Ⅱ-6. Particle Physics

RIKEN Accel. Prog. Rep. 51 (2018)

Exact algebraic separability criterion for two-qubit systems†
K. Fujikawa∗1 and C. H. Oh∗2A conceptually simpler proof of the separability criterion for two-qubit systems, which is referred to as“Hefei inequality” in literature,1) is analyzed. Thisinequality gives a necessary and suﬃcient separability criterion for any mixed two-qubit system unlike
the Bell-CHSH inequality2,3) which cannot test mixedstates such as the Werner state4) when regarded asa separability criterion. The original derivation ofthis inequality1) emphasized the uncertainty relation ofcomplementary observables; however, we show that theuncertainty relation does not play any role in the actual
derivation and that the Peres-Hodrodecki condition5)
is solely responsible for the inequality. Our derivation, which contains technically novel aspects such as
an analogy to the Dirac equation, sheds light on thisinequality and on the fundamental issue of the to extent to which the uncertainty relation can provide atest of entanglement. This separability criterion is illustrated for an exact treatment of the Werner state.
Our starting point is the fact that the general puretwo-qubit states are brought to the standard form by
the Schmidt decomposition|Φ⟩ = (u ⊗ v)[s1 |+⟩ ⊗ |−⟩ − s2 eiδ |−⟩ ⊗ |+⟩]with|+⟩ =(10), |−⟩ =(01)

,(1)

(2)

and real numbers s21 + s22 = 1 and δ. Namely, thestates are parametrized by s1 , s2 , δ and two unitarymatrices u and v. It is then shown that this system isrepresented formally in terms of a 4-dimensional Diracnotation.
We then obtain the inequalities (separability criterion)⟨P− ⟩2ρ ≥ ⟨γ3 γ0 P− ⟩2ρ + ⟨γ2 γ0 P− ⟩2ρ + ⟨γ1 γ0 P− ⟩2ρ ,⟨P+ ⟩2ρ ≥ ⟨γ3 γ0 P+ ⟩2ρ + ⟨γ2 γ0 P− ⟩2ρ + ⟨γ1 γ0 P− ⟩2ρ , (3)(4)

for example, using the Peres-Horodecki criterion without referring to the uncertainty relations.
As for the test of the Werner state4) which is deﬁnedbyρw =†∗1∗21(1 − β)1 + β|ψs ⟩⟨ψs |41 ≥ β + 2β.

(6)We thus conclude that the separability condition of theWerner state is equivalent to1,
(7)
3
which agrees with the result of a more explicit analysisof ρw .4) This in particular implies that β > 31 stands
for an inseparable state.
We have re-analyzed one of the representative inequalities proposed in Ref. 1) and have shown thatthe uncertainty relations cannot be alternative to thePeres-Horodecki condition in the analysis of entanglement for general two-qubit systems. The “Hefei inequality,” however, stands for a rare algebraic criterion
that is applicable to any mixed state that cannot betested by the Bell-CHSH inequality in general,2,3) as
was illustrated by an exact treatment of the Wernerstate.
β≤Here, in comparison with the criterions of separability of two-qubit systems, we brieﬂy mention a corresponding test of the separability of systems with two
continuous degrees of freedom.6–8) In the problem of
two-party continuum case with two-dimensional continuous phase space freedom (p, q) in each party, it ispossible to re-formulate the problem such that8)(1) the uncertainty relation leads to a necessary condition for separable two-party systems,(2) the derived condition is suﬃcient to prove theseparability of two-party Gaussian systems.
Namely, the uncertainty relation without referringto the Peres-Horodecki criterion5) provides a necessary and suﬃcient separability condition for two-party
Gaussian systems.
Referenceswhere P± = (1 ± γ5 )/2 and⟨γ3 γ0 P± ⟩ρ ≡ Trγ3 γ0 P± ρ,√
with the singlet state |ψs ⟩ = (1/ 2)[|+⟩|−⟩ − |−⟩|+⟩],
we obtain(5)Condensed from the article in K. Fujikawa and C. H. Oh,Mod. Phys. Lett. A 32, 1750070 (2017).
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