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SUMMARY:Karol Życzkowski (Jagiellonian University Cracow\, Poland\; Cente
 r for Theoretical Physics\, PAS\, Warsaw)
DTSTART:20211129T170000Z
DTEND:20211129T180000Z
DTSTAMP:20260423T024611Z
UID:QM3/55
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/QM3/55/">Mul
 ti-unitary matrices and their quantum applications</a>\nby Karol Życzkows
 ki (Jagiellonian University Cracow\, Poland\; Center for Theoretical Physi
 cs\, PAS\, Warsaw) as part of Quantum Matter meets Maths (IST\, Lisbon)\n\
 n\nAbstract\nIn the space of bipartite unitary gates one distinguishes the
  set of local gates\, formed by a tensor product\, $U=V_A \\otimes V_B$. A
 nother distinguished set contains gates of extremal non-locality\, which m
 aximize the entropy of entanglement defined by the operator Schmidt decomp
 osition of a unitary gate $U$. If the reshuffled matrix\, $U^R$\, is also 
 unitary the matrix $U$ belongs to this class and is called dual unitary. T
 he matrix $S$ corresponding to the SWAP operation is strongly non-local an
 d dual unitary\, but it does not change entanglement of any state it acts 
 on. To describe creation of entanglement in the system one defines entangl
 ing power of a gate. Its absolute maximum is achieved for any dual unitary
  gate $U$\, such that its partial transpose $U^{\\Gamma}$ is also unitary.
  These gates\, called two-unitary\, do not exist for dimension $d=2^2$\, b
 ut exist for $d=3^2$. We present an analytical construction of such a gate
  U of order $d=6^2=36$\, which leads to a solution of the quantum version 
 of the famous problem of $36$ officers of Euler [1]. It implies a pair of 
 quantum orthogonal Latin squares of order six and an Absolutely Maximally 
 Entangled (AME) state of four subsystems with six levels each. It enables 
 us to construct a quhex pure nonadditive quantum error detection code usef
 ul to encode a 6-level state into a triple of such states. Using such a st
 ate one can teleport any unknown\, two-dice quantum state\, from any pair 
 of two subsystems to the lab possessing the two other dice forming the fou
 r-dice system. Our result imples that $2$-unitary gates exist for any squa
 red dimension $d=N^2$ with $N\\ge 3$. A matrix $U$ of order $d=N^k$ is cal
 led k-unitary or multiunitary if it remains unitary after any of possible 
 (2k choose k) reordering of the matrix. Any such a matrix leads to an AME 
 state of $2k$ subsystems of size $N$. A simple example of a $3$-unitary ma
 trix $U$ of order $2^3=8$ corresponds to an AME(6\,2) state of six qubits.
 \n\nReferences:\n\n[1] S.A Rather\, A.Burchardt\, W. Bruzda\, G. Rajchel-M
 ieldzioć\, A. Lakshminarayan and K. Życzkowski\, Thirty-six entangled of
 ficers of Euler\, preprint arXiv:2104.05122\n
LOCATION:https://researchseminars.org/talk/QM3/55/
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