量子ビットα|0〉+β|1〉の複素数αとβについて

泰人 河野, 善尚 白木

Masuo Suzuki, Kuroda, Akira, Seiji Miyashita

Monte Carlo法,特に平衡状態のMonte Carlo法は諸量Xのensemble average, =(<Σ>___<<α|><α|Xe^<-βH>|α>)/(<Σ>___<<α|><α|e^<-βH|α>)(ここでHは系のノヽミルトニアン,|α>は系の1状態,β=1/kT),の|α>についての和が実行困難の時,状態|α>を<α|exp(=βH)|α>に比例する確率でとりだし,それによって, _ =<Σ>___<<α|>_ <α'|X|α'>/ _ はMonte Carlo法によるensemble average,|α>_ はMonte Calroの1 configuration)として求めるものである。状態|α>を<α|exp(-βH)α>に比例する確率でとり出す方法として,kinetic modelのように状態を<αlexp(-βH)|α>に従ってstepさせ,|α>が<αlexp(-βH)α>に従って分布しているようなensemble,つまり平衡状態を作りだし,そこでの平均をとる方法が普通行なわれている。しかし,すべが対角化されている古典系では,<αlHlα>=E_α,<αlexp(-βH)lα>=e^<-βE_α>であるのに対し、量子系では,Hの固有状態でない|α>について<αlexp(-βH)α>が一般には計算不可能なためMonte Carlo法は困難であると考えられ、現在まで実行されていない。そこで量子系でのMonte Carlo法について一つの方法を提案する.

Ole Keller

In the search for a relativistic wave equation for a free particle of rest mass m one must demand that the energy(E)-momentum (p) [dispersion] relation E2 = m2c4 + p2c2 (25.1) is satisfied. By the substitutions E → i~ ∂ ∂t , p→ −i~∇ (25.2) in Eq. (25.1) one obtains an operator, which when acting on a scalar wave function, gives us the Klein-Gordon wave equation for a spinless particle. The Klein-Gordon equation is of second order in the space (r) and time (t) coordinates. If one seeks a covariant first-order differential equation, r and t must play symmetric roles. A natural starting point therefore would be a linear dispersion relation of the form E = βmc2 + cα · p, (25.3) where β and α = (α1, α2, α3) are real and dimensionless. Since Eq. (25.3) must be compatible with Eq. (25.1), squaring of Eq. (25.3) leads to the following relations [already given in Eqs. (10.77)-(10.79)]: β2 = 1, (25.4) αiβ + βαi = 0, (25.5) αiαj + αjαi = 2δij , (25.6) where i, j = 1, 2, 3 (x, y, z). It is clear from Eqs. (25.4)-(25.6) that β and α cannot be numbers, but one can find matrices which satisfy these equations. Since the Hamiltonian appearing upon a substitution of the connections in (25.2) into Eq. (25.3) must be Hermitian, the αi and β matrices also have to be Hermitian, i.e., α†i = αi, i = 1− 3, (25.7) β† = β. (25.8) One can show that the matrices must have even rank and that the rank must be at least 4, see, e.g., [88]. The Dirac equation, which describes the quantum mechanics of a spin1/2 elementary particle, e.g., an electron, relates to a four-dimensional realization of Eqs. (25.4)-(25.8). Representations of a given rank are not unique.

茂 山下

薫 竹内

J. L. Manríquez-Zepeda, Juvenal Rueda-Paz, Manuél Ávila +1 more

We study both pentapartite GHZ and W-class states in the noninertial frame and explore their entanglement properties by carrying out the negativities including 1-4, 2-3, and 1-1 tangles, the whole entanglement measures such as algebraic and geometric averages π5 and Π5, and von Neumann entropy. We illustrate graphically the difference between the pentapartite GHZ and W-class states. We find that all 1-4, 2-3 tangles and the whole entanglements, which are observer dependent, degrade more quickly as the number of accelerated qubits increases. The entanglements of these quantities still exist even at the infinite acceleration limit. We also notice that all 1-1 tangles of pentapartite GHZ state Nαβ=NαIβ=NαIβI=0 where α,β∈(A,B,C,D,E), whereas all 1-1 tangles of the W-class state Nαβ,NαIβ and NαIβI are unequal to zero, e.g., Nαβ=0.12111 but NαIβ and NαIβI disappear at r>0.61548 and r>0.38671, respectively. We notice that the entanglement of the pentapartite GHZ and W-class quantum systems decays faster as the number of accelerated particles increases. Moreover, we also illustrate the difference of von Neumann entropy between them and find that the entropy in the pentapartite W-class state is greater than that of GHZ state. The von Neumann entropy in the pentapartite case is more unstable than those of tripartite and tetrapartite subsystems in the noninertial frame.