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A long open problem of quantum information has finally bitten the dust ...
Thirty-six entangled officers of Euler
Suhail Ahmad Rather, Adam Burchardt, Wojciech Bruzda, Grzegorz Rajchel-Mieldzioć, Arul Lakshminarayan, Karol Życzkowski
The negative solution to the famous problem of 36 officers of Euler implies that there are no two orthogonal Latin squares of order six. We show that the problem has a solution, provided the officers are entangled, and construct orthogonal quantum Latin squares of this size. As a consequence, we find an Absolutely Maximally Entangled state AME(4,6) of four subsystems with six levels each, equivalently a 2-unitary matrix of size 36, which maximizes the entangling power among all bipartite unitary gates of this dimension, or a perfect tensor with four indices, each running from one to six. This result allows us to construct a pure non-additive quhex quantum error detection code ((3,6,2))6, which saturates the Singleton bound and allows one to encode a 6-level state into a triple of such states.
S. Aravinda, Suhail Ahmad Rather, Arul Lakshminarayan
Deterministic classical dynamical systems have an ergodic hierarchy, from ergodic through mixing, to Bernoulli systems that are "as random as a coin-toss". Dual-unitary circuits have been recently introduced as solvable models of many-body nonintegrable quantum chaotic systems having a hierarchy of ergodic properties. We extend this to include the apex of a putative quantum ergodic hierarchy which is Bernoulli, in the sense that correlations of single and two-particle observables vanish at space-time separated points. We derive a condition based on the entangling power ep(U) of the basic two-particle unitary building block, U, of the circuit, that guarantees mixing, and when maximized, corresponds to Bernoulli circuits. Additionally we show, both analytically and numerically, how local-averaging over single-particle unitaries leads to an identification of the average mixing rate as being determined solely by the entangling power
ep(U). The same considerations extend to disordered dual-unitary circuits as well. Finally we provide several, both analytical and numerical, ways to construct dual-unitary operators covering the entire possible range of entangling power. We construct a coupled quantum cat map which is dual-unitary for all local dimensions and a 2-unitary or perfect tensor for odd local dimensions, and can be used to build Bernoulli circuits. These constructions and results pave the way for a systematic extension of quantities studied in the dual-unitary circuits as well as provide natural generalizations to the non-dual cases.
Sreeram PG, Vaibhav Madhok, Arul Lakshminarayan
The out-of-time-ordered correlators (OTOC) and the Loschmidt echo are two measures that are now widely being explored to characterize sensitivity to perturbations and information scrambling in complex quantum systems. Studying few qubits systems collectively modelled as a kicked top, we solve exactly the three- and four- qubit cases, giving analytical results for the OTOC and the Loschmidt echo. While we may not expect such few-body systems to display semiclassical features, we find that there are clear signatures of the exponential growth of OTOC even in systems with as low as 4 qubits in appropriate regimes, paving way for possible experimental measurements. We explain qualitatively how classical phase space structures like fixed points and periodic orbits have an influence on these quantities and how our results compare to the large-spin kicked top model. Finally we point to a peculiar case at the border of quantum-classical correspondence which is solvable for any number of qubits and yet has signatures of exponential sensitivity in a rudimentary form.
Sivaprasad Omanakuttan, Arul Lakshminarayan
Coined discrete-time quantum walks are studied using simple deterministic dynamical systems as coins whose classical limit can range from being integrable to chaotic. It is shown that a Loschmidt echo like fidelity plays a central role and when the coin is chaotic this is approximately the characteristic function of a classical random walker. Thus the classical binomial distribution arises as a limit of the quantum walk and the walker exhibits diffusive growth before eventually becoming ballistic. The coin-walker entanglement growth is shown to be logarithmic in time as in the case of many-body localization and coupled kicked rotors, and saturates to a value that depends on the relative coin and walker space dimensions. In a coin dominated scenario, the chaos can thermalize the quantum walk to typical random states such that the entanglement saturates at the Haar averaged Page value, unlike in a walker dominated case when atypical states seem to be produced.
Published as Phys. Rev. E 103, 012207 (2021)
Suhail Ahmad Rather, S. Aravinda, Arul Lakshminarayan
Maximally entangled bipartite unitary operators or gates find various applications from quantum information to being building blocks of minimal models of many-body quantum chaos, and have been referred to as "dual unitaries". Dual unitary operators that can create the maximum average entanglement when acting on product states have to satisfy additional constraints. These have been called "2-unitaries" and are examples of perfect tensors that can be used to construct absolutely maximally entangled states of four parties. Hitherto, no systematic method exists, in any local dimension, which result in the formation of such special classes of unitary operators. We outline an iterative protocol, a nonlinear map on the space of unitary operators, that creates ensembles whose members are arbitrarily close to being dual unitaries, while for qutrits and ququads we find that a slightly modified protocol yields a plethora of 2-unitaries. We further characterize the dual unitary operators via their entangling power and the 2-unitaries via the distribution of entanglement created from unentangled states.
Published as Phys. Rev. Lett. 125, 070501 (2020)