes
drastically) without requiring any reduction in security.
University of Calgary | Presentation | 2011-06-28 | B. Fortescue, 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 |

Classical capacity of unspeakable phase information via quantum systemsReference frame alignment, such as synchronization of clocks or alignment of an orthogonal triple of axis, is the communication of unspeakable information and requires physical systems with particular degrees of freedom, such as optical phase or angular momentum. An important property of alignment protocols is the ability to reliably and efficiently transmit such unspeakable information. We derive the optimal rate of transmission of information for the case where two parties wish to align their respective local phase references via the exchange of photons. The rate of transmission depends on the variance of the quantum mechanical state used to encode phase information, and can be achieved by performing collective measurements on N photons.
University of Calgary | Presentation | 2011-07-06 | M. Skotiniotis, G. Gour |

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 |

Transformation vs. estimation strategies in resource theories of asymmetry University of Calgary | Presentation | 2013-07-02 | V. Narasimhachar, G. Gour |

The resource theory of quantum reference frames University of Calgary | Presentation | 2008-08-24 | W. R. Spekkens, G. Gour |

Polygamy of distributed entanglement University of Calgary | Presentation | 2009-06-03 | F. Buscemi, G. Gour, J. Kim |

Polygamy of distributed entanglement University of Calgary | Presentation | 2009-08-20 | F. Buscemi, G. Gour, J. Kim |

Polygamy of distributed entanglement University of Calgary | Presentation | 2009-08-26 | F. Buscemi, G. Gour, J. Kim |

Operational interpretation of the G-asymmetry for Abelian groupsIn a reference frame alignment protocol the sender, Alice, prepares a quantum system in a state |psi>, that serves as a token of her reference frame, and sends this system to a receiver, Bob, who performs a measurement and learns about the reference frame. We derive the state and measurement that maximize the accessible information in a reference frame alignment protocol. We show that in the limit where a large number of systems are sent, the accessible information per copy equals the Holevo bound. The latter was shown to be equal to the relative entropy of frameness, or G-asymmetry, of the state |psi>, a measure of resourcefulness analogous to the relative entropy of entanglement. We show that for a reference frame alignment protocol, associated with a finite abelian group, Z_N, or the continuous group U(1), associated with the important case of photon number super-selection, the rate of accessible information is quantified by the linearized, regularized G-asymmetry. Our result provides an information theoretic operational interpretation for the G-asymmetry that has been thus far lacking. University of Calgary | Presentation | 2012-02-27 | M. Skotiniotis, G. Gour |

Quantum entanglement: properties and evolutionEntanglement is a key ingredient in a quantum computer, allowing for quantum
algorithms that perform exponentially faster than any classical counterpart, such
as factoring large numbers. In this talk I will gently introduce the concept of entanglement as a notion of quantum correlations and discuss some of its properties. I will then show how to explicitly quantify the ``decay" of entanglement due
external noise during a physical process (aka decoherence). The talk is intended
to be self-contained and no prior exposure to quantum mechanics is required. University of Calgary | Presentation | 2012-05-04 | V. Gheorghiu, G. Gour |

Operational Interpretation of the G-asymmetry for Abelian groupsWe determine the quantum states and measurements that optimize the accessible information in a reference frame alignment protocol associated with the groups $U(1)$, corresponding to a phase reference, and $\mathbb{Z}_M$, the cyclic group of $M$ elements. Our result provides
an operational interpretation for the $G$-asymmetry which is information-theoretic and which was thus far lacking. In particular, we show that in the limit of many copies of the bounded-size quantum reference frame, the accessible information approaches the Holevo bound. This implies that
the rate of alignment of reference frames, measured by the (linearized) accessible information per system, is equal to the regularized, linearized $G$-asymmetry. The latter quantity is equal to the variance in the case where $G=U(1)$.
Quite surprisingly, for the case where $G=\mathbb{Z}_{M}$ and $M\geq4$, it is equal to a quantity that is not additive in general, but instead can be superadditive under tensor product of two distinct bounded-size reference frames.
This remarkable phenomenon is purely quantum and has no classical analog. University of Calgary | Presentation | 2012-06-07 | M. Skotiniotis, G. Gour |

Operational interpretation of the G-asymmetry for abelian groupsWe determine the quantum states and measurements that optimize the accessible information in a reference frame alignment protocol associated with the groups $U(1)$, corresponding to a phase reference, and $\mathbb{Z}_M$, the cyclic group of $M$ elements. Our result provides an operational interpretation for the $G$-asymmetry which is information-theoretic and which was thus far lacking. In particular, we show that in the limit of many copies of the bounded-size quantum reference frame, the accessible information approaches the Holevo bound. This implies that the rate of alignment of reference frames, measured by the (linearized) accessible information per system, is equal to the regularized, linearized $G$-asymmetry. The latter quantity is equal to the variance in the case where $G=U(1)$. Quite surprisingly, for the case where $G=\mathbb{Z}_{M}$ and $M\geq 4$, it is equal to a quantity that is not additive in general, but instead can be superadditive under tensor product of two distinct bounded-size reference frames. This remarkable phenomenon is purely quantum and has no classical analog. University of Calgary | Presentation | 2012-06-11 | M. Skotiniotis, 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 |

Transformation vs. estimation strategies in resource theories of asymmetry University of Calgary, University of California, San Diego | Presentation | 2013-06-25 | V. Narasimhachar, 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 |

Additive bounds of minimum output entropies for unital channels and an exact qubit formula University of Calgary | Publication | 2015-03-01 | M. Fukuda, G. Gour |

The general structure of quantum resource theories University of Calgary | Publication | 2015-03-01 | F. G. Brandão, G. Gour |

Computable conversion witness for bipartite entanged states better than negativity University of Calgary | Publication | 2014-10-01 | M. Girard, G. Gour |

On convex optimization problems in quantum information theory University of Calgary | Publication | 2014-03-01 | M. Girard, G. Gour, S. Friedland |

Local extrema of entropy functions under tensor products University of Calgary | Publication | 2011-09-01 | S. Friedland, G. Gour, A. Roy |

Polygamy of distributed entanglement University of Calgary | Publication | 2009-07-01 | F. Buscemi, G. Gour, J. Kim |

Stability theorem of depolarizing channels for the minimal output quantum Rényi entropiesWe show that the stability theorem of the depolarizing channel holds for the output quantum p-R\'enyi entropy for p≥2 or p=1, which is an extension of the well known case p=2. As an application, we present a protocol in which Bob determines whether Alice prepares a pure quantum state close to a product state. In the protocol, Alice transmits to Bob multiple copies of a pure state through a depolarizing channel, and Bob estimates its output quantum p-R\'enyi entropy. By using our stability theorem, we show that Bob can determine whether her preparation is appropriate. University of Calgary, University of Alberta | Publication | 2016-01-01 | E. Bae, G. Gour, S. Lee, J. Park, B. C. Sanders |

Resource theory under conditioned thermal operations University of Calgary | Publication | 2017-01-01 | V. Narasimhachar, G. Gour |

Quantum relative Lorenz curves University of Calgary | Publication | 2017-01-01 | F. Buscemi, G. Gour |

Entanglement monotones and transformations of symmetric bipartite states University of Calgary | Publication | 2017-01-01 | M. Girard, G. Gour |

Numerical estimation of the relative entropy of entanglement University of Calgary | Publication | 2010-11-01 | Y. Zinchenko, S. Friedland, G. Gour |

Universal Uncertainty Relations University of Calgary | Publication | 2013-12-01 | S. Friedland, V. Gheorghiu, G. Gour |

Entanglement Sharing SchemeEntanglement is a necessary resource for quantum information tasks such as teleportation, dense coding, and Ekert QKD scheme. In entanglement sharing scheme, one share of a bipartite entangled pair is encoded and distributed to untrusted players in a way that they must collaborate in groups to unlock the entanglement. I show how to use quantum error correcting codes to share maximally entangled states between a dealer and collaborating groups of players by exploiting quantum secret sharing concepts and techniques.
University of Calgary | Presentation | 2012-02-09 | H. R. Choi, B. Fortescue, G. Gour, B. C. Sanders |

Communication of information in the absence of a shared frame of referenceIn a communication protocol the sender, Alice, encodes classical messages by preparing a quantum system in a particular state and sending it to the receiver, Bob, who decodes the message by an appropriate quantum measurement. Implicit in the protocol is the assumption that whatever the physical encoding employed by Alice, whether it is the spin of particle, or the energy levels of an atom, is known to Bob. This assumption amounts to Alice and Bob sharing a common reference frame relative to which the states of physical systems are described. The lack of a shared frame of reference imposes severe restrictions on many communication and computational tasks. We obtain the optimal protocols for two cases: where invariant subspaces are available and where they are not.
University of Calgary | Presentation | 2011-03-09 | M. Skotiniotis, A. Roy, G. Gour, B. C. Sanders |

Universal uncertainty relationsUncertainty relations lie at the core of quantum mechanics and are a direct manifestation of the non-commutative structure of the theory. They impose intrinsic limitations on the precision with which physical properties can be simultaneously determined. The modern work on uncertainty relations employs \\emph{entropic measures} to quantify the lack of knowledge associated with measuring non-commuting observables. However there is no fundamental reason for using entropies as quantifiers; any functional relation that characterizes the uncertainty of the measurement outcomes defines an uncertainty relation. Starting from a simple requirement any reasonable measure of uncertainty has to satisfy, we show that Schur-concave functions are the most general uncertainty quantifiers. We then discover a novel fine-grained uncertainty relation written in terms of a majorization relation, which generates an infinite family of distinct scalar uncertainty relations via the application of uncertainty quantifiers. Our relation is universally valid and captures the essence of uncertainty in quantum mechanics. University of Calgary, University of California, San Diego | Presentation | 2013-03-06 | S. Friedland, V. Gheorghiu, G. Gour |

Universal uncertainty relationsUncertainty relations lie at the core of quantum mechanics and are a direct manifestation of the non-commutative structure of the theory. They impose intrinsic limitations on the precision with which physical properties can be simultaneously determined. The modern work on uncertainty relations employs *entropic measures* to quantify the lack of knowledge associated with measuring non-commuting observables. However there is no fundamental reason for using entropies as quantifiers; any functional relation that characterizes the uncertainty of the measurement outcomes defines an uncertainty relation. Starting from a simple requirement any reasonable measure of uncertainty has to satisfy, we show that Schur-concave functions are the most general uncertainty quantifiers. We then discover a novel fine-grained uncertainty relation written in terms of a majorization relation, which generates an infinite family of distinct scalar uncertainty relations via the application of uncertainty quantifiers. Our relation is universally valid and captures the essence of uncertainty in quantum mechanics. University of Calgary, University of California, San Diego | Presentation | 2013-03-20 | S. Friedland, V. Gheorghiu, G. Gour |

Universal uncertainty relationsUncertainty relations lie at the core of quantum mechanics and are a direct manifestation of the non-commutative structure of the theory. They impose intrinsic limitations on the precision with which physical properties can be simultaneously determined. The modern work on uncertainty relations employs \\emph{entropic measures} to quantify the lack of knowledge associated with measuring non-commuting observables. However there is no fundamental reason for using entropies as quantifiers; any functional relation that characterizes the uncertainty of the measurement outcomes defines an uncertainty relation. Starting from a simple requirement any reasonable measure of uncertainty has to satisfy, we show that Schur-concave functions are the most general uncertainty quantifiers. We then discover a novel fine-grained uncertainty relation written in terms of a majorization relation, which generates an infinite family of distinct scalar uncertainty relations via the application of uncertainty quantifiers. Our relation is universally valid and captures the essence of uncertainty in quantum mechanics.\r\n\r\nThis work is in collaboration with Gilad Gour (IQST) and Shmuel Friedland (Univ. of Illinois at Chicago). The talk will be self contained and no prior exposure to quantum mechanics is required. University of Calgary, University of California, San Diego | Presentation | 2013-03-21 | S. Friedland, V. Gheorghiu, G. Gour |

Improving bounds on distillable entanglement University of Calgary | Presentation | 2013-07-03 | M. Girard, S. Friedland, G. Gour |

Quantifying the resourcefulness of quantum reference frames University of Calgary | Presentation | 2010-05-26 | Y. Sanders, B. Fortescue, G. Gour |

Entanglement-enhanced classical communication without a shared frame of referenceTwo parties, Alice and Bob, share a communication channel but lack a shared reference frame.
Alice's task is to communicate a message to Bob, and she does so by preparing an object in a state
that represents the message, for example as a rotation, and transmitting this object to Bob who
measures the state of the object to reveal the message. Due to the lack of a shared reference frame,
Bob may not be able to perform the appropriate measurement to learn the message. For example
Bob may be lacking the reference angle against which to measure the rotation. Here we tackle the
problem of how two parties, lacking a shared reference frame, could prepare and measure a message
in order to communicate successfully. We deem a prepare-and-measure procedure to be successful
if it minimizes the average error over all received messages.
In our communication protocol the parties circumvent the lack of a shared reference frame by
preparing and sending two objects such that the message is the relative transformation parameter
from the state of the rst object into the state of the second object. Bob performs joint measurements
on the pair of received objects to infer the message from the measurement outcomes. Our aim is
to devise a prepare-and-measure scheme that ensures the highest average success rate for sending
messages as relative transformation parameters between two objects.
We use Schur's lemmas, group representation theory, and quantum estimation theory to derive
optimal measurements given constraints imposed on Alice's preparations. We can nd closed-form
solutions for prepare-and-measure schemes for some constraints and employ numerical methods to
obtain optimal protocols in the more general cases. In particular we discover that, whereas preparing
objects in an entangled state is sucient for success, entanglement is not always necessary. Our
theory lays the groundwork for circumventing a lack of reference frames between parties by sending
messages through the parameter of a relative transformation between two objects. University of Calgary | Presentation | 2010-08-26 | M. Skotiniotis, A. Roy, G. Gour, B. C. Sanders |

Entanglement sharing schemes via quantum error-correcting codes University of Calgary | Presentation | 2011-06-17 | H. R. Choi, B. Fortescue, G. Gour, B. C. Sanders |

Entanglement sharing protocolsEntanglement sharing schemes are important as a tool for secure quantum communication in a
network where some subsets of players are authorized to access the transmitted quantum information and other subsets must be denied any quantum information. We conjecture that every stabilizer error correcting code is an entanglement sharing scheme. We test this conjecture with known codes including Shor's 9-qubit code, Steane's 7-qubit code and the 5-qubit code. If our conjecture is true, then we can use existing stabilizer error correcting codes as candidates for entanglement sharing rather than having to construct entanglement sharing schemes ab initio. University of Calgary | Presentation | 2011-11-04 | H. R. Choi, B. Fortescue, G. Gour, B. C. Sanders |

Improving bounds on distillable entanglement University of Calgary, University of California, San Diego | Presentation | 2013-06-25 | M. Girard, S. Friedland, G. Gour |

Universal Uncertainty Relations University of Calgary | Publication | 2013-12-01 | S. Friedland, V. Gheorghiu, G. Gour |

Graph states and ramp schemes for quantum secret sharingI will discuss our recent work in developing new protocols for quantum secret sharing (QSS), a cryptographic scheme in which an encoded quantum "secret" is divided between several "players" such that only certain subsets of players may recover it. We have found a class of protocols based on graph states which allow for efficient (i.e. player states of the same dimension as the secret) QSS for states of prime dimension. We have also found examples of "ramp" schemes for QSS, in which the efficiency can be improved by sacrificing some security. I will discuss these and the use of shared entanglement as a measure of the players' information about the secret.
University of Calgary | Presentation | 2011-02-24 | B. Fortescue, A. Keet, D. Markham, G. Gour, B. C. Sanders |

New directions in quantum secret sharing University of Calgary | Presentation | 2010-11-19 | B. Fortescue, A. Keet, D. Markham, G. Gour, B. C. Sanders |