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Next Q+ Hangout is on November 22nd, 2 pm GMT.

As usual, if you are watching with a group and want to reserve a seat in the hangout then leave a comment. We also encourage individuals interested in active participation - which typically involves asking questions after the talk - to join the hangout. Otherwise you can watch on the livestream. Details follow.

Speaker: Matthias Christandl, University of Copenhagen.

Title: On the tensor rank of networks of entangled pairs: tensor surgery and the laser method

Abstract: We prove upper bounds on the tensor rank of networks of entangled pairs. Any graph defines such a network by associating an entangled pair to each edge of the graph. We present two methods. First, we introduce a surgery-like procedure to transform a good decomposition of a well-chosen tensor into a good decomposition of a tensor of interest. We illustrate the method with surgery on the cycle graph, which corresponds to the iterated matrix multiplication tensor and obtain the first nontrivial rank results for large odd cycles and optimal asymptotic rank results for all cycles. Second, we generalize Strassen’s laser method to higher- order tensors in order to show a nontrivial upper bound on the asymptotic rank for the complete graph. “Per edge” this improves on the best upper bound on the matrix multiplication exponent [LG14], for four or more vertices. In entanglement theory, our results amount to protocols for creating a network of entangled pairs from GHZ states by SLOCC. In communication complexity theory, our results imply new bounds on the nondeterministic quantum communication of equality games. Our work is inspired and tightly connected with the vast body of research on matrix multiplication.

Based on joint work with Péter Vrana and Jeroen Zuiddam (http://arxiv.org/abs/1603.03757 and https://arxiv.org/abs/1609.07476)

Livestream link: https://www.youtube.com/watch?v=OLrOt8RlLS8

As usual, if you are watching with a group and want to reserve a seat in the hangout then leave a comment. We also encourage individuals interested in active participation - which typically involves asking questions after the talk - to join the hangout. Otherwise you can watch on the livestream. Details follow.

Speaker: Matthias Christandl, University of Copenhagen.

Title: On the tensor rank of networks of entangled pairs: tensor surgery and the laser method

Abstract: We prove upper bounds on the tensor rank of networks of entangled pairs. Any graph defines such a network by associating an entangled pair to each edge of the graph. We present two methods. First, we introduce a surgery-like procedure to transform a good decomposition of a well-chosen tensor into a good decomposition of a tensor of interest. We illustrate the method with surgery on the cycle graph, which corresponds to the iterated matrix multiplication tensor and obtain the first nontrivial rank results for large odd cycles and optimal asymptotic rank results for all cycles. Second, we generalize Strassen’s laser method to higher- order tensors in order to show a nontrivial upper bound on the asymptotic rank for the complete graph. “Per edge” this improves on the best upper bound on the matrix multiplication exponent [LG14], for four or more vertices. In entanglement theory, our results amount to protocols for creating a network of entangled pairs from GHZ states by SLOCC. In communication complexity theory, our results imply new bounds on the nondeterministic quantum communication of equality games. Our work is inspired and tightly connected with the vast body of research on matrix multiplication.

Based on joint work with Péter Vrana and Jeroen Zuiddam (http://arxiv.org/abs/1603.03757 and https://arxiv.org/abs/1609.07476)

Livestream link: https://www.youtube.com/watch?v=OLrOt8RlLS8

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Next Q+ Hangout is on November 22nd, 2 pm GMT.

As usual, if you are watching with a group and want to reserve a seat in the hangout then leave a comment. We also encourage individuals interested in active participation - which typically involves asking questions after the talk - to join the hangout. Otherwise you can watch on the livestream. Details follow.

Speaker: Matthias Christandl, University of Copenhagen.

Title: On the tensor rank of networks of entangled pairs: tensor surgery and the laser method

Abstract: We prove upper bounds on the tensor rank of networks of entangled pairs. Any graph defines such a network by associating an entangled pair to each edge of the graph. We present two methods. First, we introduce a surgery-like procedure to transform a good decomposition of a well-chosen tensor into a good decomposition of a tensor of interest. We illustrate the method with surgery on the cycle graph, which corresponds to the iterated matrix multiplication tensor and obtain the first nontrivial rank results for large odd cycles and optimal asymptotic rank results for all cycles. Second, we generalize Strassen’s laser method to higher- order tensors in order to show a nontrivial upper bound on the asymptotic rank for the complete graph. “Per edge” this improves on the best upper bound on the matrix multiplication exponent [LG14], for four or more vertices. In entanglement theory, our results amount to protocols for creating a network of entangled pairs from GHZ states by SLOCC. In communication complexity theory, our results imply new bounds on the nondeterministic quantum communication of equality games. Our work is inspired and tightly connected with the vast body of research on matrix multiplication.

Based on joint work with Péter Vrana and Jeroen Zuiddam (http://arxiv.org/abs/1603.03757 and https://arxiv.org/abs/1609.07476)

Livestream link: https://www.youtube.com/watch?v=OLrOt8RlLS8

As usual, if you are watching with a group and want to reserve a seat in the hangout then leave a comment. We also encourage individuals interested in active participation - which typically involves asking questions after the talk - to join the hangout. Otherwise you can watch on the livestream. Details follow.

Speaker: Matthias Christandl, University of Copenhagen.

Title: On the tensor rank of networks of entangled pairs: tensor surgery and the laser method

Abstract: We prove upper bounds on the tensor rank of networks of entangled pairs. Any graph defines such a network by associating an entangled pair to each edge of the graph. We present two methods. First, we introduce a surgery-like procedure to transform a good decomposition of a well-chosen tensor into a good decomposition of a tensor of interest. We illustrate the method with surgery on the cycle graph, which corresponds to the iterated matrix multiplication tensor and obtain the first nontrivial rank results for large odd cycles and optimal asymptotic rank results for all cycles. Second, we generalize Strassen’s laser method to higher- order tensors in order to show a nontrivial upper bound on the asymptotic rank for the complete graph. “Per edge” this improves on the best upper bound on the matrix multiplication exponent [LG14], for four or more vertices. In entanglement theory, our results amount to protocols for creating a network of entangled pairs from GHZ states by SLOCC. In communication complexity theory, our results imply new bounds on the nondeterministic quantum communication of equality games. Our work is inspired and tightly connected with the vast body of research on matrix multiplication.

Based on joint work with Péter Vrana and Jeroen Zuiddam (http://arxiv.org/abs/1603.03757 and https://arxiv.org/abs/1609.07476)

Livestream link: https://www.youtube.com/watch?v=OLrOt8RlLS8

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Reminder: Next Q+ hangout is on the 22nd of November! Comment on the original post to reserve a spot.

Next Q+ Hangout is on November 22nd, 2 pm GMT.

As usual, if you are watching with a group and want to reserve a seat in the hangout then leave a comment. We also encourage individuals interested in active participation - which typically involves asking questions after the talk - to join the hangout. Otherwise you can watch on the livestream. Details follow.

Speaker: Matthias Christandl, University of Copenhagen.

Title: On the tensor rank of networks of entangled pairs: tensor surgery and the laser method

Abstract: We prove upper bounds on the tensor rank of networks of entangled pairs. Any graph defines such a network by associating an entangled pair to each edge of the graph. We present two methods. First, we introduce a surgery-like procedure to transform a good decomposition of a well-chosen tensor into a good decomposition of a tensor of interest. We illustrate the method with surgery on the cycle graph, which corresponds to the iterated matrix multiplication tensor and obtain the first nontrivial rank results for large odd cycles and optimal asymptotic rank results for all cycles. Second, we generalize Strassen’s laser method to higher- order tensors in order to show a nontrivial upper bound on the asymptotic rank for the complete graph. “Per edge” this improves on the best upper bound on the matrix multiplication exponent [LG14], for four or more vertices. In entanglement theory, our results amount to protocols for creating a network of entangled pairs from GHZ states by SLOCC. In communication complexity theory, our results imply new bounds on the nondeterministic quantum communication of equality games. Our work is inspired and tightly connected with the vast body of research on matrix multiplication.

Based on joint work with Péter Vrana and Jeroen Zuiddam (http://arxiv.org/abs/1603.03757 and https://arxiv.org/abs/1609.07476)

Livestream link: https://www.youtube.com/watch?v=OLrOt8RlLS8

As usual, if you are watching with a group and want to reserve a seat in the hangout then leave a comment. We also encourage individuals interested in active participation - which typically involves asking questions after the talk - to join the hangout. Otherwise you can watch on the livestream. Details follow.

Speaker: Matthias Christandl, University of Copenhagen.

Title: On the tensor rank of networks of entangled pairs: tensor surgery and the laser method

Abstract: We prove upper bounds on the tensor rank of networks of entangled pairs. Any graph defines such a network by associating an entangled pair to each edge of the graph. We present two methods. First, we introduce a surgery-like procedure to transform a good decomposition of a well-chosen tensor into a good decomposition of a tensor of interest. We illustrate the method with surgery on the cycle graph, which corresponds to the iterated matrix multiplication tensor and obtain the first nontrivial rank results for large odd cycles and optimal asymptotic rank results for all cycles. Second, we generalize Strassen’s laser method to higher- order tensors in order to show a nontrivial upper bound on the asymptotic rank for the complete graph. “Per edge” this improves on the best upper bound on the matrix multiplication exponent [LG14], for four or more vertices. In entanglement theory, our results amount to protocols for creating a network of entangled pairs from GHZ states by SLOCC. In communication complexity theory, our results imply new bounds on the nondeterministic quantum communication of equality games. Our work is inspired and tightly connected with the vast body of research on matrix multiplication.

Based on joint work with Péter Vrana and Jeroen Zuiddam (http://arxiv.org/abs/1603.03757 and https://arxiv.org/abs/1609.07476)

Livestream link: https://www.youtube.com/watch?v=OLrOt8RlLS8

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Next Q+ Hangout is on November 8th!

As usual, if you are watching with a group and want to reserve a seat in the hangout then leave a comment. We also encourage individuals interested in active participation---which typically involves asking questions after the talk---to join the hangout. Otherwise you can watch on the livestream. Details follow.

Speaker: Miguel Navascues

Title: Entanglement and Nonlocality of 1D Macroscopic Systems

Abstract: We consider the problem of certifying entanglement and nonlocality in one-dimensional macroscopic systems when just averaged near-neighbor correlators are available. We map this question to the characterization of the set of all quantum states (distributions) which admit a separable (classical) Translation-Invariant (TI) extension to infinitely many sites. We advance the first problem by constructing a family of witnesses which can detect entanglement in infinite TI states from the nearest-neighbor reduced density matrix. Along the way, we identify a set of separable two-qubit states which only admit entangled TI extensions. For nonlocality detection, we show that the set of classical TI boxes forms a polytope and devise a general procedure to generate all Bell inequalities which characterize it. Through a Matrix Product State-based method, we show how some of them can be violated by distant parties conducting identical measurements on an infinite TI quantum state.

As usual, if you are watching with a group and want to reserve a seat in the hangout then leave a comment. We also encourage individuals interested in active participation---which typically involves asking questions after the talk---to join the hangout. Otherwise you can watch on the livestream. Details follow.

Speaker: Miguel Navascues

Title: Entanglement and Nonlocality of 1D Macroscopic Systems

Abstract: We consider the problem of certifying entanglement and nonlocality in one-dimensional macroscopic systems when just averaged near-neighbor correlators are available. We map this question to the characterization of the set of all quantum states (distributions) which admit a separable (classical) Translation-Invariant (TI) extension to infinitely many sites. We advance the first problem by constructing a family of witnesses which can detect entanglement in infinite TI states from the nearest-neighbor reduced density matrix. Along the way, we identify a set of separable two-qubit states which only admit entangled TI extensions. For nonlocality detection, we show that the set of classical TI boxes forms a polytope and devise a general procedure to generate all Bell inequalities which characterize it. Through a Matrix Product State-based method, we show how some of them can be violated by distant parties conducting identical measurements on an infinite TI quantum state.

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Reminder: Q+ Hangout with Miguel Navascues happens this week on Tuesday at 2 pm GMT. Livestream here: https://www.youtube.com/watch?v=awvdeBtcubY

Next Q+ Hangout is on November 8th, 2 pm GMT.

As usual, if you are watching with a group and want to reserve a seat in the hangout then leave a comment. We also encourage individuals interested in active participation---which typically involves asking questions after the talk---to join the hangout. Otherwise you can watch on the livestream. Details follow.

Speaker: Miguel Navascues

Title: Entanglement and Nonlocality of 1D Macroscopic Systems

Abstract: We consider the problem of certifying entanglement and nonlocality in one-dimensional macroscopic systems when just averaged near-neighbor correlators are available. We map this question to the characterization of the set of all quantum states (distributions) which admit a separable (classical) Translation-Invariant (TI) extension to infinitely many sites. We advance the first problem by constructing a family of witnesses which can detect entanglement in infinite TI states from the nearest-neighbor reduced density matrix. Along the way, we identify a set of separable two-qubit states which only admit entangled TI extensions. For nonlocality detection, we show that the set of classical TI boxes forms a polytope and devise a general procedure to generate all Bell inequalities which characterize it. Through a Matrix Product State-based method, we show how some of them can be violated by distant parties conducting identical measurements on an infinite TI quantum state.

Livestream link: https://www.youtube.com/watch?v=awvdeBtcubY

As usual, if you are watching with a group and want to reserve a seat in the hangout then leave a comment. We also encourage individuals interested in active participation---which typically involves asking questions after the talk---to join the hangout. Otherwise you can watch on the livestream. Details follow.

Speaker: Miguel Navascues

Title: Entanglement and Nonlocality of 1D Macroscopic Systems

Abstract: We consider the problem of certifying entanglement and nonlocality in one-dimensional macroscopic systems when just averaged near-neighbor correlators are available. We map this question to the characterization of the set of all quantum states (distributions) which admit a separable (classical) Translation-Invariant (TI) extension to infinitely many sites. We advance the first problem by constructing a family of witnesses which can detect entanglement in infinite TI states from the nearest-neighbor reduced density matrix. Along the way, we identify a set of separable two-qubit states which only admit entangled TI extensions. For nonlocality detection, we show that the set of classical TI boxes forms a polytope and devise a general procedure to generate all Bell inequalities which characterize it. Through a Matrix Product State-based method, we show how some of them can be violated by distant parties conducting identical measurements on an infinite TI quantum state.

Livestream link: https://www.youtube.com/watch?v=awvdeBtcubY

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Reservations are still open for the Hangout next week! Comment on the original post to reserve.

Next Q+ Hangout is on November 8th, 2 pm GMT.

As usual, if you are watching with a group and want to reserve a seat in the hangout then leave a comment. We also encourage individuals interested in active participation---which typically involves asking questions after the talk---to join the hangout. Otherwise you can watch on the livestream. Details follow.

Speaker: Miguel Navascues

Title: Entanglement and Nonlocality of 1D Macroscopic Systems

Abstract: We consider the problem of certifying entanglement and nonlocality in one-dimensional macroscopic systems when just averaged near-neighbor correlators are available. We map this question to the characterization of the set of all quantum states (distributions) which admit a separable (classical) Translation-Invariant (TI) extension to infinitely many sites. We advance the first problem by constructing a family of witnesses which can detect entanglement in infinite TI states from the nearest-neighbor reduced density matrix. Along the way, we identify a set of separable two-qubit states which only admit entangled TI extensions. For nonlocality detection, we show that the set of classical TI boxes forms a polytope and devise a general procedure to generate all Bell inequalities which characterize it. Through a Matrix Product State-based method, we show how some of them can be violated by distant parties conducting identical measurements on an infinite TI quantum state.

Livestream link: https://www.youtube.com/watch?v=awvdeBtcubY

As usual, if you are watching with a group and want to reserve a seat in the hangout then leave a comment. We also encourage individuals interested in active participation---which typically involves asking questions after the talk---to join the hangout. Otherwise you can watch on the livestream. Details follow.

Speaker: Miguel Navascues

Title: Entanglement and Nonlocality of 1D Macroscopic Systems

Abstract: We consider the problem of certifying entanglement and nonlocality in one-dimensional macroscopic systems when just averaged near-neighbor correlators are available. We map this question to the characterization of the set of all quantum states (distributions) which admit a separable (classical) Translation-Invariant (TI) extension to infinitely many sites. We advance the first problem by constructing a family of witnesses which can detect entanglement in infinite TI states from the nearest-neighbor reduced density matrix. Along the way, we identify a set of separable two-qubit states which only admit entangled TI extensions. For nonlocality detection, we show that the set of classical TI boxes forms a polytope and devise a general procedure to generate all Bell inequalities which characterize it. Through a Matrix Product State-based method, we show how some of them can be violated by distant parties conducting identical measurements on an infinite TI quantum state.

Livestream link: https://www.youtube.com/watch?v=awvdeBtcubY

Next Q+ Hangout is on November 8th, 2 pm GMT.

As usual, if you are watching with a group and want to reserve a seat in the hangout then leave a comment. We also encourage individuals interested in active participation---which typically involves asking questions after the talk---to join the hangout. Otherwise you can watch on the livestream. Details follow.

Speaker: Miguel Navascues

Title: Entanglement and Nonlocality of 1D Macroscopic Systems

Abstract: We consider the problem of certifying entanglement and nonlocality in one-dimensional macroscopic systems when just averaged near-neighbor correlators are available. We map this question to the characterization of the set of all quantum states (distributions) which admit a separable (classical) Translation-Invariant (TI) extension to infinitely many sites. We advance the first problem by constructing a family of witnesses which can detect entanglement in infinite TI states from the nearest-neighbor reduced density matrix. Along the way, we identify a set of separable two-qubit states which only admit entangled TI extensions. For nonlocality detection, we show that the set of classical TI boxes forms a polytope and devise a general procedure to generate all Bell inequalities which characterize it. Through a Matrix Product State-based method, we show how some of them can be violated by distant parties conducting identical measurements on an infinite TI quantum state.

Livestream link: https://www.youtube.com/watch?v=awvdeBtcubY

As usual, if you are watching with a group and want to reserve a seat in the hangout then leave a comment. We also encourage individuals interested in active participation---which typically involves asking questions after the talk---to join the hangout. Otherwise you can watch on the livestream. Details follow.

Speaker: Miguel Navascues

Title: Entanglement and Nonlocality of 1D Macroscopic Systems

Abstract: We consider the problem of certifying entanglement and nonlocality in one-dimensional macroscopic systems when just averaged near-neighbor correlators are available. We map this question to the characterization of the set of all quantum states (distributions) which admit a separable (classical) Translation-Invariant (TI) extension to infinitely many sites. We advance the first problem by constructing a family of witnesses which can detect entanglement in infinite TI states from the nearest-neighbor reduced density matrix. Along the way, we identify a set of separable two-qubit states which only admit entangled TI extensions. For nonlocality detection, we show that the set of classical TI boxes forms a polytope and devise a general procedure to generate all Bell inequalities which characterize it. Through a Matrix Product State-based method, we show how some of them can be violated by distant parties conducting identical measurements on an infinite TI quantum state.

Livestream link: https://www.youtube.com/watch?v=awvdeBtcubY

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Next Q+ Hangout is on the 12th of April folks!

As usual, if you are watching with a group and want to reserve a seat in the hangout then leave a comment. We also encourage individuals interested in active participation---which typically involves asking questions after the talk---to join the hangout. Otherwise you can watch on the livestream. Details follow.

Title: Size-driven quantum phase transitions

Speaker: David Perez-Garcia, Universidad Complutense de Madrid

Abstract: Most of the theoretical knowledge about quantum many body systems comes from performing numerical simulations. One tries to capture the relevant physical features of a system by extrapolating to the large system size the knowledge obtained in the analysis of an increasing sequence of finite-size systems, which must be small enough for the computer to be capable of giving an answer in a reasonable amount of time. In this work we show simple examples that totally defeat any such approach. More concretely, we construct translationally invariant quantum spin models on the 2D square lattice with reasonably small local dimension exhibiting the following surprising feature that we refer to as a "size-driven phase transition"': For all system sizes smaller than a threshold value L, the system has a unique ground state with product structure and a constant spectral gap to the first excited state, which also has product structure. However, for all system sizes larger than L, the system has topological quantum order, meaning a finite number of ground states which are locally indistinguishable, a finite spectral gap and first excited states with anyonic statistics. Moreover, we construct examples (all of them with local dimension smaller than 10) for which the threshold size L can occur at essentially any order of magnitude. From sizes that are reachable within current experimental setups and numerical simulations (L=15 or L=84) to sizes that are beyond any present or future capability, such as L> 10^35000.

As usual, if you are watching with a group and want to reserve a seat in the hangout then leave a comment. We also encourage individuals interested in active participation---which typically involves asking questions after the talk---to join the hangout. Otherwise you can watch on the livestream. Details follow.

Title: Size-driven quantum phase transitions

Speaker: David Perez-Garcia, Universidad Complutense de Madrid

Abstract: Most of the theoretical knowledge about quantum many body systems comes from performing numerical simulations. One tries to capture the relevant physical features of a system by extrapolating to the large system size the knowledge obtained in the analysis of an increasing sequence of finite-size systems, which must be small enough for the computer to be capable of giving an answer in a reasonable amount of time. In this work we show simple examples that totally defeat any such approach. More concretely, we construct translationally invariant quantum spin models on the 2D square lattice with reasonably small local dimension exhibiting the following surprising feature that we refer to as a "size-driven phase transition"': For all system sizes smaller than a threshold value L, the system has a unique ground state with product structure and a constant spectral gap to the first excited state, which also has product structure. However, for all system sizes larger than L, the system has topological quantum order, meaning a finite number of ground states which are locally indistinguishable, a finite spectral gap and first excited states with anyonic statistics. Moreover, we construct examples (all of them with local dimension smaller than 10) for which the threshold size L can occur at essentially any order of magnitude. From sizes that are reachable within current experimental setups and numerical simulations (L=15 or L=84) to sizes that are beyond any present or future capability, such as L> 10^35000.

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Next Q+ Hangout is on the 12th of April folks!

As usual, if you are watching with a group and want to reserve a seat in the hangout then leave a comment. We also encourage individuals interested in active participation---which typically involves asking questions after the talk---to join the hangout. Otherwise you can watch on the livestream. Details follow.

Title: Size-driven quantum phase transitions

Speaker: David Perez-Garcia, Universidad Complutense de Madrid

Abstract: Most of the theoretical knowledge about quantum many body systems comes from performing numerical simulations. One tries to capture the relevant physical features of a system by extrapolating to the large system size the knowledge obtained in the analysis of an increasing sequence of finite-size systems, which must be small enough for the computer to be capable of giving an answer in a reasonable amount of time. In this work we show simple examples that totally defeat any such approach. More concretely, we construct translationally invariant quantum spin models on the 2D square lattice with reasonably small local dimension exhibiting the following surprising feature that we refer to as a "size-driven phase transition"': For all system sizes smaller than a threshold value L, the system has a unique ground state with product structure and a constant spectral gap to the first excited state, which also has product structure. However, for all system sizes larger than L, the system has topological quantum order, meaning a finite number of ground states which are locally indistinguishable, a finite spectral gap and first excited states with anyonic statistics. Moreover, we construct examples (all of them with local dimension smaller than 10) for which the threshold size L can occur at essentially any order of magnitude. From sizes that are reachable within current experimental setups and numerical simulations (L=15 or L=84) to sizes that are beyond any present or future capability, such as L> 10^35000.

As usual, if you are watching with a group and want to reserve a seat in the hangout then leave a comment. We also encourage individuals interested in active participation---which typically involves asking questions after the talk---to join the hangout. Otherwise you can watch on the livestream. Details follow.

Title: Size-driven quantum phase transitions

Speaker: David Perez-Garcia, Universidad Complutense de Madrid

Abstract: Most of the theoretical knowledge about quantum many body systems comes from performing numerical simulations. One tries to capture the relevant physical features of a system by extrapolating to the large system size the knowledge obtained in the analysis of an increasing sequence of finite-size systems, which must be small enough for the computer to be capable of giving an answer in a reasonable amount of time. In this work we show simple examples that totally defeat any such approach. More concretely, we construct translationally invariant quantum spin models on the 2D square lattice with reasonably small local dimension exhibiting the following surprising feature that we refer to as a "size-driven phase transition"': For all system sizes smaller than a threshold value L, the system has a unique ground state with product structure and a constant spectral gap to the first excited state, which also has product structure. However, for all system sizes larger than L, the system has topological quantum order, meaning a finite number of ground states which are locally indistinguishable, a finite spectral gap and first excited states with anyonic statistics. Moreover, we construct examples (all of them with local dimension smaller than 10) for which the threshold size L can occur at essentially any order of magnitude. From sizes that are reachable within current experimental setups and numerical simulations (L=15 or L=84) to sizes that are beyond any present or future capability, such as L> 10^35000.

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