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International Journal of Theoretical Physics

Uncertainty Inequality for the Entanglement of Both Two-rebits and Two-qubits States

Manuél Ávila

Published Jul 19, 2019• 1 citation
Publisher
DOI
10.1007/s10773-019-04198-1

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Qubit-Qudit Separability/PPT-Probability Analyses and Lovas-Andai Formula Extensions to Induced Measures

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We begin by seeking the qubit-qutrit and rebit-retrit counterparts to the now well-established Hilbert-Schmidt separability probabilities for (the 15-dimensional convex set of) two-qubits of $\frac{8}{33} = \frac{2^3}{3 \cdot 11} \approx 0.242424$ and (the 9-dimensional) two-rebits of $\frac{29}{64} =\frac{29}{2^6} \approx 0.453125$. Based in part on extensive numerical computations, we advance the possibilities of a qubit-qutrit value of $\frac{27}{1000} = (\frac{3}{10})^3 =\frac{3^3}{2^3 \cdot 5^3} = 0.027$ and a rebit-retrit one of $\frac{860}{6561} =\frac{2^2 \cdot 5 \cdot 43}{3^8} \approx 0.131078$. These four values for $2 \times m$ systems ($m=2,3$) suggest certain numerator/denominator sequences involving powers of $m$, which we further investigate for $m>3$. Additionally, we find that the Hilbert-Schmidt separability/PPT-probabilities for the two-rebit, rebit-retrit and two-retrit $X$-states all equal $\frac{16}{3 π^2} \approx 0.54038$, as well as more generally, that the probabilities based on induced measures are equal across these three sets of $X$-states. Then, we extend the generalized two-qubit framework introduced by Lovas and Andai from Hilbert-Schmidt measures to induced ones. For instance, while the Lovas-Andai two-qubit function is $\frac{1}{3} \varepsilon^2 (4 -\varepsilon^2)$, yielding $\frac{8}{33}$, its $k=1$ induced measure counterpart is $\frac{1}{4} \varepsilon ^2 \left(3-\varepsilon ^2\right)^2$, yielding $\frac{61}{143} =\frac{61}{11 \cdot 13} \approx 0.426573$, where $\varepsilon$ is a singular-value ratio. We investigate, in these regards, the possibility of extending the previously-obtained "Lovas-Andai master formula".

arXiv:1803.10680•Mar 28, 2018•1 citation
The three-tangle is a measure of three-way entanglement in a system of three qubits. For a pure state, it can be understood as the residual entanglement not accounted for by pairwise entanglements between individual qubits. Here we define and evaluate the analogous quantity for three rebits (that is, binary systems in the real-amplitude variant of quantum theory). We find that the resulting formula is the same as in the complex case, except that an overall absolute value sign is missing. As a result, the rebit three-tangle can be negative, expressing the possibility of non-monogamous entanglement in real-amplitude quantum theory (for entanglement based on the convex-roof construction). We then relate the entanglement among three rebits to the entanglement of two qubits, by re-expressing the two-qubit state as a three-rebit state in the ubit model. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to '50 years of Bell's theorem'.
We begin by seeking the qubit-qutrit and rebit-retrit counterparts to the now well-established Hilbert-Schmidt separability probabilities for (the 15-dimensional convex set of) two-qubits of 833=233⋅11≈0.242424\frac{8}{33} = \frac{2^3}{3 \cdot 11} \approx 0.242424338​=3⋅1123​≈0.242424 and (the 9-dimensional) two-rebits of 2964=2926≈0.453125\frac{29}{64} =\frac{29}{2^6} \approx 0.4531256429​=2629​≈0.453125. Based in part on extensive numerical computations, we advance the possibilities of a qubit-qutrit value of 271000=(310)3=3323⋅53=0.027\frac{27}{1000} = (\frac{3}{10})^3 =\frac{3^3}{2^3 \cdot 5^3} = 0.027100027​=(103​)3=23⋅5333​=0.027 and a rebit-retrit one of 8606561=22⋅5⋅4338≈0.131078\frac{860}{6561} =\frac{2^2 \cdot 5 \cdot 43}{3^8} \approx 0.1310786561860​=3822⋅5⋅43​≈0.131078. These four values for 2×m2 \times m2×m systems (m=2,3m=2,3m=2,3) suggest certain numerator/denominator sequences involving powers of mmm, which we further investigate for m>3m>3m>3. Additionally, we find that the Hilbert-Schmidt separability/PPT-probabilities for the two-rebit, rebit-retrit and two-retrit XXX-states all equal 163π2≈0.54038\frac{16}{3 π^2} \approx 0.540383π216​≈0.54038, as well as more generally, that the probabilities based on induced measures are equal across these three sets of XXX-states. Then, we extend the generalized two-qubit framework introduced by Lovas and Andai from Hilbert-Schmidt measures to induced ones. For instance, while the Lovas-Andai two-qubit function is 13ε2(4−ε2)\frac{1}{3} \varepsilon^2 (4 -\varepsilon^2)31​ε2(4−ε2), yielding 833\frac{8}{33}338​, its k=1k=1k=1 induced measure counterpart is 14ε2(3−ε2)2\frac{1}{4} \varepsilon ^2 \left(3-\varepsilon ^2\right)^241​ε2(3−ε2)2, yielding 61143=6111⋅13≈0.426573\frac{61}{143} =\frac{61}{11 \cdot 13} \approx 0.42657314361​=11⋅1361​≈0.426573, where ε\varepsilonε is a singular-value ratio. We investigate, in these regards, the possibility of extending the previously-obtained "Lovas-Andai master formula".