Algorithmic Pseudorandomness in Quantum Setups

Abstract

Many experimental setups in quantum physics use pseudorandomness in places where the theory requires randomness. In this Letter we show that the use of pseudorandomness instead of proper randomness in quantum setups has potentially observable consequences. First, we present a new loophole for Bell-like experiments: if some of the parties choose their measurements pseudorandomly, then the computational resources of the local model have to be limited in order to have a proper observation of nonlocality. Second, we show that no amount of pseudorandomness is enough to produce a mixed state by computably choosing pure states from some basis.

Randomness? What Randomness?

“…(B.4)This motivates the concept of a martingale as a function d : 2 * → [0, ∞) that satisfiesd(σ0) + d(σ1) = 2d(σ), (B.5)for each σ ∈ 2 * . Both the betting strategy itself and the payoff (for given x ∈ 2 N ) can be reconstructed from d: the bet on the N + 1'th digit x N +1 is given byb(x N +1 = 0) = 1 2 d(x N 0); (B.6) b(x N +1 = 1) = 1 2 d(x N 1), (B 7). and after N bets the punter owns d(x |N).…”

Algorithmic independence of initial condition and dynamical law in thermodynamics and causal inference

“…12 For quantum systems, note that [13] discusses a condition that can also indicate common causes. algorithmic information [57,58] and causality [7][8][9][10][11][12][13], our results may also point new directions for research in the foundations of quantum physics.…”

Distinguishing computable mixtures of quantum states

“…In this Section we extend the results from [1]. We will consider a scenario in which there are two boxes providing qudits to an observer named Bob.…”

“…Before going to Bob's protocol, let us first note that if we fix a basis B and only allow the computer to pick eigenstates from such basis, a slight modification of the protocol from [1] allows Bob to distinguish between the boxes. Namely, if instead of alternating between measuring σ x and measuring σ z as in [1], Bob measures the outputs of both boxes in the B basis, the d-ary sequence associated with the box which has the computer will be computable and the other, according to quantum mechanics, independent tosses of a fair coin and so Martin-Löf random. Hence, our previous result applies.…”

“…In this work we extend the ideas from [1] in two ways. First, we generalize the theoretical result by showing that any preparation performed by a computer intended to generate the maximally mixed state, and not just those in which the states are chosen from a predefined basis, can be distinguished from the maximally mixed state.…”