Abstract
Quantum entanglement – the phenomenon where distant particles can be correlated in ways that cannot be explained by classical physics – has mystified scientists since the1930s, when quantum theory was beginning to emerge. Investigation into fundamental questions about quantum entanglement has continually propelled seismic shifts in our understanding of nature. Examples include Einstein, Podolsky and Rosen’s famous 1935 paper about the incompleteness of quantum mechanics, and John Bell’s refutation of EPR’s argument, 29 years later, via an experiment to demonstrate the non-classicality of quantum entanglement.
More recently, the field of quantum computing has motivated researchers to study entanglement in information processing contexts. One question of deep interest concerns the computability of nonlocal games, which are mathematical abstractions of Bell’s experiments. The question is simple: is there an algorithm to compute the optimal winning probability of a quantum game – or at least, approximate it? In this paper, I will discuss a remarkable connection between the complexity of nonlocal games and classes in the arithmetical hierarchy. In particular, different versions of the nonlocal games computability problem neatly line up with the problems of deciding \(\varSigma _1^0\), \(\varPi _1^0\), and \(\varPi _2^0\) sentences, respectively.
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Notes
- 1.
This non-communication constraint is used to model the situation when Alice and Bob are separated far from each other, and relativity prevents Alice and Bob from instantaneously signaling to each other.
- 2.
Formally, a commuting operator strategy consists of a unit state \(\psi \) defined on a separable Hilbert space \(\mathcal {H}\) (which is in general infinite-dimensional), and measurement operators \(\{A^x_a\}_{x,a}\) and \(\{B^y_b\}\) such that for all x, y, \(\sum _a A^x_a = \sum _b B^y_b = \mathbb {I}\) where \(\mathbb {I}\) denotes the identity operator on \(\mathcal {H}\), and furthermore Alice’s and Bob’s operators must commute with each other: \(A^x_a B^y_b = B^y_b A^x_a\) for all x, y, a, b. The probability of producing answers (a, b) given questions (x, y) is given by \(\psi ^* A^x_a B^y_b \psi \). The essential difference between this model of strategies and the quantum strategies defined above is that (a) the dimension of the Hilbert space may be infinite, and (b) there is not necessarily a tensor product structure in the Hilbert space.
- 3.
For the curious: \(\beta _d\) is defined to be the smallest number such that the nonnegativity of \(\beta _d - \omega _{co}(G)\) admits a degree-d sum-of-squares polynomial in noncommuting variables. Each \(\beta _d\) can be computed in finite time using the semidefinite programming hierarchies of [6, 9].
- 4.
Formally, these are sentences over a first-order language that uses binary strings as its universe, and can encode the behavior of Turing machines.
- 5.
We assume some natural distance measure between strategies; such as the sum of the \(\ell _2\) distances between the states and the measurements.
- 6.
In the terminology of computability theory, this is stating that the recursively enumerable languages and co-recursively enumerable languages are incomparable sets.
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Yuen, H. (2021). Einstein Meets Turing: The Computability of Nonlocal Games. In: De Mol, L., Weiermann, A., Manea, F., Fernández-Duque, D. (eds) Connecting with Computability. CiE 2021. Lecture Notes in Computer Science(), vol 12813. Springer, Cham. https://doi.org/10.1007/978-3-030-80049-9_47
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