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We give constructions of such states where <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>k<\/mml:mi><\/mml:math> is not too far from the optimal <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>n<\/mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>\/<\/mml:mo><\/mml:mrow><mml:mn>2<\/mml:mn><\/mml:math> while the individual parties need to hold only a constant number of qubits. In the special case when each party holds only one qubit, we describe a family of <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>n<\/mml:mi><\/mml:math>-qubit states with <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>k<\/mml:mi><\/mml:math> proportional to <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>log<\/mml:mi><mml:mo>&amp;#x2061;<\/mml:mo><mml:mi>n<\/mml:mi><\/mml:math> based on Reed-Muller codes, as well as small numerically found examples for <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>k<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>2<\/mml:mn><\/mml:math> and <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>k<\/mml:mi><mml:mo>=<\/mml:mo><mml:mn>3<\/mml:mn><\/mml:math>. We also prove some lower bounds, for example showing that if <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>k<\/mml:mi><mml:mo>=<\/mml:mo><mml:mi>n<\/mml:mi><mml:mrow class=\"MJX-TeXAtom-ORD\"><mml:mo>\/<\/mml:mo><\/mml:mrow><mml:mn>2<\/mml:mn><\/mml:math> then the parties must have at least <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi mathvariant=\"normal\">&amp;#x03A9;<\/mml:mi><mml:mo stretchy=\"false\">(<\/mml:mo><mml:mi>log<\/mml:mi><mml:mo>&amp;#x2061;<\/mml:mo><mml:mi>log<\/mml:mi><mml:mo>&amp;#x2061;<\/mml:mo><mml:mi>n<\/mml:mi><mml:mo stretchy=\"false\">)<\/mml:mo><\/mml:math> qubits each.<\/jats:p>","DOI":"10.22331\/q-2024-05-14-1348","type":"journal-article","created":{"date-parts":[[2024,5,14]],"date-time":"2024-05-14T13:15:52Z","timestamp":1715692552000},"page":"1348","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":6,"title":["Generating <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>k<\/mml:mi><\/mml:math> EPR-pairs from an <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>n<\/mml:mi><\/mml:math>-party resource state"],"prefix":"10.22331","volume":"8","author":[{"given":"Sergey","family":"Bravyi","sequence":"first","affiliation":[{"name":"IBM Quantum, IBM T.J. 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