2018/03/28 by Paul B. Slater, Slater, Paul B.
Computer Science · Mathematics · #15B52 #60B20 #81P16 #81P40 #81P45 #Benford’s Law and Fraud Detection #Computability, Logic, AI Algorithms #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Quantum Computing Algorithms and Architecture #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.1803.10680
openalex publication_date 2018/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We begin by seeking the qubit-qutrit and rebit-retrit counterparts to the now\nwell-established Hilbert-Schmidt separability probabilities for (the\n15-dimensional convex set of) two-qubits of \(8)/(33) = \(23)/(3 \⋅\n11) \≈ 0.242424 and (the 9-dimensional) two-rebits of \(29)/(64)\n=\(29)/(26) \≈ 0.453125. Based in part on extensive numerical\ncomputations, we advance the possibilities of a qubit-qutrit value of\n\(27)/(1000) = (\(3)/(10))3 =\(33)/(23 \⋅ 53) = 0.027 and a\nrebit-retrit one of \(860)/(6561) =\(22 \⋅ 5 \⋅ 43)/(38) \≈\n0.131078. These four values for 2 \× m systems (m=2,3) suggest certain\nnumerator/denominator sequences involving powers of m, which we further\ninvestigate for m>3. Additionally, we find that the Hilbert-Schmidt\nseparability/PPT-probabilities for the two-rebit, rebit-retrit and two-retrit\nX-states all equal \(16)/(3 \π2) \≈ 0.54038, as well as more\ngenerally, that the probabilities based on induced measures are equal across\nthese three sets of X-states. Then, we extend the generalized two-qubit\nframework introduced by Lovas and Andai from Hilbert-Schmidt measures to\ninduced ones. For instance, while the Lovas-Andai two-qubit function is\n\(1)/(3) \ε2 (4 -\ε2), yielding \(8)/(33), its\nk=1 induced measure counterpart is \(1)/(4) \ε 2\n\(3-\ε 2\)2, yielding \(61)/(143) =\(61)/(11 \⋅\n13) \≈ 0.426573, where \ε is a singular-value ratio. We\ninvestigate, in these regards, the possibility of extending the\npreviously-obtained "Lovas-Andai master formula".\n