The nonnegativity of the determinant of the partial transpose of a two-qubit (4 x 4) density matrix is both a necessary and sufficient condition for its separability. While the determinant is restricted to the interval [0,1/256], the determinant of the partial transpose can range over [-1/16,1/256], with negative values corresponding to entangled states. We report here the exact values of the first nine moments of the probability distribution of the partial transpose over this interval, with respect to the Hilbert-Schmidt (metric volume element) measure on the nine-dimensional convex set of real two-qubit density matrices. Rational functions C_{2 j}(m), yielding the coefficients of the 2j-th power of even polynomials occurring at intermediate steps in our derivation of the m-th moment, emerge. These functions possess poles at finite series of consecutive half-integers (m=-3/2,-1/2,...,(2j-1)/2), and certain (trivial) roots at finite series of consecutive natural numbers (m=0, 1,...). Additionally, the (nontrivial) dominant roots of C_{2 j}(m) approach the same half-integer values (m = (2 j-1)/2, (2 j-3)/2,...), as j increases. The first two moments (mean and variance) found--when employed in the one-sided Chebyshev inequality--give an upper bound of 30397/34749 = 0.874759 on the separability probability of real two-qubit density matrices. We are able to report general formulas for the m-th moment of the Hilbert-Schmidt probability distribution of the density matrix determinant over [0,1/256], in the real, complex and quaternionic two-qubit cases.The nonnegativity of the determinant of the partial transpose of a two-qubit (4 x 4) density matrix is both a necessary and sufficient condition for its separability. While the determinant is restricted to the interval [0,1/256], the determinant of the partial transpose can range over [-1/16,1/256], with negative values corresponding to entangled states. We report here the exact values of the first nine moments of the probability distribution of the partial transpose over this interval, with respect to the Hilbert-Schmidt (metric volume element) measure on the nine-dimensional convex set of real two-qubit density matrices. Rational functions C_{2 j}(m), yielding the coefficients of the 2j-th power of even polynomials occurring at intermediate steps in our derivation of the m-th moment, emerge. These functions possess poles at finite series of consecutive half-integers (m=-3/2,-1/2,...,(2j-1)/2), and certain (trivial) roots at finite series of consecutive natural numbers (m=0, 1,...). Additionally, the (nontrivial) dominant roots of C_{2 j}(m) approach the same half-integer values (m = (2 j-1)/2, (2 j-3)/2,...), as j increases. The first two moments (mean and variance) found--when employed in the one-sided Chebyshev inequality--give an upper bound of 30397/34749 = 0.874759 on the separability probability of real two-qubit density matrices. We are able to report general formulas for the m-th moment of the Hilbert-Schmidt probability distribution of the density matrix determinant over [0,1/256], in the real, complex and quaternionic two-qubit cases.

arXiv:1003.3839Mar 19, 20100 citations

Superposition Principle for Qubit States in the Spin-Projection Mean Representation

Andrey Yu. Fedorov, Vladimir I. Man’ko

arXiv:1909.09896Journal of Russian Laser ResearchJan 1, 20221 citation

Visualization of all two-qubit states via partial-transpose moments

Lin Zhang, Yi Shen, Hua Xiang +2 more

Efficiently detecting entanglement based on measurable quantities is a basic problem for quantum information processing. Recently, the measurable quantities called partial-transpose (PT) moments have been proposed to detect and characterize entanglement. Recently [L. Zhang et al., Ann. Phys. (Berlin) 534, 2200289 (2022)], we have already identified the two-dimensional region, comprised of the second and third PT moments, corresponding to two-qubit entangled states, and described the whole region for all two-qubit states. In the present paper, we visualize the three-dimensional (3D) region corresponding to all two-qubit states by further involving the fourth PT moment (the last one for two-qubit states). The characterization of this 3D region can finally be achieved by optimizing some polynomials. Furthermore, we identify the dividing surface which separates the two parts of the whole 3D region corresponding to entangled and separable states, respectively. Due to the measurability of PT moments, we obtain a complete and operational criterion for the detection of two-qubit entanglement.Efficiently detecting entanglement based on measurable quantities is a basic problem for quantum information processing. Recently, the measurable quantities called partial-transpose (PT) moments have been proposed to detect and characterize entanglement. Recently [L. Zhang et al., Ann. Phys. (Berlin) 534, 2200289 (2022)], we have already identified the two-dimensional region, comprised of the second and third PT moments, corresponding to two-qubit entangled states, and described the whole region for all two-qubit states. In the present paper, we visualize the three-dimensional (3D) region corresponding to all two-qubit states by further involving the fourth PT moment (the last one for two-qubit states). The characterization of this 3D region can finally be achieved by optimizing some polynomials. Furthermore, we identify the dividing surface which separates the two parts of the whole 3D region corresponding to entangled and separable states, respectively. Due to the measurability of PT moments, we obtain a complete and operational criterion for the detection of two-qubit entanglement.

arXiv:2307.05305Physical Review AJul 11, 20233 citations

Observable Correlations in Two-Qubit States

Shunlong Luo, Qiang Zhang

Journal of Statistical PhysicsJul 1, 200941 citations

The Inverse Eigenvalue Problem for Partial Transposes of Two-Qubit States

Ruoting Dou, Shengjun Wu, Zeng-Bing Chen

For a bipartite state $ρ$ , information about the spectrum of its partial transpose $ρ^{Γ_B}$ can be inferred from measurements on multiple copies of $ρ$ , without full state tomography. This raises a natural question: which eigenvalue lists can arise as $\operatorname{spec}(ρ^{Γ_B})$ for a density operator $ρ$ ? We completely solve this inverse eigenvalue problem for two qubits. Every nonnegative trace-one spectrum is realized as $\operatorname{spec}(ρ^{Γ_B})$ by some PPT state $ρ$ , whereas an ordered candidate eigenvalue list $(x,y,z,-q)$ , with $x\ge y\ge z\ge0$ , $q>0$ , and $x+y+z-q=1$ , is realized by an NPT state iff $q\le y$ and $qy\le xz$ . Sufficiency in the latter case is established by an explicit $X$ state whose quantum steering ellipsoid has center $c=(y-q)/(1-z)$ and normalized volume $V/V_{\max}(c)=qy/(xz)$ , providing a geometric interpretation of the inequalities $q\le y$ and $qy\le xz$ as the allowed ellipsoid-center region and the fixed-center volume bound. Beyond this geometric picture, the two-qubit inverse theorem also yields exact negativity bounds from the two lowest nontrivial PT moments. Given fixed values of $p_2=Tr[(ρ^{Γ_B})^2]$ and $p_3=Tr[(ρ^{Γ_B})^3]$ , we determine the exact minimum and maximum negativity over all two-qubit states subject to these moment constraints. When no PPT state is consistent with the pair $(p_2,p_3)$ , the minimum is attained either at $x=y$ or $qy=xz$ , while the maximum is attained either at $y=z$ or $q=y$ . Finally, we show how the two-qubit inequalities persist as necessary constraints for the inverse eigenvalue problem in qubit--qudit systems.For a bipartite state $ρ$, information about the spectrum of its partial transpose $ρ^{Γ_B}$ can be inferred from measurements on multiple copies of $ρ$, without full state tomography. This raises a natural question: which eigenvalue lists can arise as $\operatorname{spec}(ρ^{Γ_B})$ for a density operator $ρ$? We completely solve this inverse eigenvalue problem for two qubits. Every nonnegative trace-one spectrum is realized as $\operatorname{spec}(ρ^{Γ_B})$ by some PPT state $ρ$, whereas an ordered candidate eigenvalue list $(x,y,z,-q)$, with $x\ge y\ge z\ge0$, $q>0$, and $x+y+z-q=1$, is realized by an NPT state iff $q\le y$ and $qy\le xz$. Sufficiency in the latter case is established by an explicit $X$ state whose quantum steering ellipsoid has center $c=(y-q)/(1-z)$ and normalized volume $V/V_{\max}(c)=qy/(xz)$, providing a geometric interpretation of the inequalities $q\le y$ and $qy\le xz$ as the allowed ellipsoid-center region and the fixed-center volume bound. Beyond this geometric picture, the two-qubit inverse theorem also yields exact negativity bounds from the two lowest nontrivial PT moments. Given fixed values of $p_2=Tr[(ρ^{Γ_B})^2]$ and $p_3=Tr[(ρ^{Γ_B})^3]$, we determine the exact minimum and maximum negativity over all two-qubit states subject to these moment constraints. When no PPT state is consistent with the pair $(p_2,p_3)$, the minimum is attained either at $x=y$ or $qy=xz$, while the maximum is attained either at $y=z$ or $q=y$. Finally, we show how the two-qubit inequalities persist as necessary constraints for the inverse eigenvalue problem in qubit--qudit systems.

quant-pharXiv:2608.29068Sep 1, 20260 citations

For a bipartite state

ρ

, information about the spectrum of its partial transpose

ρ^{Γ_B}

can be inferred from measurements on multiple copies of

ρ

, without full state tomography. This raises a natural question: which eigenvalue lists can arise as

\operatorname{spec}(ρ^{Γ_B})

for a density operator

ρ

? We completely solve this inverse eigenvalue problem for two qubits. Every nonnegative trace-one spectrum is realized as

\operatorname{spec}(ρ^{Γ_B})

by some PPT state

ρ

, whereas an ordered candidate eigenvalue list

(x,y,z,-q)

, with

x\ge y\ge z\ge0

q>0

, and

x+y+z-q=1

, is realized by an NPT state iff

q\le y

and

qy\le xz

. Sufficiency in the latter case is established by an explicit

X

state whose quantum steering ellipsoid has center

c=(y-q)/(1-z)

and normalized volume

V/V_{\max}(c)=qy/(xz)

, providing a geometric interpretation of the inequalities

q\le y

and

qy\le xz

as the allowed ellipsoid-center region and the fixed-center volume bound. Beyond this geometric picture, the two-qubit inverse theorem also yields exact negativity bounds from the two lowest nontrivial PT moments. Given fixed values of

p_2=Tr[(ρ^{Γ_B})^2]

and

p_3=Tr[(ρ^{Γ_B})^3]

, we determine the exact minimum and maximum negativity over all two-qubit states subject to these moment constraints. When no PPT state is consistent with the pair

(p_2,p_3)

, the minimum is attained either at

x=y

qy=xz

, while the maximum is attained either at

y=z

q=y

. Finally, we show how the two-qubit inequalities persist as necessary constraints for the inverse eigenvalue problem in qubit--qudit systems.