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• W4387160832 abstract "Abstract Associative memories are devices storing information that can be fully retrieved given partial disclosure of it. We examine a toy model of associative memory and the ultimate limitations to which it is subjected within the framework of general probabilistic theories (GPTs), which represent the most general class of physical theories satisfying some basic operational axioms. We ask ourselves how large the dimension of a GPT should be so that it can accommodate 2 m states with the property that any N of them are perfectly distinguishable. Call <?CDATA $d(N,m)$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML overflow=scroll> <mml:mi>d</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:math> the minimal such dimension. Invoking an old result by Danzer and Grünbaum, we prove that <?CDATA $d(2,m) = m+1$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML overflow=scroll> <mml:mi>d</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:math> , to be compared with <?CDATA $O(2^m)$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML overflow=scroll> <mml:mi>O</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>m</mml:mi> </mml:msup> <mml:mo stretchy=false>)</mml:mo> </mml:math> when the GPT is required to be either classical or quantum. This yields an example of a task where GPTs outperform both classical and quantum theory exponentially. More generally, we resolve the case of fixed N and asymptotically large m , proving that <?CDATA $d(N,m) unicode{x2A7D} m^{1+o_N(1)}$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML overflow=scroll> <mml:mi>d</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mtext>⩽</mml:mtext> <mml:msup> <mml:mi>m</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>o</mml:mi> <mml:mi>N</mml:mi> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> </mml:msup> </mml:math> (as <?CDATA $mtoinfty$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML overflow=scroll> <mml:mi>m</mml:mi> <mml:mo stretchy=false>→</mml:mo> <mml:mi mathvariant=normal>∞</mml:mi> </mml:math> ) for every <?CDATA $Nunicode{x2A7E} 2$?> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML overflow=scroll> <mml:mi>N</mml:mi> <mml:mtext>⩾</mml:mtext> <mml:mn>2</mml:mn> </mml:math> , which yields again an exponential improvement over classical and quantum theories. Finally, we develop a numerical approach to the general problem of finding the largest N -wise mutually distinguishable set for a given GPT, which can be seen as an instance of the maximum clique problem on N -regular hypergraphs." @default.
• W4387160832 created "2023-09-30" @default.
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• W4387160832 date "2023-10-13" @default.
• W4387160832 modified "2023-10-15" @default.
• W4387160832 title "A post-quantum associative memory" @default.
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• W4387160832 doi "https://doi.org/10.1088/1751-8121/acfeb7" @default.
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