Gustav0-Freind/100m · Datasets at Hugging Face

text stringlengths 1 5.12k
"
" In 1993, Lucien Hardy demonstrated a logical proof of quantum nonlocality that, like the GHZ proof is a possibilistic proof. The state involved is often called the GHZ state. It starts with the observation that the state formula_50 defined below can be written in a few suggestive ways:
"
" where, as above, formula_52.
"
" The experiment consists of this entangled state being shared between two experimenters, each of whom has the ability to measure either with respect to the basis formula_53 or formula_54. We see that if they each measure with respect to formula_53, then they never see the outcome formula_56. If one measures with respect to formula_53 and the other formula_54, they never see the outcomes formula_59 formula_60 However, sometimes they see the outcome formula_61 when measuring with respect to formula_54, since formula_63
"
" This leads to the paradox: having the outcome formula_64 we conclude that if one of the experimenters had measured with respect to the formula_53 basis instead, the outcome must have been formula_66 or formula_67, since formula_68 and formula_69 are impossible. But then, if they had both measured with respect to the formula_53 basis, by locality the result must have been formula_56, which is also impossible.
"
" The work of Bancal et al. generalizes Bell’s result by proving that correlations achievable in quantum theory are also incompatible with a large class of superluminal hidden variable models. In this framework, faster-than-light signaling is precluded. However, the choice of settings of one party can influence hidden variables at another party’s distant location, if there is enough time for a superluminal influence (of finite, but otherwise unknown speed) to propagate from one point to the other. In this scenario, any bipartite experiment revealing Bell nonlocality can just provide lower bounds on the hidden influence’s propagation speed. Quantum experiments with three or more parties can, nonetheless, disprove all such non-local hidden variable models.
"
" The random variables measured in a general experiment can depend on each other in complicated ways. In the field of causal inference, such dependencies are represented via Bayesian networks: directed acyclic graphs where each node represents a variable and an edge from a variable to another signifies that the former influences the latter and not otherwise, see the figure.
"
" In a standard bipartite Bell experiment, Alice’s (Bob’s) setting formula_29 (formula_73), together with her (his) local variable formula_31 (formula_32), influence her (his) local outcome formula_13 (formula_77). Bell’s theorem can thus be interpreted as a separation between the quantum and classical predictions in a type of causal structures with just one hidden node formula_78. Similar separations have been established in other types of causal structures. The characterization of the boundaries for classical correlations in such extended Bell scenarios is challenging, but there exist complete practical computational methods to achieve it.
"
" In the media and popular science, quantum nonlocality is often portrayed as being equivalent to entanglement. However, this is not the case. Quantum entanglement can be defined only within the formalism of quantum mechanics, i.e., it is a model-dependent property. In contrast, nonlocality refers to the impossibility of a description of observed statistics in terms of a local hidden variable model, so it is independent of the physical model used to describe the experiment.
"
" It is true that for any pure entangled state there exists a choice of measurements that produce Bell nonlocal correlations, but the situation is more complex for mixed states. While any Bell nonlocal state must be entangled, there exist (mixed) entangled states which do not produce Bell nonlocal correlations (although, operating on several copies of some of such states, or carrying out local post-selections, it is possible to witness nonlocal effects). In addition, reasonably simple examples of Bell inequalities have been found for which the quantum state giving the largest violation is never a maximally entangled state, showing that entanglement is, in some sense, not even proportional to nonlocality.
"
" As shown, the statistics achievable by two or more parties conducting experiments in a classical system are constrained in a non-trivial way. Analogously, the statistics achievable by separate observers in a quantum theory also happen to be restricted. The first derivation of a non-trivial statistical limit on the set of quantum correlations, due to B. Tsirelson, is known as Tsirelson's bound.
"
" Consider the CHSH Bell scenario detailed before, but this time assume that, in their experiments, Alice and Bob are preparing and measuring quantum systems. In that case, the CHSH parameter can be shown to be bounded by
"
" Mathematically, a box formula_44 admits a quantum realization if and only if there exists a pair of Hilbert spaces formula_81, a normalized vector formula_82 and projection operators formula_83 such that
"
" In the following, the set of such boxes will be called formula_89. Contrary to the classical set of correlations, when viewed in probability space, formula_89 is not a polytope. On the contrary, it contains both straight and curved boundaries. In addition, formula_89 is not closed: this means that there exist boxes formula_44 which can be arbitrarily well approximated by quantum systems but are themselves not quantum.
"
" In the above definition, the space-like separation of the two parties conducting the Bell experiment was modeled by imposing that their associated operator algebras act on different factors formula_81 of the overall Hilbert space formula_94 describing the experiment. Alternatively, one could model space-like separation by imposing that these two algebras commute. This leads to a different definition:
"
" formula_44 admits a field quantum realization if and only if there exists a Hilbert space formula_96, a normalized vector formula_97 and projection operators formula_98 such that
"
" Call formula_106 the set of all such correlations formula_44.
"
" How does this new set relate to the more conventional formula_89 defined above? It can be proven that formula_106 is closed. Moreover, formula_110, where formula_111 denotes the closure of formula_89. Tsirelson’s problem consists in deciding whether the inclusion relation formula_110 is strict, i.e., whether or not formula_114. This problem only appears in infinite dimensions: when the Hilbert space formula_96 in the definition of formula_106 is constrained to be finite-dimensional, the closure of the corresponding set equals formula_111.
"
" Tsirelson’s problem can be shown equivalent to Connes embedding problem, a famous conjecture in the theory of operator algebras.
"
" Since the dimensions of formula_118 and formula_119 are, in principle, unbounded, determining whether a given box formula_44 admits a quantum realization is a complicated problem. In fact, the dual problem of establishing whether a quantum box can have a perfect score at a non-local game is known to be undecidable. Moreover, the problem of deciding whether formula_44 can be approximated by a quantum system with precision formula_122 is NP-hard. Characterizing quantum boxes is equivalent to characterizing the cone of completely positive semidefinite matrices under a set of linear constraints.
"
" For small fixed dimensions formula_123, one can explore, using variational methods, whether formula_44 can be realized in a bipartite quantum system formula_125, with formula_126, formula_127. That method, however, can just be used to prove the realizability of formula_44, and not its unrealizability with quantum systems.
"
" To prove unrealizability, the most known method is the Navascués-Pironio-Acín (NPA) hierarchy. This is an infinite decreasing sequence of sets of correlations formula_129 with the properties:
"
" The NPA hierarchy thus provides a computational characterization, not of formula_89, but of formula_106. If Tsirelson’s problem is solved in the affirmative, namely, formula_140, then the above two methods would provide a practical characterization of formula_111. If, on the contrary, formula_142, then a new method to detect the non-realizability of the correlations in formula_143 is needed.
"
" The works listed above describe what the quantum set of correlations looks like, but they don’t explain why. Are quantum correlations unavoidable, even in post-quantum physical theories, or on the contrary, could there exist correlations outside formula_111 which nonetheless do not lead to any unphysical operational behavior?
"
" In their seminal 1994 paper, Popescu and Rorhlich explore whether quantum correlations can be explained by appealing to relativistic causality alone. Namely, whether any hypothetical box formula_145 would allow building a device capable of transmitting information faster than the speed of light. At the level of correlations between two parties, Einstein’s causality translates in the requirement that Alice’s measurement choice should not affect Bob’s statistics, and viceversa. Otherwise, Alice (Bob) could signal Bob (Alice) instantaneously by choosing her (his) measurement setting formula_29 formula_147 appropriately. Mathematically, Popescu and Rohrlich’s no-signalling conditions are:
"
" Like the set of classical boxes, when represented in probability space, the set of no-signalling boxes forms a polytope. Popescu and Rohrlich identified a box formula_44 that, while complying with the no-signalling conditions, violates Tsirelson’s bound, and is thus unrealizable in quantum physics. Dubbed the PR-box, it can be written as:
"
" formula_151
"
" Here formula_39 take values in formula_43, and formula_154 denotes the sum modulo two. It can be verified that the CHSH value of this box is 4 (as opposed to the Tsirelson bound of formula_49). This box had been identified earlier, by Rastall and Khalfin and Tsirelson.
"
" In view of this mismatch, Popescu and Rohrlich pose the problem of identifying a physical principle, stronger than the no-signalling conditions, that allows deriving the set of quantum correlations. Several proposals followed:
"
" All these principles can be experimentally falsified under the assumption that we can decide if two or more events are space-like separated. This sets this research program aside from the axiomatic reconstruction of quantum mechanics via Generalized Probabilistic Theories.
"
" The works above rely on the implicit assumption that any physical set of correlations must be closed under wirings. This means that any effective box built by combining the inputs and outputs of a number of boxes within the considered set must also belong to the set. Closure under wirings does not seem to enforce any limit on the maximum value of CHSH. However, it is not a void principle: on the contrary, in it is shown that many simple, intuitive families of sets of correlations in probability space happen to violate it.
"
" Originally, it was unknown whether any of these principles (or a subset thereof) was strong enough to derive all the constraints defining formula_111. This state of affairs continued for some years until the construction of the almost quantum set formula_182. formula_182 is a set of correlations that is closed under wirings and can be characterized via semidefinite programming. It contains all correlations in formula_110, but also some non-quantum boxes formula_133. Remarkably, all boxes within the almost quantum set are shown to be compatible with the principles of NTCC, NANLC, ML and LO. There is also numerical evidence that almost quantum boxes also comply with IC. It seems, therefore, that, even when the above principles are taken together, they do not suffice to single out the quantum set in the simplest Bell scenario of two parties, two inputs and two outputs.
"
" Nonlocality can be exploited to conduct quantum information tasks which do not rely on the knowledge of the inner workings of the prepare-and-measurement apparatuses involved in the experiment. The security or reliability of any such protocol just depends on the strength of the experimentally measured correlations formula_44. These protocols are termed device-independent.
"
" The first device-independent protocol proposed was device-independent Quantum Key Distribution (QKD). In this primitive, two distant parties, Alice and Bob, are distributed an entangled quantum state, that they probe, thus obtaining the statistics formula_44. Based on how non-local the box formula_44 happens to be, Alice and Bob estimate how much knowledge an external quantum adversary Eve (the eavesdropper) could possess on the value of Alice and Bob’s outputs. This estimation allows them to devise a reconciliation protocol at the end of which Alice and Bob share a perfectly correlated one-time pad of which Eve has no information whatsoever. The one-time pad can then be used to transmit a secret message through a public channel. Although the first security analyses on device-independent QKD relied on Eve carrying out a specific family of attacks, all such protocols have been recently proven unconditionally secure.
"
" Nonlocality can be used to certify that the outcomes of one of the parties in a Bell experiment are partially unknown to an external adversary. By feeding a partially random seed to several non-local boxes, and, after processing the outputs, one can end up with a longer (potentially unbounded) string of comparable randomness or with a shorter but more random string. This last primitive can be proven impossible in a classical setting.
"
" Sometimes, the box formula_44 shared by Alice and Bob is such that it only admits a unique quantum realization. This means that there exist measurement operators formula_190 and a quantum state formula_191 giving rise to formula_44 such that any other physical realization formula_193 of formula_44 is connected to formula_195 via local unitary transformations. This phenomenon, that can be interpreted as an instance of device-independent quantum tomography, was first pointed out by Tsirelson and named self-testing by Mayers and Yao. Self-testing is known to be robust against systematic noise, i.e., if the experimentally measured statistics are close enough to formula_44, one can still determine the underlying state and measurement operators up to error bars.
"
" The degree of non-locality of a quantum box formula_44 can also provide lower bounds on the Hilbert space dimension of the local systems accessible to Alice and Bob. This problem is equivalent to deciding the existence of a matrix with low completely positive semidefinite rank. Finding lower bounds on the Hilbert space dimension based on statistics happens to be a hard task, and current general methods only provide very low estimates. However, a Bell scenario with five inputs and three outputs suffices to provide arbitrarily high lower bounds on the underlying Hilbert space dimension. Quantum communication protocols which assume a knowledge of the local dimension of Alice and Bob’s systems, but otherwise do not make claims on the mathematical description of the preparation and measuring devices involved are termed semi-device independent protocols. Currently, there exist semi-device independent protocols for quantum key distribution and randomness expansion.
"
"
"
"= = = Martin Henfield = = =
"
"
"
" Martin Henfield is a British TV and radio presenter and media specialist. Henfield has worked as a reporter, producer, editor and senior manager in BBC Radio and TV for 26 years. He ran BBC GMR radio station for 5 years and presented BBC ""North West Tonight"" in the 1990s.
"
" He also presented the ""Vague News"" section of Mark and Lards afternoon show on BBC Radio 1 in the late 1990s.
"
" He now works in public presentation and media training. Henfield has an identical twin brother, Michael, who has also worked in radio and media training for many years and used to be a journalism lecturer at the University of Salford. He is married to a University of Central Lancashire journalism lecturer, Maggie, a former features editor for the Manchester Evening News.
"
" Henfield presented his first ever television commercial, promoting participation in the vote for the Manchester congestion charge in November 2008. which is currently being investigated by Ofcom after complaints questioning the impartiality of the advert which leaned towards the pro side of the debate.
"
"
"
"= = = Tarumirim = = =
"
"
"
" Tarumirim is a municipality in east Minas Gerais state, Brazil. It is located in the Vale do Rio Doce region and its population was approximately 14,000 inhabitants in 2004 (IBGE).
"
"
"
"= = = Leslie A. Miller = = =
"
"

"

" In 1993, Lucien Hardy demonstrated a logical proof of quantum nonlocality that, like the GHZ proof is a possibilistic proof. The state involved is often called the GHZ state. It starts with the observation that the state formula_50 defined below can be written in a few suggestive ways:

"

" where, as above, formula_52.

"

" The experiment consists of this entangled state being shared between two experimenters, each of whom has the ability to measure either with respect to the basis formula_53 or formula_54. We see that if they each measure with respect to formula_53, then they never see the outcome formula_56. If one measures with respect to formula_53 and the other formula_54, they never see the outcomes formula_59 formula_60 However, sometimes they see the outcome formula_61 when measuring with respect to formula_54, since formula_63

"

" This leads to the paradox: having the outcome formula_64 we conclude that if one of the experimenters had measured with respect to the formula_53 basis instead, the outcome must have been formula_66 or formula_67, since formula_68 and formula_69 are impossible. But then, if they had both measured with respect to the formula_53 basis, by locality the result must have been formula_56, which is also impossible.

"

" The work of Bancal et al. generalizes Bell’s result by proving that correlations achievable in quantum theory are also incompatible with a large class of superluminal hidden variable models. In this framework, faster-than-light signaling is precluded. However, the choice of settings of one party can influence hidden variables at another party’s distant location, if there is enough time for a superluminal influence (of finite, but otherwise unknown speed) to propagate from one point to the other. In this scenario, any bipartite experiment revealing Bell nonlocality can just provide lower bounds on the hidden influence’s propagation speed. Quantum experiments with three or more parties can, nonetheless, disprove all such non-local hidden variable models.

"

" The random variables measured in a general experiment can depend on each other in complicated ways. In the field of causal inference, such dependencies are represented via Bayesian networks: directed acyclic graphs where each node represents a variable and an edge from a variable to another signifies that the former influences the latter and not otherwise, see the figure.

"

" In a standard bipartite Bell experiment, Alice’s (Bob’s) setting formula_29 (formula_73), together with her (his) local variable formula_31 (formula_32), influence her (his) local outcome formula_13 (formula_77). Bell’s theorem can thus be interpreted as a separation between the quantum and classical predictions in a type of causal structures with just one hidden node formula_78. Similar separations have been established in other types of causal structures. The characterization of the boundaries for classical correlations in such extended Bell scenarios is challenging, but there exist complete practical computational methods to achieve it.

"

" In the media and popular science, quantum nonlocality is often portrayed as being equivalent to entanglement. However, this is not the case. Quantum entanglement can be defined only within the formalism of quantum mechanics, i.e., it is a model-dependent property. In contrast, nonlocality refers to the impossibility of a description of observed statistics in terms of a local hidden variable model, so it is independent of the physical model used to describe the experiment.

"

" It is true that for any pure entangled state there exists a choice of measurements that produce Bell nonlocal correlations, but the situation is more complex for mixed states. While any Bell nonlocal state must be entangled, there exist (mixed) entangled states which do not produce Bell nonlocal correlations (although, operating on several copies of some of such states, or carrying out local post-selections, it is possible to witness nonlocal effects). In addition, reasonably simple examples of Bell inequalities have been found for which the quantum state giving the largest violation is never a maximally entangled state, showing that entanglement is, in some sense, not even proportional to nonlocality.

"

" As shown, the statistics achievable by two or more parties conducting experiments in a classical system are constrained in a non-trivial way. Analogously, the statistics achievable by separate observers in a quantum theory also happen to be restricted. The first derivation of a non-trivial statistical limit on the set of quantum correlations, due to B. Tsirelson, is known as Tsirelson's bound.

"

" Consider the CHSH Bell scenario detailed before, but this time assume that, in their experiments, Alice and Bob are preparing and measuring quantum systems. In that case, the CHSH parameter can be shown to be bounded by

"

" Mathematically, a box formula_44 admits a quantum realization if and only if there exists a pair of Hilbert spaces formula_81, a normalized vector formula_82 and projection operators formula_83 such that

"

" In the following, the set of such boxes will be called formula_89. Contrary to the classical set of correlations, when viewed in probability space, formula_89 is not a polytope. On the contrary, it contains both straight and curved boundaries. In addition, formula_89 is not closed: this means that there exist boxes formula_44 which can be arbitrarily well approximated by quantum systems but are themselves not quantum.

"

" In the above definition, the space-like separation of the two parties conducting the Bell experiment was modeled by imposing that their associated operator algebras act on different factors formula_81 of the overall Hilbert space formula_94 describing the experiment. Alternatively, one could model space-like separation by imposing that these two algebras commute. This leads to a different definition:

"

" formula_44 admits a field quantum realization if and only if there exists a Hilbert space formula_96, a normalized vector formula_97 and projection operators formula_98 such that

"

" Call formula_106 the set of all such correlations formula_44.

"

" How does this new set relate to the more conventional formula_89 defined above? It can be proven that formula_106 is closed. Moreover, formula_110, where formula_111 denotes the closure of formula_89. Tsirelson’s problem consists in deciding whether the inclusion relation formula_110 is strict, i.e., whether or not formula_114. This problem only appears in infinite dimensions: when the Hilbert space formula_96 in the definition of formula_106 is constrained to be finite-dimensional, the closure of the corresponding set equals formula_111.

"

" Tsirelson’s problem can be shown equivalent to Connes embedding problem, a famous conjecture in the theory of operator algebras.

"

" Since the dimensions of formula_118 and formula_119 are, in principle, unbounded, determining whether a given box formula_44 admits a quantum realization is a complicated problem. In fact, the dual problem of establishing whether a quantum box can have a perfect score at a non-local game is known to be undecidable. Moreover, the problem of deciding whether formula_44 can be approximated by a quantum system with precision formula_122 is NP-hard. Characterizing quantum boxes is equivalent to characterizing the cone of completely positive semidefinite matrices under a set of linear constraints.

"

" For small fixed dimensions formula_123, one can explore, using variational methods, whether formula_44 can be realized in a bipartite quantum system formula_125, with formula_126, formula_127. That method, however, can just be used to prove the realizability of formula_44, and not its unrealizability with quantum systems.

"

" To prove unrealizability, the most known method is the Navascués-Pironio-Acín (NPA) hierarchy. This is an infinite decreasing sequence of sets of correlations formula_129 with the properties:

"

" The NPA hierarchy thus provides a computational characterization, not of formula_89, but of formula_106. If Tsirelson’s problem is solved in the affirmative, namely, formula_140, then the above two methods would provide a practical characterization of formula_111. If, on the contrary, formula_142, then a new method to detect the non-realizability of the correlations in formula_143 is needed.

"

" The works listed above describe what the quantum set of correlations looks like, but they don’t explain why. Are quantum correlations unavoidable, even in post-quantum physical theories, or on the contrary, could there exist correlations outside formula_111 which nonetheless do not lead to any unphysical operational behavior?

"

" In their seminal 1994 paper, Popescu and Rorhlich explore whether quantum correlations can be explained by appealing to relativistic causality alone. Namely, whether any hypothetical box formula_145 would allow building a device capable of transmitting information faster than the speed of light. At the level of correlations between two parties, Einstein’s causality translates in the requirement that Alice’s measurement choice should not affect Bob’s statistics, and viceversa. Otherwise, Alice (Bob) could signal Bob (Alice) instantaneously by choosing her (his) measurement setting formula_29 formula_147 appropriately. Mathematically, Popescu and Rohrlich’s no-signalling conditions are:

"

" Like the set of classical boxes, when represented in probability space, the set of no-signalling boxes forms a polytope. Popescu and Rohrlich identified a box formula_44 that, while complying with the no-signalling conditions, violates Tsirelson’s bound, and is thus unrealizable in quantum physics. Dubbed the PR-box, it can be written as:

"

" formula_151

"

" Here formula_39 take values in formula_43, and formula_154 denotes the sum modulo two. It can be verified that the CHSH value of this box is 4 (as opposed to the Tsirelson bound of formula_49). This box had been identified earlier, by Rastall and Khalfin and Tsirelson.

"

" In view of this mismatch, Popescu and Rohrlich pose the problem of identifying a physical principle, stronger than the no-signalling conditions, that allows deriving the set of quantum correlations. Several proposals followed:

"

" All these principles can be experimentally falsified under the assumption that we can decide if two or more events are space-like separated. This sets this research program aside from the axiomatic reconstruction of quantum mechanics via Generalized Probabilistic Theories.

"

" The works above rely on the implicit assumption that any physical set of correlations must be closed under wirings. This means that any effective box built by combining the inputs and outputs of a number of boxes within the considered set must also belong to the set. Closure under wirings does not seem to enforce any limit on the maximum value of CHSH. However, it is not a void principle: on the contrary, in it is shown that many simple, intuitive families of sets of correlations in probability space happen to violate it.

"

" Originally, it was unknown whether any of these principles (or a subset thereof) was strong enough to derive all the constraints defining formula_111. This state of affairs continued for some years until the construction of the almost quantum set formula_182. formula_182 is a set of correlations that is closed under wirings and can be characterized via semidefinite programming. It contains all correlations in formula_110, but also some non-quantum boxes formula_133. Remarkably, all boxes within the almost quantum set are shown to be compatible with the principles of NTCC, NANLC, ML and LO. There is also numerical evidence that almost quantum boxes also comply with IC. It seems, therefore, that, even when the above principles are taken together, they do not suffice to single out the quantum set in the simplest Bell scenario of two parties, two inputs and two outputs.

"

" Nonlocality can be exploited to conduct quantum information tasks which do not rely on the knowledge of the inner workings of the prepare-and-measurement apparatuses involved in the experiment. The security or reliability of any such protocol just depends on the strength of the experimentally measured correlations formula_44. These protocols are termed device-independent.

"

" The first device-independent protocol proposed was device-independent Quantum Key Distribution (QKD). In this primitive, two distant parties, Alice and Bob, are distributed an entangled quantum state, that they probe, thus obtaining the statistics formula_44. Based on how non-local the box formula_44 happens to be, Alice and Bob estimate how much knowledge an external quantum adversary Eve (the eavesdropper) could possess on the value of Alice and Bob’s outputs. This estimation allows them to devise a reconciliation protocol at the end of which Alice and Bob share a perfectly correlated one-time pad of which Eve has no information whatsoever. The one-time pad can then be used to transmit a secret message through a public channel. Although the first security analyses on device-independent QKD relied on Eve carrying out a specific family of attacks, all such protocols have been recently proven unconditionally secure.

"

" Nonlocality can be used to certify that the outcomes of one of the parties in a Bell experiment are partially unknown to an external adversary. By feeding a partially random seed to several non-local boxes, and, after processing the outputs, one can end up with a longer (potentially unbounded) string of comparable randomness or with a shorter but more random string. This last primitive can be proven impossible in a classical setting.

"

" Sometimes, the box formula_44 shared by Alice and Bob is such that it only admits a unique quantum realization. This means that there exist measurement operators formula_190 and a quantum state formula_191 giving rise to formula_44 such that any other physical realization formula_193 of formula_44 is connected to formula_195 via local unitary transformations. This phenomenon, that can be interpreted as an instance of device-independent quantum tomography, was first pointed out by Tsirelson and named self-testing by Mayers and Yao. Self-testing is known to be robust against systematic noise, i.e., if the experimentally measured statistics are close enough to formula_44, one can still determine the underlying state and measurement operators up to error bars.

"

" The degree of non-locality of a quantum box formula_44 can also provide lower bounds on the Hilbert space dimension of the local systems accessible to Alice and Bob. This problem is equivalent to deciding the existence of a matrix with low completely positive semidefinite rank. Finding lower bounds on the Hilbert space dimension based on statistics happens to be a hard task, and current general methods only provide very low estimates. However, a Bell scenario with five inputs and three outputs suffices to provide arbitrarily high lower bounds on the underlying Hilbert space dimension. Quantum communication protocols which assume a knowledge of the local dimension of Alice and Bob’s systems, but otherwise do not make claims on the mathematical description of the preparation and measuring devices involved are termed semi-device independent protocols. Currently, there exist semi-device independent protocols for quantum key distribution and randomness expansion.

"

"= = = Martin Henfield = = =

"

" Martin Henfield is a British TV and radio presenter and media specialist. Henfield has worked as a reporter, producer, editor and senior manager in BBC Radio and TV for 26 years. He ran BBC GMR radio station for 5 years and presented BBC ""North West Tonight"" in the 1990s.

"

" He also presented the ""Vague News"" section of Mark and Lards afternoon show on BBC Radio 1 in the late 1990s.

"

" He now works in public presentation and media training. Henfield has an identical twin brother, Michael, who has also worked in radio and media training for many years and used to be a journalism lecturer at the University of Salford. He is married to a University of Central Lancashire journalism lecturer, Maggie, a former features editor for the Manchester Evening News.

"

" Henfield presented his first ever television commercial, promoting participation in the vote for the Manchester congestion charge in November 2008. which is currently being investigated by Ofcom after complaints questioning the impartiality of the advert which leaned towards the pro side of the debate.

"

"= = = Tarumirim = = =

"

" Tarumirim is a municipality in east Minas Gerais state, Brazil. It is located in the Vale do Rio Doce region and its population was approximately 14,000 inhabitants in 2004 (IBGE).

"

"= = = Leslie A. Miller = = =

"