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ФИЗИКА ЭЛЕМЕНТАРНЫХ ЧАСТИЦ И АТОМНОГО ЯДРА

О параметризации пространства запутанности пары кубитов: Аннотации

A. Khvedelidze, Dimitar M. Mladenov, Astghik Torosyan

Published Mar 24, 2026• 0 citations
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Abstract

A method that allows increasing a computational efficiency of evaluation of nonlocal characteristics of a pair of qubits is described. The method is based on the construction of coordinates on a generic section of $2$-qubit’s entanglement space E2×2 represented as the direct product of an ordered $3$-dimensional simplex and the double coset $SU(2) × SU(2)\text{ \ }SU(4)/T^3$. Within this framework, the subset $S\mathcal{E}_{2×2} ⊂ \mathcal{E}_{2×2}$ corresponding to the rank-$4$ separable $2$-qubit states is described as a semialgebraic variety given by a system of $3$rd and $4$th order polynomial inequalities in eigenvalues of the density matrix, whereas the polynomials coefficients are trigonometric functions defined over a direct product of two regular octahedra.

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A method allowing to increase a computational efficiency of evaluation of non-local characteristics of a pair of qubits is described. The method is based on the construction of coordinates on a generic section of 2-qubit’s entanglement space $${{\mathcal{E}}_{{2 \times 2}}}$$ represented as the direct product of an ordered 3-dimensional simplex and the double coset $${\text{SU(2)}} \times {\text{SU(2)}}{\kern 1pt} \backslash {\kern 1pt} {\text{SU(4)}}{\kern 1pt} /{\kern 1pt} {{{\text{T}}}^{{\text{3}}}}{\kern 1pt} .$$ Within this framework, the subset $$\mathcal{S}{{\mathcal{E}}_{{2 \times 2}}} \subset {{\mathcal{E}}_{{2 \times 2}}}$$ corresponding to the rank-4 separable 2-qubit states is described as a semialgebraic variety given by a system of 3rd and 4th order polynomial inequalities in eigenvalues of the density matrix, whereas the polynomials coefficients are trigonometric functions defined over a direct product of two regular octahedra.

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Exploring High-dimensional Hilbert spaces by Quantum Optics

Andrea Chiuri, Paolo Mataloni

In the case of mixed states ρAB of the composite system A ⊗ B, the previous relation generalizes to ρAB = ∑ k pkρ A k ⊗ρk , where pk are probabilities and ρk ’s (ρk ’s) are generic density matrix of the system A (B). The entanglement derives from “subtle” nonlocal correlations between the parts of a quantum system and combines three basic structural elements of quantum theory: the superposition principle, the quantum non-separability property and the exponential scaling of the state space with the number of partitions. This unique resource can be used to perform computational and cryptographic tasks otherwise impossible with classical systems. An entangled state shared by two or more separated parties is a valuable resource for fundamental quantum protocols, such as quantum teleportation [2], quantum computing [3] and quantum cryptography. By using entangled states we can deeply investigate the nonlocal properties of the quantum world. In fact, an entangled system exhibits correlations between two (or more) parties, which cannot be reproduced by a classical theory or, in general, by a Local Hidden Variables Theory (LHVT). Quantum optics represents an excellent experimental test bench for various novel concepts introduced within the framework of quantum information theory. Quantum states of photons, generally produced by a probabilistic process, namely spontaneous parametric down conversion (SPDC), may be easily and accurately manipulated using linear and nonlinear optical devices and measured by efficient single-photon detectors. Photonic qubits represent the typical carrier of quantum information and can be distributed over long distances, either in free-space or in low-loss optical fibers. Two-particle experiments correspond to operate with qubits belonging to a (2 × 2) Hilbert space. Many quantum information tasks and fundamental tests of quantum mechanics, such as the simulation of properties of quantum systems, the realization of quantum algorithms of increasing complexity and the investigation of the quantum world at a mesoscopic level, deal with a large number of qubits. In order to take full advantage of the possibilities offered by quantum mechanics, more qubits must be added to a quantum state. Two approaches may be followed to increase the number of qubits. The first one consists of increasing the number of entangled particles. In this way, multi-qubit entangled states are created by distributing the qubits between the particles so that each particle carries one qubit. A second strategy consists of encoding more than one qubit in each particle, by exploiting different degrees of freedom (DOFs) of the photon such as the polarization, the path, the emission time, the orbital angular momentum. This has been used to create two-photon fourand six-qubit states and up to five-photon ten-qubit graph states. The entanglement of two particles in different DOFs corresponds to socalled hyperentanglement. It is worth noting that multi-qubit hyperentanglement allows to achieve higher

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Qubit-Qubit and Qubit-Qutrit Separability Functions and Probabilities

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We list in increasing order -- 1/3, 3/8, 2/5, 135 pi/1024, 16/(3 pi^2), 3 pi/16, 5/8, 105 pi/512, 2 - 435 pi/1024, 11/16, 1 -- a number of exact two-qubit Hilbert-Schmidt (HS) separability probabilities, we are able to compute. Each probability corresponds to a specific scenario -- a class of 4 x 4 density matrices (rho) with an m-subset (m<6) of its six off-diagonal pairs of symmetrically located entries set to zero. Initially, we consider only scenarios in which the (6-m) non-nullified pairs are real, but then permit them to be complex (as well as quaternionic) in nature also. The general analytical strategy we implement is based on the Bloore density matrix parameterization (J. Phys. A, 9, 2059 [1976]), allowing us to conveniently reduce the dimensionalities of required integrations. For each scenario, we identify a certain univariate ``separability function'' S_{scenario}(nu), where nu = rho_{11} rho_{44}/(rho_{22} rho_{33}). The integral over nu in [0,infty] of the product of this function with a scenario-specific (marginal) jacobian function J_{scenario}(nu) yields the HS {separable} volume (V^{HS}_{sep}). The ratio of V^{HS}_{sep} to the HS total (entangled and non-entangled) volume, gives us the HS scenario-specific separability probability. Among the possible forms that we have so far determined for S_{scenario}(nu) are piecewise combinations of c, c sqrt{nu} and c nu for nu in [0,1] and (in a dual manner) c, c/sqrt{nu} and c/nu for nu in [1,infty]. We also obtain bivariate separability functions S_{scenario)^{6 x 6}(nu_{1},nu_{2}} in the qubit-qutrit case, involving 6 x 6 density matrices, with ratio variables, nu_{1}= rho_{11} rho_{55}/(rho_{22} rho_{44}), and nu_{2}= rho_{22} rho_{66}/(rho_{33} rho_{55}). Additionally, we investigate parallel two-qutrit and three-qubit problems.

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Мера запутанности, создаваемой квантовыми операциями: Аннотации

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A measure of entanglement production by quantum operations is suggested. This measure is general, being valid for operations over pure states as well as over mixed states, for equilibrium as well as for nonequilibrium processes. The measure of entanglement production satisfies all properties typical of such a characteristic. Systems of arbitrary nature can be treated, described by field operators, spin operators, or any other operators, which is realized by defining generalized correlation matrices. Particular cases of entanglement production are considered.

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Constraints on SU(2) ⊗ SU(2) invariant polynomials for a pair of entangled qubits

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We discuss the entanglement properties of two qubits in terms of polynomial invariants of the adjoint action of SU(2) ⊕ SU(2) group on the space of density matrices $\mathfrak{P}_ +$ . Since elements of $\mathfrak{P}_ +$ are Hermitian, non-negative fourth-order matrices with unit trace, the space of density matrices represents a semi-algebraic subset, $\mathfrak{P}_ + \in \mathbb{R}^{15}$ . We define $\mathfrak{P}_ +$ explicitly with the aid of polynomial inequalities in the Casimir operators of the enveloping algebra of SU(4) group. Using this result the optimal integrity basis for polynomial SU(2) ⊕ SU(2) invariants is proposed and the well-known Peres-Horodecki separability criterion for 2-qubit density matrices is given in the form of polynomial inequalities in three SU(4) Casimir invariants and two SU(2) ⊕ SU(2) scalars; namely, determinants of the so-called correlation and the Schlienz-Mahler entanglement matrices.

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К вопросу об аналитическом описании локально эквивалентных квантовых состояний системы двух кубитов

Додин Дмитрий Валерьевич

The double-qubit quantum system is considered. The original parameterization based on vector presentation of state are proposed. We consider the states of quantum system which are equivalent to local unitary transformation. The analytical description of manifold of these states are constructed. The general case of mix state are investigated. The questions of relationship this manifold with properties and measure of entanglement are discussed.

•Jan 1, 2011•0 citations
A method that allows increasing a computational efficiency of evaluation of nonlocal characteristics of a pair of qubits is described. The method is based on the construction of coordinates on a generic section of 222-qubit’s entanglement space E2×2 represented as the direct product of an ordered 333-dimensional simplex and the double coset SU(2)×SU(2)  SU(4)/T3SU(2) × SU(2)\text{ \ }SU(4)/T^3SU(2)×SU(2)  SU(4)/T3. Within this framework, the subset SE2×2⊂E2×2S\mathcal{E}_{2×2} ⊂ \mathcal{E}_{2×2}SE2×2​⊂E2×2​ corresponding to the rank-444 separable 222-qubit states is described as a semialgebraic variety given by a system of 333rd and 444th order polynomial inequalities in eigenvalues of the density matrix, whereas the polynomials coefficients are trigonometric functions defined over a direct product of two regular octahedra.
A method allowing to increase a computational efficiency of evaluation of non-local characteristics of a pair of qubits is described. The method is based on the construction of coordinates on a generic section of 2-qubit’s entanglement space E2×2{{\mathcal{E}}_{{2 \times 2}}}E2×2​ represented as the direct product of an ordered 3-dimensional simplex and the double coset SU(2)×SU(2)\SU(4)/T3.{\text{SU(2)}} \times {\text{SU(2)}}{\kern 1pt} \backslash {\kern 1pt} {\text{SU(4)}}{\kern 1pt} /{\kern 1pt} {{{\text{T}}}^{{\text{3}}}}{\kern 1pt} .SU(2)×SU(2)\SU(4)/T3. Within this framework, the subset SE2×2⊂E2×2\mathcal{S}{{\mathcal{E}}_{{2 \times 2}}} \subset {{\mathcal{E}}_{{2 \times 2}}}SE2×2​⊂E2×2​ corresponding to the rank-4 separable 2-qubit states is described as a semialgebraic variety given by a system of 3rd and 4th order polynomial inequalities in eigenvalues of the density matrix, whereas the polynomials coefficients are trigonometric functions defined over a direct product of two regular octahedra.
In the case of mixed states ρAB of the composite system A ⊗ B, the previous relation generalizes to ρAB = ∑ k pkρ A k ⊗ρk , where pk are probabilities and ρk ’s (ρk ’s) are generic density matrix of the system A (B). The entanglement derives from “subtle” nonlocal correlations between the parts of a quantum system and combines three basic structural elements of quantum theory: the superposition principle, the quantum non-separability property and the exponential scaling of the state space with the number of partitions. This unique resource can be used to perform computational and cryptographic tasks otherwise impossible with classical systems. An entangled state shared by two or more separated parties is a valuable resource for fundamental quantum protocols, such as quantum teleportation [2], quantum computing [3] and quantum cryptography. By using entangled states we can deeply investigate the nonlocal properties of the quantum world. In fact, an entangled system exhibits correlations between two (or more) parties, which cannot be reproduced by a classical theory or, in general, by a Local Hidden Variables Theory (LHVT). Quantum optics represents an excellent experimental test bench for various novel concepts introduced within the framework of quantum information theory. Quantum states of photons, generally produced by a probabilistic process, namely spontaneous parametric down conversion (SPDC), may be easily and accurately manipulated using linear and nonlinear optical devices and measured by efficient single-photon detectors. Photonic qubits represent the typical carrier of quantum information and can be distributed over long distances, either in free-space or in low-loss optical fibers. Two-particle experiments correspond to operate with qubits belonging to a (2 × 2) Hilbert space. Many quantum information tasks and fundamental tests of quantum mechanics, such as the simulation of properties of quantum systems, the realization of quantum algorithms of increasing complexity and the investigation of the quantum world at a mesoscopic level, deal with a large number of qubits. In order to take full advantage of the possibilities offered by quantum mechanics, more qubits must be added to a quantum state. Two approaches may be followed to increase the number of qubits. The first one consists of increasing the number of entangled particles. In this way, multi-qubit entangled states are created by distributing the qubits between the particles so that each particle carries one qubit. A second strategy consists of encoding more than one qubit in each particle, by exploiting different degrees of freedom (DOFs) of the photon such as the polarization, the path, the emission time, the orbital angular momentum. This has been used to create two-photon fourand six-qubit states and up to five-photon ten-qubit graph states. The entanglement of two particles in different DOFs corresponds to socalled hyperentanglement. It is worth noting that multi-qubit hyperentanglement allows to achieve higher
We list in increasing order -- 1/3, 3/8, 2/5, 135 pi/1024, 16/(3 pi^2), 3 pi/16, 5/8, 105 pi/512, 2 - 435 pi/1024, 11/16, 1 -- a number of exact two-qubit Hilbert-Schmidt (HS) separability probabilities, we are able to compute. Each probability corresponds to a specific scenario -- a class of 4 x 4 density matrices (rho) with an m-subset (m<6) of its six off-diagonal pairs of symmetrically located entries set to zero. Initially, we consider only scenarios in which the (6-m) non-nullified pairs are real, but then permit them to be complex (as well as quaternionic) in nature also. The general analytical strategy we implement is based on the Bloore density matrix parameterization (J. Phys. A, 9, 2059 [1976]), allowing us to conveniently reduce the dimensionalities of required integrations. For each scenario, we identify a certain univariate ``separability function'' S_{scenario}(nu), where nu = rho_{11} rho_{44}/(rho_{22} rho_{33}). The integral over nu in [0,infty] of the product of this function with a scenario-specific (marginal) jacobian function J_{scenario}(nu) yields the HS {separable} volume (V^{HS}_{sep}). The ratio of V^{HS}_{sep} to the HS total (entangled and non-entangled) volume, gives us the HS scenario-specific separability probability. Among the possible forms that we have so far determined for S_{scenario}(nu) are piecewise combinations of c, c sqrt{nu} and c nu for nu in [0,1] and (in a dual manner) c, c/sqrt{nu} and c/nu for nu in [1,infty]. We also obtain bivariate separability functions S_{scenario)^{6 x 6}(nu_{1},nu_{2}} in the qubit-qutrit case, involving 6 x 6 density matrices, with ratio variables, nu_{1}= rho_{11} rho_{55}/(rho_{22} rho_{44}), and nu_{2}= rho_{22} rho_{66}/(rho_{33} rho_{55}). Additionally, we investigate parallel two-qutrit and three-qubit problems.
A measure of entanglement production by quantum operations is suggested. This measure is general, being valid for operations over pure states as well as over mixed states, for equilibrium as well as for nonequilibrium processes. The measure of entanglement production satisfies all properties typical of such a characteristic. Systems of arbitrary nature can be treated, described by field operators, spin operators, or any other operators, which is realized by defining generalized correlation matrices. Particular cases of entanglement production are considered.
We discuss the entanglement properties of two qubits in terms of polynomial invariants of the adjoint action of SU(2) ⊕ SU(2) group on the space of density matrices P+\mathfrak{P}_ +P+​ . Since elements of P+\mathfrak{P}_ +P+​ are Hermitian, non-negative fourth-order matrices with unit trace, the space of density matrices represents a semi-algebraic subset, P+∈R15\mathfrak{P}_ + \in \mathbb{R}^{15}P+​∈R15 . We define P+\mathfrak{P}_ +P+​ explicitly with the aid of polynomial inequalities in the Casimir operators of the enveloping algebra of SU(4) group. Using this result the optimal integrity basis for polynomial SU(2) ⊕ SU(2) invariants is proposed and the well-known Peres-Horodecki separability criterion for 2-qubit density matrices is given in the form of polynomial inequalities in three SU(4) Casimir invariants and two SU(2) ⊕ SU(2) scalars; namely, determinants of the so-called correlation and the Schlienz-Mahler entanglement matrices.