What is Nonclassical about Uncertainty Relations?

Catani, Lorenzo; Leifer, Matthew; Scala, Giovanni; Schmid, David; Spekkens, Robert W.

doi:10.1103/physrevlett.129.240401

Cited by 12 publications

(3 citation statements)

References 46 publications

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“…We have chosen to adopt that of generalised contextuality [20], which from hereon we will often often refer to as simply 'contextuality'. This is a modern version of contextuality that is commonly adopted in current foundations research [45,46,47,48]. Like the more orthodox notion of Kochen-Specker contextuality [49], generalised contextuality has also been connected to speedup in quantum computation [27], where it has been shown to be necessary for computational advantage.…”

Section: Generalised Contextualitymentioning

confidence: 99%

Contextuality and inductive bias in quantum machine learning

Bowles¹,

Wright²,

Farkas³

et al. 2023

Preprint

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Generalisation in machine learning often relies on the ability to encode structures present in data into an inductive bias of the model class. To understand the power of quantum machine learning, it is therefore crucial to identify the types of data structures that lend themselves naturally to quantum models. In this work we look to quantum contextuality-a form of nonclassicality with links to computational advantage-for answers to this question. We introduce a framework for studying contextuality in machine learning, which leads us to a definition of what it means for a learning model to be contextual. From this, we connect a central concept of contextuality, called operational equivalence, to the ability of a model to encode a linearly conserved quantity in its label space. A consequence of this connection is that contextuality is tied to expressivity: contextual model classes that encode the inductive bias are generally more expressive than their noncontextual counterparts. To demonstrate this, we construct an explicit toy learning problem-based on learning the payoff behaviour of a zero-sum game-for which this is the case. By leveraging tools from geometric quantum machine learning, we then describe how to construct quantum learning models with the associated inductive bias, and show through our toy problem that they outperform their corresponding classical surrogate models. This suggests that understanding learning problems of this form may lead to useful insights about the power of quantum machine learning.

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Section: Generalised Contextualitymentioning

confidence: 99%

Contextuality and inductive bias in quantum machine learning

Bowles¹,

Wright²,

Farkas³

et al. 2023

Preprint

View full text Add to dashboard Cite

show abstract

“…Moreover, knowing the maximal classical and quantum bound for a more general Bell scenario [39,57,58], for future research, one can extend the method we presented, which is also useful to test the criteria of nonclassicality and determine whether the trend of robustness is a general property that does not depend on the particular qubit-qubit Bell scenario, but it can also be applied for different notions of nonclassicality as well as generalised noncontextuality [59,60].…”

Section: Realising the Fidelity Constraintmentioning

confidence: 99%

On the Fidelity Robustness of CHSH–Bell Inequality via Filtered Random States

Mandarino

Scala

2023

Entropy

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The theorem developed by John Bell constituted the starting point of a revolution that translated a philosophical question about the nature of reality into the broad and intense field of research of the quantum information technologies. We focus on a system of two qubits prepared in a random, mixed state, and we study the typical behavior of their nonlocality via the CHSH–Bell inequality. Afterward, motivated by the necessity of accounting for inefficiency in the state preparation, we address to what extent states close enough to one with a high degree of nonclassicality can violate local realism with a previously chosen experimental setup.

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“…Besides the Robertson and Schrödinger relations, many other uncertainty relations are known. Indeed, since uncertainty relations have found applications in quantum information science [ 9 – 15 ] and quantum foundations [ 16 , 17 ], proving new ones has become something of a sport. The two most common classes of uncertainty relations are those based on entropy [ 18 ] and those based on standard deviations [ 4 , 5 , 19 ].…”

Section: Introductionmentioning

confidence: 99%

Uncertainty from the Aharonov–Vaidman identity

Leifer

2023

Quantum Stud.: Math. Found.

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In this article, I show how the Aharonov–Vaidman identity $$A\left| \psi \right\rangle = \left\langle A \right\rangle \left| \psi \right\rangle + \Delta A \left| \psi ^{\perp }_A \right\rangle $$ A ψ = A ψ + Δ A ψ A ⊥ can be used to prove relations between the standard deviations of observables in quantum mechanics. In particular, I review how it leads to a more direct and less abstract proof of the Robertson uncertainty relation $$\Delta A \Delta B \ge \frac{1}{2} \left| \left\langle [A,B] \right\rangle \right| $$ Δ A Δ B ≥ 1 2 [ A , B ] than the textbook proof. I discuss the relationship between these two proofs and show how the Cauchy–Schwarz inequality can be derived from the Aharonov–Vaidman identity. I give Aharonov–Vaidman based proofs of the Maccone–Pati uncertainty relations and show how the Aharonov–Vaidman identity can be used to handle propagation of uncertainty in quantum mechanics. Finally, I show how the Aharonov–Vaidman identity can be extended to mixed states and discuss how to generalize the results to the mixed case.

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What is Nonclassical about Uncertainty Relations?

Cited by 12 publications

References 46 publications

Contextuality and inductive bias in quantum machine learning

Contextuality and inductive bias in quantum machine learning

On the Fidelity Robustness of CHSH–Bell Inequality via Filtered Random States

Uncertainty from the Aharonov–Vaidman identity

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