We address various aspects of a widely used tool in quantum information theory: the Choi-Jamio lk... more We address various aspects of a widely used tool in quantum information theory: the Choi-Jamio lkowski isomorphism [A. Jamio lkowski, Rep. Math. Phys., 3, 275 (1972)]. We review different versions of the isomorphism, their properties and propose a unified description that combines them all. To this end, we identify the physical reason for the appearance of the (basis-dependent) operation of transposition in the isomorphism as used in Choi's theorem [M.-D. Choi, Lin. Alg. Appl., 10, 285 (1975)]. This requires a careful distinction between Jordan algebras and the different C *-algebras they arise from, which are distinguished by their order of composition. Physically, the latter encodes a choice of time orientation in the respective algebras, which relates to a number of recent results, including a characterisation of quantum from more general nonsignalling bipartite correlations [M. Frembs and A. Döring, arXiv:2204.11471] and a classification of bipartite entanglement [M. Frembs, arXiv:2207.00024].
Linear constraint systems (LCS) have proven to be a surprisingly prolific tool in the study of no... more Linear constraint systems (LCS) have proven to be a surprisingly prolific tool in the study of nonclassical correlations and various related issues in quantum foundations. Many results are known for the Boolean case, yet the generalisation to systems of odd dimension is largely open. In particular, it is not known whether there exist LCS in odd dimension, which admit finite-dimensional quantum, but no classical solutions. Here, we approach this question from a computational perspective. We observe that every deterministic, non-adaptive measurement-based quantum computation (MBQC) with linear sideprocessing defines a LCS. Moreover, the measurement operators of such a MBQC almost define a quantum solution to the respective LCS: the only difference is that measurement operators generally only commute with respect to the resource state of the MBQC. This raises the question whether this state-dependence can be lifted in certain cases, thus providing examples of quantum solutions to LCS in odd dimension. Our main result asserts that no such examples arise within a large extension of the Pauli group for p odd prime, which naturally arises from and is universal for computation in deterministic, non-adaptive MBQC with linear side-processing.
We provide a new perspective on the close relationship between entanglement and time. Our main fo... more We provide a new perspective on the close relationship between entanglement and time. Our main focus is on bipartite entanglement, where this connection is foreshadowed both in the positive partial transpose criterion due to Peres [A. Peres, Phys. Rev. Lett., 77, 1413 (1996)] and in the classification of quantum within more general non-signalling bipartite correlations [M. Frembs and A. Döring, http://arxiv.org/abs/2204.11471]. Extracting the relevant common features, we identify a necessary and sufficient condition for bipartite entanglement in terms of a compatibility condition with respect to time orientations in local observable algebras, which express the dynamics in the respective subsystems. We discuss the relevance of the latter in the broader context of von Neumann algebras and the thermodynamical notion of time naturally arising within the latter.
We introduce a formalism for the evaluation of counterfactual queries in the framework of quantum... more We introduce a formalism for the evaluation of counterfactual queries in the framework of quantum causal models, by generalising the three-step procedure of abduction, action, and prediction in Pearl's classical formalism of counterfactuals [1]. To this end, we define a suitable extension of Pearl's notion of a 'classical structural causal model', which we denote analogously by 'quantum structural causal model'. We show that every classical (probabilistic) structural causal model can be extended to a quantum structural causal model, and prove that counterfactual queries that can be formulated within a classical structural causal model agree with their corresponding queries in the quantum extension-but the latter is more expressive. Counterfactuals in quantum causal models come in different forms: we distinguish between active and passive counterfactual queries, depending on whether or not an intervention is to be performed in the action step. This is in contrast to the classical case, where counterfactuals are always interpreted in the active sense. As a consequence of this distinction, we observe that quantum causal models break the connection between causal and counterfactual dependence that exists in the classical case: (passive) quantum counterfactuals allow counterfactual dependence without causal dependence. This illuminates an important distinction between classical and quantum causal models, which underlies the fact that the latter can reproduce quantum correlations that violate Bell inequalities while being faithful to the relativistic causal structure.
Contextuality-the obstruction to describing quantum mechanics in a classical statistical way-has ... more Contextuality-the obstruction to describing quantum mechanics in a classical statistical way-has been proposed as a resource that powers quantum computing. The measurement-based model provides a concrete manifestation of contextuality as a computational resource, as follows. If local measurements on a multi-qubit state can be used to evaluate nonlinear boolean functions with only linear control processing, then this computation constitutes a proof of strong contextuality-the possible local measurement outcomes cannot all be pre-assigned. However, this connection is restricted to the special case when the local measured systems are qubits, which have unusual properties from the perspective of contextuality. A single qubit cannot allow for a proof of contextuality, unlike higher-dimensional systems, and multiple qubits can allow for stateindependent contextuality with only Pauli observables, again unlike higher-dimensional generalisations. Here we identify precisely that strong non-locality is necessary in a qudit measurement-based computation (MBC) that evaluates high-degree polynomial functions with only linear control. We introduce the concept of local universality, which places a bound on the space of output functions accessible under the constraint of single-qudit measurements. Thus, the partition of a physical system into subsystems plays a crucial role for the increase in computational power. A prominent feature of our setting is that the enabling resources for qubit and qudit MBC are of the same underlying nature, avoiding the pathologies associated with qubit contextuality.
For certain restricted computational tasks, quantum mechanics provides a provable advantage over ... more For certain restricted computational tasks, quantum mechanics provides a provable advantage over any possible classical implementation. Several of these results have been proven using the framework of measurement-based quantum computation (MBQC), where nonlocality and more generally contextuality have been identified as necessary resources for certain quantum computations. Here, we consider the computational power of MBQC in more detail by refining its resource requirements, both on the allowed operations and the number of accessible qubits. More precisely, we identify which Boolean functions can be computed in non-adaptive MBQC, with local operations contained within a finite level in the Clifford hierarchy. Moreover, for non-adaptive MBQC restricted to certain subtheories such as stabiliser MBQC, we compute the minimal number of qubits required to compute a given Boolean function. Our results point towards hierarchies of resources that more sharply characterise the power of MBQC beyond the binary of contextuality vs non-contextuality.
Linear constraint systems (LCS) have proven to be a surprisingly prolific tool in the study of no... more Linear constraint systems (LCS) have proven to be a surprisingly prolific tool in the study of nonclassical correlations and various related issues in quantum foundations. Many results are known for the Boolean case, yet the generalisation to systems of odd dimension is largely open. In particular, it is not known whether there exist LCS in odd dimension, which admit finite-dimensional quantum, but no classical solutions. Here, we approach this question from a computational perspective. We observe that every deterministic, non-adaptive measurement-based quantum computation (MBQC) with linear sideprocessing defines a LCS. Moreover, the measurement operators of such a MBQC almost define a quantum solution to the respective LCS: the only difference is that measurement operators generally only commute with respect to the resource state of the MBQC. This raises the question whether this state-dependence can be lifted in certain cases, thus providing examples of quantum solutions to LCS in odd dimension. Our main result asserts that no such examples arise within a large extension of the Pauli group for p odd prime, which naturally arises from and is universal for computation in deterministic, non-adaptive MBQC with linear side-processing.
Gleason's theorem [25] is an important result in the foundations of quantum mechanics, where it j... more Gleason's theorem [25] is an important result in the foundations of quantum mechanics, where it justifies the Born rule as a mathematical consequence of the quantum formalism. Formally, it presents a key insight into the projective geometry of Hilbert spaces, showing that finitely additive measures on the projection lattice P(H) extend to positive linear functionals on the algebra of
For certain restricted computational tasks, quantum mechanics provides a provable advantage over ... more For certain restricted computational tasks, quantum mechanics provides a provable advantage over any possible classical implementation. Several of these results have been proven using the framework of measurement-based quantum computation (MBQC), where non-locality and more generally contextuality have been identified as necessary resources for certain quantum computations. Here, we consider the computational power of MBQC in more detail by refining its resource requirements, both on the allowed operations and the number of accessible qubits. More precisely, we identify which Boolean functions can be computed in non-adaptive MBQC, with local operations contained within a finite level in the Clifford hierarchy. Moreover, for non-adaptive MBQC restricted to certain subtheories such as stabiliser MBQC, we compute the minimal number of qubits required to compute a given Boolean function. Our results point towards hierarchies of resources that more sharply characterise the power of MBQC beyond the binary of contextuality vs non-contextuality.
We provide a new perspective on the close relationship between entanglement and time. Our main fo... more We provide a new perspective on the close relationship between entanglement and time. Our main focus is on bipartite entanglement, where this connection is foreshadowed both in the positive partial transpose criterion due to Peres [A. Peres, Phys. Rev. Lett., 77, 1413 (1996)] and in the classification of quantum within more general non-signalling bipartite correlations [M. Frembs and A. Döring, http://arxiv.org/abs/2204.11471]. Extracting the relevant common features, we identify a necessary and sufficient condition for bipartite entanglement in terms of a compatibility condition with respect to time orientations in local observable algebras, which express the dynamics in the respective subsystems. We discuss the relevance of the latter in the broader context of von Neumann algebras and the thermodynamical notion of time naturally arising within the latter.
Bell's seminal paper shows that some correlations in quantum theory are not reconcilable with... more Bell's seminal paper shows that some correlations in quantum theory are not reconcilable with hidden variables and the classical notion of locality. Yet, a weaker notion of locality, known as no-signalling, survives the no-go-result. Here, we study its restrictiveness by considering the full set of local quantum observables. This leads to a much larger set of no-signalling constraints than usually considered, which is enough to exclude PR-boxes and other types of super-quantum correlations. We emphasise the close connection with contextuality, in particular, we show how no-signalling arises naturally from context composition. Finally, we prove a one-to-one correspondence between quantum states and collections of (non-signalling) probability distributions over product contexts under an additional consistency condition between time arrows in subsystems.
Characterising quantum correlations from physical principles is a central problem in the field of... more Characterising quantum correlations from physical principles is a central problem in the field of quantum information theory. Entanglement breaks bounds on correlations put by Bell's theorem, thus challenging the notion of local causality as a physical principle. A natural relaxation is to study no-signalling as a constraint on joint probability distributions. It has been shown that when considered with respect to so-called locally quantum observables, bipartite non-signalling correlations never exceed their quantum counterparts; still, such correlations generally do not derive from quantum states. This leaves open the search for additional principles which identify quantum states within the larger set of (collections of) non-signalling joint probability distributions over locally quantum observables. Here, we suggest a natural generalisation of no-signalling in the form of no-disturbance to dilated systems. We prove that non-signalling joint probability distributions satisfying this extension correspond with bipartite quantum states up to a choice of time orientation in subsystems.
Contextuality - the obstruction to describing quantum mechanics in a classical statistical way - ... more Contextuality - the obstruction to describing quantum mechanics in a classical statistical way - has been proposed as a resource that powers quantum computing. The measurement-based model provides a concrete manifestation of contextuality as a computational resource, as follows. If local measurements on a multi-qubit state can be used to evaluate non-linear boolean functions with only linear control processing, then this computation constitutes a proof of strong contextuality - the possible local measurement outcomes cannot all be pre-assigned. However, this connection is restricted to the special case when the local measured systems are qubits, which have unusual properties from the perspective of contextuality. A single qubit cannot allow for a proof of contextuality, unlike higher-dimensional systems, and multiple qubits can allow for state-independent contextuality with only Pauli observables, again unlike higher-dimensional generalisations. Here we identify precisely that stron...
Contextuality is a key feature of quantum mechanics, as was first brought to light by Bohr and la... more Contextuality is a key feature of quantum mechanics, as was first brought to light by Bohr and later realised more technically by Kochen and Specker. Isham and Butterfield put contextuality at the heart of their topos-based formalism and gave a reformulation of the Kochen-Specker theorem in the language of presheaves. Here, we broaden this perspective considerably (partly drawing on existing, but scattered results) and show that apart from the Kochen-Specker theorem, also Wigner's theorem, Gleason's theorem, and Bell's theorem relate fundamentally to contextuality. We provide reformulations of the theorems using the language of presheaves over contexts and give general versions valid for von Neumann algebras. This shows that a very substantial part of the structure of quantum theory is encoded by contextuality.
Contextuality-the obstruction to describing quantum mechanics in a classical statistical way-has ... more Contextuality-the obstruction to describing quantum mechanics in a classical statistical way-has been proposed as a resource that powers quantum computing. The measurement-based model provides a concrete manifestation of contextuality as a computational resource, as follows. If local measurements on a multi-qubit state can be used to evaluate nonlinear boolean functions with only linear control processing, then this computation constitutes a proof of strong contextuality-the possible local measurement outcomes cannot all be pre-assigned. However, this connection is restricted to the special case when the local measured systems are qubits, which have unusual properties from the perspective of contextuality. A single qubit cannot allow for a proof of contextuality, unlike higher-dimensional systems, and multiple qubits can allow for stateindependent contextuality with only Pauli observables, again unlike higher-dimensional generalisations. Here we identify precisely that strong non-locality is necessary in a qudit measurement-based computation (MBC) that evaluates high-degree polynomial functions with only linear control. We introduce the concept of local universality, which places a bound on the space of output functions accessible under the constraint of single-qudit measurements. Thus, the partition of a physical system into subsystems plays a crucial role for the increase in computational power. A prominent feature of our setting is that the enabling resources for qubit and qudit MBC are of the same underlying nature, avoiding the pathologies associated with qubit contextuality.
We address various aspects of a widely used tool in quantum information theory: the Choi-Jamio lk... more We address various aspects of a widely used tool in quantum information theory: the Choi-Jamio lkowski isomorphism [A. Jamio lkowski, Rep. Math. Phys., 3, 275 (1972)]. We review different versions of the isomorphism, their properties and propose a unified description that combines them all. To this end, we identify the physical reason for the appearance of the (basis-dependent) operation of transposition in the isomorphism as used in Choi's theorem [M.-D. Choi, Lin. Alg. Appl., 10, 285 (1975)]. This requires a careful distinction between Jordan algebras and the different C *-algebras they arise from, which are distinguished by their order of composition. Physically, the latter encodes a choice of time orientation in the respective algebras, which relates to a number of recent results, including a characterisation of quantum from more general nonsignalling bipartite correlations [M. Frembs and A. Döring, arXiv:2204.11471] and a classification of bipartite entanglement [M. Frembs, arXiv:2207.00024].
Linear constraint systems (LCS) have proven to be a surprisingly prolific tool in the study of no... more Linear constraint systems (LCS) have proven to be a surprisingly prolific tool in the study of nonclassical correlations and various related issues in quantum foundations. Many results are known for the Boolean case, yet the generalisation to systems of odd dimension is largely open. In particular, it is not known whether there exist LCS in odd dimension, which admit finite-dimensional quantum, but no classical solutions. Here, we approach this question from a computational perspective. We observe that every deterministic, non-adaptive measurement-based quantum computation (MBQC) with linear sideprocessing defines a LCS. Moreover, the measurement operators of such a MBQC almost define a quantum solution to the respective LCS: the only difference is that measurement operators generally only commute with respect to the resource state of the MBQC. This raises the question whether this state-dependence can be lifted in certain cases, thus providing examples of quantum solutions to LCS in odd dimension. Our main result asserts that no such examples arise within a large extension of the Pauli group for p odd prime, which naturally arises from and is universal for computation in deterministic, non-adaptive MBQC with linear side-processing.
We provide a new perspective on the close relationship between entanglement and time. Our main fo... more We provide a new perspective on the close relationship between entanglement and time. Our main focus is on bipartite entanglement, where this connection is foreshadowed both in the positive partial transpose criterion due to Peres [A. Peres, Phys. Rev. Lett., 77, 1413 (1996)] and in the classification of quantum within more general non-signalling bipartite correlations [M. Frembs and A. Döring, http://arxiv.org/abs/2204.11471]. Extracting the relevant common features, we identify a necessary and sufficient condition for bipartite entanglement in terms of a compatibility condition with respect to time orientations in local observable algebras, which express the dynamics in the respective subsystems. We discuss the relevance of the latter in the broader context of von Neumann algebras and the thermodynamical notion of time naturally arising within the latter.
We introduce a formalism for the evaluation of counterfactual queries in the framework of quantum... more We introduce a formalism for the evaluation of counterfactual queries in the framework of quantum causal models, by generalising the three-step procedure of abduction, action, and prediction in Pearl's classical formalism of counterfactuals [1]. To this end, we define a suitable extension of Pearl's notion of a 'classical structural causal model', which we denote analogously by 'quantum structural causal model'. We show that every classical (probabilistic) structural causal model can be extended to a quantum structural causal model, and prove that counterfactual queries that can be formulated within a classical structural causal model agree with their corresponding queries in the quantum extension-but the latter is more expressive. Counterfactuals in quantum causal models come in different forms: we distinguish between active and passive counterfactual queries, depending on whether or not an intervention is to be performed in the action step. This is in contrast to the classical case, where counterfactuals are always interpreted in the active sense. As a consequence of this distinction, we observe that quantum causal models break the connection between causal and counterfactual dependence that exists in the classical case: (passive) quantum counterfactuals allow counterfactual dependence without causal dependence. This illuminates an important distinction between classical and quantum causal models, which underlies the fact that the latter can reproduce quantum correlations that violate Bell inequalities while being faithful to the relativistic causal structure.
Contextuality-the obstruction to describing quantum mechanics in a classical statistical way-has ... more Contextuality-the obstruction to describing quantum mechanics in a classical statistical way-has been proposed as a resource that powers quantum computing. The measurement-based model provides a concrete manifestation of contextuality as a computational resource, as follows. If local measurements on a multi-qubit state can be used to evaluate nonlinear boolean functions with only linear control processing, then this computation constitutes a proof of strong contextuality-the possible local measurement outcomes cannot all be pre-assigned. However, this connection is restricted to the special case when the local measured systems are qubits, which have unusual properties from the perspective of contextuality. A single qubit cannot allow for a proof of contextuality, unlike higher-dimensional systems, and multiple qubits can allow for stateindependent contextuality with only Pauli observables, again unlike higher-dimensional generalisations. Here we identify precisely that strong non-locality is necessary in a qudit measurement-based computation (MBC) that evaluates high-degree polynomial functions with only linear control. We introduce the concept of local universality, which places a bound on the space of output functions accessible under the constraint of single-qudit measurements. Thus, the partition of a physical system into subsystems plays a crucial role for the increase in computational power. A prominent feature of our setting is that the enabling resources for qubit and qudit MBC are of the same underlying nature, avoiding the pathologies associated with qubit contextuality.
For certain restricted computational tasks, quantum mechanics provides a provable advantage over ... more For certain restricted computational tasks, quantum mechanics provides a provable advantage over any possible classical implementation. Several of these results have been proven using the framework of measurement-based quantum computation (MBQC), where nonlocality and more generally contextuality have been identified as necessary resources for certain quantum computations. Here, we consider the computational power of MBQC in more detail by refining its resource requirements, both on the allowed operations and the number of accessible qubits. More precisely, we identify which Boolean functions can be computed in non-adaptive MBQC, with local operations contained within a finite level in the Clifford hierarchy. Moreover, for non-adaptive MBQC restricted to certain subtheories such as stabiliser MBQC, we compute the minimal number of qubits required to compute a given Boolean function. Our results point towards hierarchies of resources that more sharply characterise the power of MBQC beyond the binary of contextuality vs non-contextuality.
Linear constraint systems (LCS) have proven to be a surprisingly prolific tool in the study of no... more Linear constraint systems (LCS) have proven to be a surprisingly prolific tool in the study of nonclassical correlations and various related issues in quantum foundations. Many results are known for the Boolean case, yet the generalisation to systems of odd dimension is largely open. In particular, it is not known whether there exist LCS in odd dimension, which admit finite-dimensional quantum, but no classical solutions. Here, we approach this question from a computational perspective. We observe that every deterministic, non-adaptive measurement-based quantum computation (MBQC) with linear sideprocessing defines a LCS. Moreover, the measurement operators of such a MBQC almost define a quantum solution to the respective LCS: the only difference is that measurement operators generally only commute with respect to the resource state of the MBQC. This raises the question whether this state-dependence can be lifted in certain cases, thus providing examples of quantum solutions to LCS in odd dimension. Our main result asserts that no such examples arise within a large extension of the Pauli group for p odd prime, which naturally arises from and is universal for computation in deterministic, non-adaptive MBQC with linear side-processing.
Gleason's theorem [25] is an important result in the foundations of quantum mechanics, where it j... more Gleason's theorem [25] is an important result in the foundations of quantum mechanics, where it justifies the Born rule as a mathematical consequence of the quantum formalism. Formally, it presents a key insight into the projective geometry of Hilbert spaces, showing that finitely additive measures on the projection lattice P(H) extend to positive linear functionals on the algebra of
For certain restricted computational tasks, quantum mechanics provides a provable advantage over ... more For certain restricted computational tasks, quantum mechanics provides a provable advantage over any possible classical implementation. Several of these results have been proven using the framework of measurement-based quantum computation (MBQC), where non-locality and more generally contextuality have been identified as necessary resources for certain quantum computations. Here, we consider the computational power of MBQC in more detail by refining its resource requirements, both on the allowed operations and the number of accessible qubits. More precisely, we identify which Boolean functions can be computed in non-adaptive MBQC, with local operations contained within a finite level in the Clifford hierarchy. Moreover, for non-adaptive MBQC restricted to certain subtheories such as stabiliser MBQC, we compute the minimal number of qubits required to compute a given Boolean function. Our results point towards hierarchies of resources that more sharply characterise the power of MBQC beyond the binary of contextuality vs non-contextuality.
We provide a new perspective on the close relationship between entanglement and time. Our main fo... more We provide a new perspective on the close relationship between entanglement and time. Our main focus is on bipartite entanglement, where this connection is foreshadowed both in the positive partial transpose criterion due to Peres [A. Peres, Phys. Rev. Lett., 77, 1413 (1996)] and in the classification of quantum within more general non-signalling bipartite correlations [M. Frembs and A. Döring, http://arxiv.org/abs/2204.11471]. Extracting the relevant common features, we identify a necessary and sufficient condition for bipartite entanglement in terms of a compatibility condition with respect to time orientations in local observable algebras, which express the dynamics in the respective subsystems. We discuss the relevance of the latter in the broader context of von Neumann algebras and the thermodynamical notion of time naturally arising within the latter.
Bell's seminal paper shows that some correlations in quantum theory are not reconcilable with... more Bell's seminal paper shows that some correlations in quantum theory are not reconcilable with hidden variables and the classical notion of locality. Yet, a weaker notion of locality, known as no-signalling, survives the no-go-result. Here, we study its restrictiveness by considering the full set of local quantum observables. This leads to a much larger set of no-signalling constraints than usually considered, which is enough to exclude PR-boxes and other types of super-quantum correlations. We emphasise the close connection with contextuality, in particular, we show how no-signalling arises naturally from context composition. Finally, we prove a one-to-one correspondence between quantum states and collections of (non-signalling) probability distributions over product contexts under an additional consistency condition between time arrows in subsystems.
Characterising quantum correlations from physical principles is a central problem in the field of... more Characterising quantum correlations from physical principles is a central problem in the field of quantum information theory. Entanglement breaks bounds on correlations put by Bell's theorem, thus challenging the notion of local causality as a physical principle. A natural relaxation is to study no-signalling as a constraint on joint probability distributions. It has been shown that when considered with respect to so-called locally quantum observables, bipartite non-signalling correlations never exceed their quantum counterparts; still, such correlations generally do not derive from quantum states. This leaves open the search for additional principles which identify quantum states within the larger set of (collections of) non-signalling joint probability distributions over locally quantum observables. Here, we suggest a natural generalisation of no-signalling in the form of no-disturbance to dilated systems. We prove that non-signalling joint probability distributions satisfying this extension correspond with bipartite quantum states up to a choice of time orientation in subsystems.
Contextuality - the obstruction to describing quantum mechanics in a classical statistical way - ... more Contextuality - the obstruction to describing quantum mechanics in a classical statistical way - has been proposed as a resource that powers quantum computing. The measurement-based model provides a concrete manifestation of contextuality as a computational resource, as follows. If local measurements on a multi-qubit state can be used to evaluate non-linear boolean functions with only linear control processing, then this computation constitutes a proof of strong contextuality - the possible local measurement outcomes cannot all be pre-assigned. However, this connection is restricted to the special case when the local measured systems are qubits, which have unusual properties from the perspective of contextuality. A single qubit cannot allow for a proof of contextuality, unlike higher-dimensional systems, and multiple qubits can allow for state-independent contextuality with only Pauli observables, again unlike higher-dimensional generalisations. Here we identify precisely that stron...
Contextuality is a key feature of quantum mechanics, as was first brought to light by Bohr and la... more Contextuality is a key feature of quantum mechanics, as was first brought to light by Bohr and later realised more technically by Kochen and Specker. Isham and Butterfield put contextuality at the heart of their topos-based formalism and gave a reformulation of the Kochen-Specker theorem in the language of presheaves. Here, we broaden this perspective considerably (partly drawing on existing, but scattered results) and show that apart from the Kochen-Specker theorem, also Wigner's theorem, Gleason's theorem, and Bell's theorem relate fundamentally to contextuality. We provide reformulations of the theorems using the language of presheaves over contexts and give general versions valid for von Neumann algebras. This shows that a very substantial part of the structure of quantum theory is encoded by contextuality.
Contextuality-the obstruction to describing quantum mechanics in a classical statistical way-has ... more Contextuality-the obstruction to describing quantum mechanics in a classical statistical way-has been proposed as a resource that powers quantum computing. The measurement-based model provides a concrete manifestation of contextuality as a computational resource, as follows. If local measurements on a multi-qubit state can be used to evaluate nonlinear boolean functions with only linear control processing, then this computation constitutes a proof of strong contextuality-the possible local measurement outcomes cannot all be pre-assigned. However, this connection is restricted to the special case when the local measured systems are qubits, which have unusual properties from the perspective of contextuality. A single qubit cannot allow for a proof of contextuality, unlike higher-dimensional systems, and multiple qubits can allow for stateindependent contextuality with only Pauli observables, again unlike higher-dimensional generalisations. Here we identify precisely that strong non-locality is necessary in a qudit measurement-based computation (MBC) that evaluates high-degree polynomial functions with only linear control. We introduce the concept of local universality, which places a bound on the space of output functions accessible under the constraint of single-qudit measurements. Thus, the partition of a physical system into subsystems plays a crucial role for the increase in computational power. A prominent feature of our setting is that the enabling resources for qubit and qudit MBC are of the same underlying nature, avoiding the pathologies associated with qubit contextuality.
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