RIKEN Accel. Prog. Rep. 50 (2017)

Ⅱ-6. Particle Physics

Separability criteria with angular and Hilbert-space averages†
K. Fujikawa,∗1 C.H. Oh,∗2 K. Umetsu,∗3 and Sixia Yu2
To characterize entanglement, the notions of separability and inseparability, which are the characteristic properties of state vectors in quantum mechanics,
are commonly used. On the other hand, the notionof local realism based on local non-contextual hiddenvariable models is also used to test properties such as
locality and reduction.1) Local hidden-variable models
are generally diﬀerent from quantum mechanics, andthus local realism tests the deviation from quantum
mechanics also.
In the experimental study of local realism, which iscommonly tested by the CHSH inequality,2) it is customary to ﬁrst conﬁrm the consistency of the measured basic correlation such as the spin correlation
a · σ ⊗ b · σ with quantum mechanics and then test
the CHSH inequality. If one conﬁrms the consistencyof a · σ ⊗ b · σ for any unit vectors a and b withquantum mechanics, one can naturally apply the criterion of quantum mechanical separability to the correlation. In the present study, we discuss this aspectof the separability test in quantum mechanics. It is
well known that the Peres criterion of the positivity ofa partial transposed density matrix gives a necessary
and suﬃcient condition for the separability of a general density matrix in d = 2 × 23) which we study inthe present paper. However, it is also well-known fromthe days of Pauli that the reconstruction of the statevector or density matrix from measured data is generally very involved.4) It is thus practically useful toderive simpler criteria that do not require precise state
reconstruction. The purpose of the present paper is toderive such separability criteria. We assume the mostgeneral separable density matrix, but we use a limited
set of two-point correlations and thus obtain only thenecessary condition of separability in general. Nevertheless, we illustrate that our criteria are very usefulwhen applied to the past experimental data of Aspect
et al. in 19815) and Sakai et al. in 2006.6)
To be speciﬁc, we study the general separable quantum mechanical statesρ=wk ρ k(1)− ψ|P (a) ⊗ 1|ψψ|1 ⊗ P (b)|ψ] = 0 (2)for two arbitrary projection operators P (a) and P (b).
In the case of a general mixed ρ, we derive the useful
criteria of separable mixed states, namelyTrρ{a · σ ⊗ b · σ} = (1/3)C cos ϕ(3)for two spin 1/2 systems, and4Trρ{P (a) ⊗ P (b)} = 1 + (1/2)C cos 2ϕ(4)for two photon systems with linear polarization projectors P (a) and P (b), respectively. Here, −1 ≤ C ≤ 1.
The basic new ingredient in our derivation of criteria(3) and (4) is that we take a geometric angular average of unit vectors a and b with ﬁxed cos ϕ = a · b. It
is shown that this angular averaging, which was originally motivated by the speciﬁc experiment,6) does not
add a new burden on the measurements by analyzing
the existing experimental data. The diﬀerence in thenumerical coeﬃcients 1/3 and 1/2 in (3) and (4) arises
from the diﬀerence in rotational properties; the spinor
rotational freedom is 3-dimensional, which agrees withthe freedom of the d = 2 Hilbert space, while the rotational freedom of the photon is two-dimensional, which
diﬀers from the freedom of the d = 2 Hilbert space because it is conﬁned in a plane perpendicular to themomentum direction. If one instead takes an averageover the states in the d = 2 Hilbert space, the formulafor the photon is replaced by4Trρ{P (a) ⊗ P (b)} = 1 + (1/3)C cos 2ϕ.

(5)This diﬀerence between the geometrical angular average and the Hilbert space average is interesting, and it
has an interesting implication on the separability issueof the entangled Werner state7) which accommodates
a speciﬁc local hidden-variable representation and thus
satisﬁes the CHSH inequality. In the photon case, theHilbert space average can judge the inseparability of
the Werner state but the geometrical angular average cannot, while the geometric average can judge theWerner state in the case of spin 1/2 systems.

kin d = 2 × 2 = 4; all the states ρk are separable purequantum states.7) If the state is a separable pure stateρ = |ψψ|, separability in (1) is quantiﬁed by theequality, which is necessary and suﬃcient,G(a, b) ≡ 4[ψ|P (a) ⊗ P (b)|ψ
†
∗1
∗2
∗3

Condensed from the article in K. Fujikawa, C.H. Oh, K.
Umetsu, and Sixia Yu, Ann. of Physics 368, 248 (2016).
RIKEN Nishina Center
Department of Physics, National University of Singapore
College of Science and Technology, Nihon University- 140 -References
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