Kavli Affiliate: David Gross
| First 5 Authors: David Gross, Paulina Goedicke, , ,
| Summary:
A perfect tensor of order $d$ is a state of four $d$-level systems that is
maximally entangled under any bipartition. These objects have attracted
considerable attention in quantum information and many-body theory. Perfect
tensors generalize the combinatorial notion of orthogonal Latin squares (OLS).
Deciding whether OLS of a given order exist has historically been a difficult
problem. The case $d=6$ proved particularly thorny, and was popularized by
Leonhard Euler in terms of a putative constellation of "36 officers". It took
more than a century to show that Euler’s puzzle has no solution. After yet
another century, its quantum generalization was resolved in the affirmative: 36
entangled officers can be suitably arranged. However, the construction and
verification of known instances relies on elaborate computer codes. (In
particular, Leonhard would have had no means of dealing with such solutions to
his own puzzle — an unsatisfactory state of affairs). In this paper, we
present the first human-made order-$6$ perfect tensors. We decompose the
Hilbert space $(mathbb{C}^6)^{otimes 2}$ of two quhexes into the direct sum
$(mathbb{C}^3)^{otimes 2}oplus(mathbb{C}^3)^{otimes 3}$ comprising
superpositions of two-qutrit and three-qutrit states. Perfect tensors arise
when certain Clifford unitaries are applied separately to the two sectors.
Technically, our construction realizes solutions to the perfect functions
ansatz recently proposed by Rather. Generalizing an observation of Bruzda and
.Zyczkowski, we show that any solution of this kind gives rise to a
two-unitary complex Hadamard matrix, of which we construct infinite families.
Finally, we sketch a formulation of the theory of perfect tensors in terms of
quasi-orthogonal decompositions of matrix algebras.
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