-design on a quantum system followed by a fixed quantum operation almost decouples, with high probability, the given system from another reference system to which it may initially have been correlated. Earlier works either did not obtain high decoupling probability, or used provably inefficient unitaries, or required catalytic entanglement for decoupling. In contrast, our approximate unitary designs always guarantee decoupling with exponentially high probability and, under certain conditions, lead to computationally efficient unitaries. As a result we conclude that, under suitable conditions, efficiently implementable approximate unitary designs achieve relative thermalisation in quantum thermodynamics with exponentially high probability. We also show the scrambling property of black hole, when the black hole evolution is according to pseudorandom approximate unitary

t

-design, as opposed to the Haar random evolution considered earlier by Hayden-Preskill.

DOI: 10.48550/arxiv.2002.00247

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Strong unitary designs in optimal depth and space

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Approximate Unitary Designs Give Rise to Quantum Channels With Super Additive Classical Holevo Capacity

Aditya Nema, Pranab Sen

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IEEE Transactions on Information TheoryApr 21, 20221 citation

We prove a new concentration result for non-catalytic decoupling by showing that, for suitably large

t

, applying a unitary chosen uniformly at random from an approximate

t

t

-design, as opposed to the Haar random evolution considered earlier by Hayden-Preskill.

We prove a new concentration result for non-catalytic decoupling by showing that, for suitably large

t

, applying a unitary chosen uniformly at random from an approximate

t

t

-design, as opposed to the Haar random evolution considered earlier by Hayden-Preskill.

k

and measurable-error tolerance, we construct strong approximate unitary

k

-designs in optimal

Θ(\log n)

all-to-all circuit depth using only the

n

original system qubits. Our new ingredient is a logarithmic-depth Pauli-mixing bound for the perfect-matching ensemble, whose layers pair the qubits uniformly at random and apply independent random two-qubit gates. This bound controls the mixed forward-reverse two-query case, which we combine with existing design and gluing results to obtain strong unitary designs of arbitrary fixed order.

n^{2/3}

-design gives rise to a quantum channel with superadditive classical Holevo capacity, where

n

is the dimension of the unitary exhibiting the Stinespring dilation of the channel superoperator. We follow the geometric functional analytic approach of Aubrun, Szarek and Werner in order to prove our result. More precisely we prove a sharp Dvoretzky-like theorem stating that, with high probability under the choice of a unitary from an approximate

t

-design, random subspaces of large dimension make a Lipschitz function take almost constant value. Such theorems were known earlier only for Haar random unitaries. We obtain our result by appealing to Low’s technique for proving concentration of measure for an approximate

t

-design, combined with a stratified analysis of the variational behaviour of Lipschitz functions on the unit sphere in high dimension. The stratified analysis is the main technical advance of this work. Haar random unitaries require at least

\Omega (n^{2})

random bits in order to describe them with good precision. In contrast, there exist exact

n^{2/3}

-designs using only

O(n^{2/3} \log n)

random bits. Thus, our work can be viewed as a partial derandomisation of Hastings’ result, and a step towards the quest of finding an explicit quantum channel with superadditive classical Holevo capacity. Finally we also show that for any

p > 1

, approximate unitary

n^{1.7}

-designs give rise to channels violating subadditivity of Rényi

p

-entropy. In addition to stratified analysis, the proof of this result uses a new technique of approximating a monotonic differentiable function defined on a closed bounded interval and its derivative by moderate degree polynomials which should be of independent interest.