Construction of the Lovas-Andai Two-Qubit Function $\tilde{\chi}_2 (\varepsilon )=\frac{1}{3} \varepsilon ^2 \left(4-\varepsilon ^2\right)$ Verifies the $\frac{8}{33}$ -Hilbert Schmidt Separability Probability ConjectureConstruction of the Lovas-Andai Two-Qubit Function $\tilde{\chi}_2 (\varepsilon )=\frac{1}{3} \varepsilon ^2 \left(4-\varepsilon ^2\right)$ Verifies the $\frac{8}{33}$-Hilbert Schmidt Separability Probability Conjecture

Paul B. Slater

We investigate relationships between two forms of Hilbert-Schmidt two-re[al]bit and two-qubit functions''--those recently advanced by Lovas and Andai (arXiv:1610.01410), and those earlier presented by Slater ({\it J. Phys. A} {\bf{40}} [2007] 14279). In the Lovas-Andai framework, the independent variable $\varepsilon \in [0,1]$ is the ratio $\sigma(V)$ of the singular values of the $2 \times 2$ matrix $V=D_2^{1/2} D_1^{-1/2}$ formed from the two $2 \times 2$ diagonal blocks ( $D_1, D_2$ ) of a randomly generated $4 \times 4$ density matrix $D$ . In the Slater setting, the independent variable $\mu$ is the diagonal-entry ratio $\sqrt{\frac{d_ {11} d_ {44}}{d_ {22} d_ {33}}}$ --with, importantly, $\mu=\varepsilon$ or $\mu=\frac{1}{\varepsilon}$ when both $D_1$ and $D_2$ are themselves diagonal. Lovas and Andai established that their two-rebit function $\tilde{\chi}_1 (\varepsilon )$ ( $\approx \varepsilon$ ) yields the previously conjectured Hilbert-Schmidt separability probability of $\frac{29}{64}$ . We are able, in the Slater framework (using cylindrical algebraic decompositions [CAD] to enforce positivity constraints), to reproduce this result. Further, we similarly obtain its new (much simpler) two-qubit counterpart, $\tilde{\chi}_2(\varepsilon) =\frac{1}{3} \varepsilon ^2 \left(4-\varepsilon ^2\right)$ . Verification of the companion conjecture of a Hilbert-Schmidt separability probability of $\frac{8}{33}$ immediately follows in the Lovas-Andai framework. We obtain the formulas for $\tilde{\chi}_1(\varepsilon)$ and $\tilde{\chi}_2(\varepsilon)$ by taking $D_1$ and $D_2$ to be diagonal, allowing us to proceed in lower (7 and 11), rather than the full (9 and 15) dimensions occupied by the convex sets of two-rebit and two-qubit states. The CAD's themselves involve 4 and 8 variables, in addition to $\mu=\varepsilon$ . We also investigate extensions of these analyses to rebit-retrit and qubit-qutrit ( $6 \times 6$ ) settings.We investigate relationships between two forms of Hilbert-Schmidt two-re[al]bit and two-qubit functions''--those recently advanced by Lovas and Andai (arXiv:1610.01410), and those earlier presented by Slater ({\it J. Phys. A} {\bf{40}} [2007] 14279). In the Lovas-Andai framework, the independent variable $\varepsilon \in [0,1]$ is the ratio $\sigma(V)$ of the singular values of the $2 \times 2$ matrix $V=D_2^{1/2} D_1^{-1/2}$ formed from the two $2 \times 2$ diagonal blocks ($D_1, D_2$) of a randomly generated $4 \times 4$ density matrix $D$. In the Slater setting, the independent variable $\mu$ is the diagonal-entry ratio $\sqrt{\frac{d_ {11} d_ {44}}{d_ {22} d_ {33}}}$--with, importantly, $\mu=\varepsilon$ or $\mu=\frac{1}{\varepsilon}$ when both $D_1$ and $D_2$ are themselves diagonal. Lovas and Andai established that their two-rebit function $\tilde{\chi}_1 (\varepsilon )$ ($\approx \varepsilon$) yields the previously conjectured Hilbert-Schmidt separability probability of $\frac{29}{64}$. We are able, in the Slater framework (using cylindrical algebraic decompositions [CAD] to enforce positivity constraints), to reproduce this result. Further, we similarly obtain its new (much simpler) two-qubit counterpart, $\tilde{\chi}_2(\varepsilon) =\frac{1}{3} \varepsilon ^2 \left(4-\varepsilon ^2\right)$. Verification of the companion conjecture of a Hilbert-Schmidt separability probability of $\frac{8}{33}$ immediately follows in the Lovas-Andai framework. We obtain the formulas for $\tilde{\chi}_1(\varepsilon)$ and $\tilde{\chi}_2(\varepsilon)$ by taking $D_1$ and $D_2$ to be diagonal, allowing us to proceed in lower (7 and 11), rather than the full (9 and 15) dimensions occupied by the convex sets of two-rebit and two-qubit states. The CAD's themselves involve 4 and 8 variables, in addition to $\mu=\varepsilon$. We also investigate extensions of these analyses to rebit-retrit and qubit-qutrit ($6 \times 6$) settings.

May 1, 20170 citations

Qubit-qudit separability/PPT-probability investigations, including Lovas-Andai formula advancements

Paul B. Slater

We begin by advancing conjectures, closely fitting numerical estimates, for the Hilbert-Schmidt separability/PPT probabilities of the 35- and 63-dimensional convex sets of ( $2 \times 3$ ) qubit-qutrit and ( $2 \times 4$ ) qubit-qudit states, as well as certain related scenarios. We reason that the underlying volume formulas might closely parallel those now well-established--via a Lovas-Andai formula--in their lower (15)-dimensional two-qubit analogue. The qubit-qutrit conjecture we obtain is $(\frac{3}{10})^3 =\frac{3^3}{2^3 \cdot 5^3} = \frac{27}{1000} =0.027$ , with the counterpart two-qubit separability probablity being $\frac{8}{33}=\frac{2^3}{3 \cdot 11} \approx 0.242424$ . In the ( $2 \times 4$ ) qubit-qudit case, we advance $\frac{16}{12375} =\frac{4^2}{3^2 \cdot 5^3 \cdot 11} \approx 0.001292929$ as the PPT-probability. The numerators of the probabilities for $2 \times m$ scenarios are, thus, possibly simply powers of $m$ . We also search for the rebit-retrit counterpart to the established two-rebit result of $\frac{29}{64} =\frac{29}{2^6} \approx 0.453125$ . Additionally, we find that the Hilbert-Schmidt separability/PPT-probabilities for the two-rebit, rebit-retrit and two-retrit $X$ -states all equal $\frac{16}{3 \pi^2} \approx 0.54038$ , 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 Lovas-Andai two-qubit framework from Hilbert-Schmidt measures to induced ones, raising the possibility of a still more general master formula.We begin by advancing conjectures, closely fitting numerical estimates, for the Hilbert-Schmidt separability/PPT probabilities of the 35- and 63-dimensional convex sets of ($2 \times 3$) qubit-qutrit and ($2 \times 4$) qubit-qudit states, as well as certain related scenarios. We reason that the underlying volume formulas might closely parallel those now well-established--via a Lovas-Andai formula--in their lower (15)-dimensional two-qubit analogue. The qubit-qutrit conjecture we obtain is $(\frac{3}{10})^3 =\frac{3^3}{2^3 \cdot 5^3} = \frac{27}{1000} =0.027$, with the counterpart two-qubit separability probablity being $\frac{8}{33}=\frac{2^3}{3 \cdot 11} \approx 0.242424$. In the ($2 \times 4$) qubit-qudit case, we advance $\frac{16}{12375} =\frac{4^2}{3^2 \cdot 5^3 \cdot 11} \approx 0.001292929$ as the PPT-probability. The numerators of the probabilities for $2 \times m$ scenarios are, thus, possibly simply powers of $m$. We also search for the rebit-retrit counterpart to the established two-rebit result of $\frac{29}{64} =\frac{29}{2^6} \approx 0.453125$. Additionally, we find that the Hilbert-Schmidt separability/PPT-probabilities for the two-rebit, rebit-retrit and two-retrit $X$-states all equal $\frac{16}{3 \pi^2} \approx 0.54038$, 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 Lovas-Andai two-qubit framework from Hilbert-Schmidt measures to induced ones, raising the possibility of a still more general master formula.

Mar 28, 20180 citations

Master Lovas–Andai and equivalent formulas verifying the

\frac{8}{33}

8 33 two-qubit Hilbert–Schmidt separability probability and companion rational-valued conjectures

Construction of the Lovas-Andai Two-Qubit Function

\tilde{\chi}_2 (\varepsilon )=\frac{1}{3} \varepsilon ^2 \left(4-\varepsilon ^2\right)

Verifies the

\frac{8}{33}

-Hilbert Schmidt Separability Probability Conjecture

\varepsilon \in [0,1]

is the ratio

\sigma(V)

of the singular values of the

2 \times 2

matrix

V=D_2^{1/2} D_1^{-1/2}

formed from the two

2 \times 2

diagonal blocks (

D_1, D_2

) of a randomly generated

4 \times 4

density matrix

D

. In the Slater setting, the independent variable

\mu

is the diagonal-entry ratio

\sqrt{\frac{d_ {11} d_ {44}}{d_ {22} d_ {33}}}

--with, importantly,

\mu=\varepsilon

\mu=\frac{1}{\varepsilon}

when both

D_1

and

D_2

are themselves diagonal. Lovas and Andai established that their two-rebit function

\tilde{\chi}_1 (\varepsilon )

(

\approx \varepsilon

) yields the previously conjectured Hilbert-Schmidt separability probability of

\frac{29}{64}

. We are able, in the Slater framework (using cylindrical algebraic decompositions [CAD] to enforce positivity constraints), to reproduce this result. Further, we similarly obtain its new (much simpler) two-qubit counterpart,

\tilde{\chi}_2(\varepsilon) =\frac{1}{3} \varepsilon ^2 \left(4-\varepsilon ^2\right)

. Verification of the companion conjecture of a Hilbert-Schmidt separability probability of

\frac{8}{33}

immediately follows in the Lovas-Andai framework. We obtain the formulas for

\tilde{\chi}_1(\varepsilon)

and

\tilde{\chi}_2(\varepsilon)

by taking

D_1

and

D_2

to be diagonal, allowing us to proceed in lower (7 and 11), rather than the full (9 and 15) dimensions occupied by the convex sets of two-rebit and two-qubit states. The CAD's themselves involve 4 and 8 variables, in addition to

\mu=\varepsilon

. We also investigate extensions of these analyses to rebit-retrit and qubit-qutrit (

6 \times 6

) settings.

We begin by advancing conjectures, closely fitting numerical estimates, for the Hilbert-Schmidt separability/PPT probabilities of the 35- and 63-dimensional convex sets of (

2 \times 3

) qubit-qutrit and (

2 \times 4

) qubit-qudit states, as well as certain related scenarios. We reason that the underlying volume formulas might closely parallel those now well-established--via a Lovas-Andai formula--in their lower (15)-dimensional two-qubit analogue. The qubit-qutrit conjecture we obtain is

(\frac{3}{10})^3 =\frac{3^3}{2^3 \cdot 5^3} = \frac{27}{1000} =0.027

, with the counterpart two-qubit separability probablity being

\frac{8}{33}=\frac{2^3}{3 \cdot 11} \approx 0.242424

. In the (

2 \times 4

) qubit-qudit case, we advance

\frac{16}{12375} =\frac{4^2}{3^2 \cdot 5^3 \cdot 11} \approx 0.001292929

as the PPT-probability. The numerators of the probabilities for

2 \times m

scenarios are, thus, possibly simply powers of

m

. We also search for the rebit-retrit counterpart to the established two-rebit result of

\frac{29}{64} =\frac{29}{2^6} \approx 0.453125

. 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 \pi^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 Lovas-Andai two-qubit framework from Hilbert-Schmidt measures to induced ones, raising the possibility of a still more general master formula.