Duality, measurements, and factorization in finite quantum systems

Vourdas, A.; Bendjaballah, C.

doi:10.1103/physreva.47.3523

Cited by 46 publications

(27 citation statements)

References 25 publications

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“…The weak mutually unbiased bases in H15 and their factorizations in terms of the mutually unbiased bases |B ; m1 in H3, and |B(2) j ; m2 in H5, according to Eq (40)…”

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confidence: 99%

Weak mutually unbiased bases

Shalaby¹,

Vourdas²

2012

J. Phys. A: Math. Theor.

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Quantum systems with variables in Z(d) are considered. The properties of lines in the Z(d) × Z(d) phase space of these systems, are studied. Weak mutually unbiased bases in these systems are defined as bases for which the overlap of any two vectors in two different bases, is equal to d −1/2 or alternatively to one of the d −1/2 i , 0 (where di is a divisor of d apart from d, 1). They are designed for the geometry of the Z(d) × Z(d) phase space, in the sense that there is a duality between the weak mutually unbiased bases and the maximal lines through the origin. In the special case of prime d, there are no divisors of d apart from 1, d and the weak mutually unbiased bases are mutually unbiased bases.form the group Sp(2, Z(d)). The cardinality of this group is dJ 2 (d).Acting with the matrix g(κ, λ|µ, ν) on a point (ρ, σ) ∈ Z(d) × Z(d) we get the point (κρ + λσ, µρ + νσ). Acting with the symplectic matrix g(κ, λ|µ, ν) on all the points of a line L(ρ, σ) we get all the points of the line L(κρ + λσ, µρ + νσ) for which we also use the notation g(κ, λ|µ, ν)L(ρ, σ).In [28] we have proved various properties for lines in Z(d) × Z(d). For completeness below we give these properties together with several new properties, but we prove only the new ones.Definition II.1. Lines L(ν, µ) through the origin, with exactly d points are called maximal lines.Proposition II.2.

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“…The weak mutually unbiased bases in H15 and their factorizations in terms of the mutually unbiased bases |B ; m1 in H3, and |B(2) j ; m2 in H5, according to Eq (40)…”

mentioning

confidence: 99%

Weak mutually unbiased bases

Shalaby¹,

Vourdas²

2012

J. Phys. A: Math. Theor.

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“…Here e = n + k where n, k are the degrees of local constancy and compact support, correspondingly, of the functions f p (x p ). The quantum formalism of Σ[ p Z(p e ), p Z(p e )] is precisely the quantum formalism for Σ[Z(n), Z(n)] with n = p e (the proof is based on the Chinese remainder theorem [12,13]). Therefore in Σ[ Z, (Q/Z)] we have a rich quantum formalism that includes all Σ[Z(n), Z(n)] as subsystems.…”

Section: G Physical Importance Of the Profinite Topology And Of The mentioning

confidence: 99%

Quantum mechanics on profinite groups and partial order

Vourdas¹

2013

J. Phys. A: Math. Theor.

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“…Good factorized Fourier transforms [34][35][36] in the context of 'fast Fourier transforms' and this has been extended to a factorization of quantum mechanics on Z(q) in [37,38]. We discuss this briefly below.…”

Section: Embedding Of the Formalism For Finite Quantum Systems Inmentioning

confidence: 99%

“…In [37,38] we have extended Good's scheme and showed that various quantities in quantum mechanics in Z(q) are factorized in terms of their counterparts in quantum mechanics on Z(p ei i ) (with i = 1, ..., ℓ). We use the notation |P ; s and |X; r for the momentum and position bases in the q-dimensional Hilbert space H of quantum mechanics on Z(q).…”

Section: A Factorization Of Quantum Mechanics On Z(q)mentioning

confidence: 99%

Quantum mechanics on ${\mathbb Q}/{\mathbb Z}$Q/Z

Vourdas

2011

Journal of Mathematical Physics

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Quantum mechanics with positions in Q/Z and momenta in b Z is considered. Displacement operators and coherent states, parity operators, Wigner and Weyl functions, and time evolution, are discussed. The restriction of the formalism to certain finite subspaces, is equivalent to Good's factorization of quantum mechanics on Z(q). I. INTRODUCTIONQuantum mechanics and quantum field theory on the field Q p of p-adic numbers have been studied by various authors (e.g., [1][2][3][4][5][6][7][8][9][10][11][12]). Applications to condensed matter physics have been discussed in [13][14][15][16]. Mathematical work related to these problems has been presented in [17][18][19][20][21]. General references on p-adic numbers are [22,23], and on the related topic of profinite groups [24,25].Recently we have studied quantum mechanics with positions in Z p (p-adic integers) and momenta in Q p /Z p , with emphasis on both the physical aspects [26], and the mathematical aspects (using inverse limits) [27]. Physical applications include the physics at the Planck scale, condensed matter, etc. Our interest is in quantum engineering of devices with p-adic arithmetic [26]. Such devices might have applications in the general area of information processing (e.g., [28]).In the present paper we extend this work and study quantum mechanics with positions in Q/Z and momenta in the Pontryagin dual group Z (where Q are the rational numbers, Z are the integers and Z is discussed below). We study the Heisenberg-Weyl group and discuss various properties of the displacement and parity operators, Wigner and Weyl functions and coherent states. Hamiltonians in this context, and the corresponding time evolution, are also studied. We show that a finite number of coupled finite quantum systems can be embedded into quantum mechanics on Q/Z and in this sense there are many realistic physical systems, where the formalism is applicable.There has been a lot of work recently (for reviews see [29][30][31][32][33]) on various aspects of finite quantum systems with positions and momenta in Z(d) (the integers modulo d). When d = p n (with p a fixed prime number), the large d limit of these systems is precisely the system with phase space Z p × (Q p /Z p ) (studied in [26,27]). When d takes all integer values, the large d limit of these systems is the system with phase space Z × (Q/Z) (studied here). This is shown in table I. The mathematical tools that bring these finite systems to the 'edge' are the inverse limit and direct limit. Using the inverse limit with the groups corresponding to momenta and the direct limit with the groups corresponding to positions, ensures the Pontryagin duality between the two groups for momenta and positions. The inverse limit of Z(p n ) is the profinite group Z p and the direct limit of Z(p n ) is Q p /Z p [27]. The inverse limit of Z(d) is the profinite group Z, and the direct limit of Z(d) is Q/Z. In this paper we only mention the inverse limits very briefly (with reference to the literature) because the emphasis is on the physical aspect...

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Duality, measurements, and factorization in finite quantum systems

Cited by 46 publications

References 25 publications

Weak mutually unbiased bases

Weak mutually unbiased bases

Quantum mechanics on profinite groups and partial order

Quantum mechanics on ${\mathbb Q}/{\mathbb Z}$Q/Z

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