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Building partially entangled states defined with even collision functions

Hiroo Azuma

Published Oct 30, 1998• 0 citations
OpenAlex

Abstract

How to build partially entangled states defined with functions whose all collisions are even are discussed. These states include a partially entangled state with a high degree of symmetry that is invariant under permutation of any two qubits and inversion between the ground state |0> and an excited state |1> for each qubit. An optimized high symmetric partially entangled state is considered to be useful for overcoming a shot noise limit of the Ramsey spectroscopy. We show that using selective rotations and Grover's inversion about average operations, we can construct this high symmetric entangled state by a network of poly(n) x 2^{n/2} quantum basic gates, where n is a number of qubits.

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How to build partially entangled states defined with functions whose all collisions are even are discussed. These states include a partially entangled state with a high degree of symmetry that is invariant under permutation of any two qubits and inversion between the ground state |0> and an excited state |1> for each qubit. An optimized high symmetric partially entangled state is considered to be useful for overcoming a shot noise limit of the Ramsey spectroscopy. We show that using selective rotations and Grover's inversion about average operations, we can construct this high symmetric entangled state by a network of poly(n) x 2^{n/2} quantum basic gates, where n is a number of qubits.
In the present study, we discuss how to build some partially entangled states of n two-state quantum systems (qubits). The optimal partially entangled state with a high degree of symmetry is considered to be useful for overcoming a shot noise limit of Ramsey spectroscopy under some decoherence. This state is invariant under permutation of any two qubits, and inversion between the ground state |0> and an excited state |1> for each qubit. We show that by using selective phase shifts in certain basis vectors and Grover's inversion about average operations, we can construct this high symmetric entangled state by (polynomial in n) ×2n/2 successive unitary transformations that are applied on two or three qubits.
Given two arbitrary pure states $ |\\phi>$ and $ |\\psi>$ of qubits or higher level states, we provide arguments in favor of states of the form $ \\frac{1}{\\sqrt{2}}(|\\psi> |\\phi> + i |\\phi> |\\psi>) $ instead of symmetric or anti-symmetric states, as natural candidates for optimally entangled states constructed from these states. We show that such states firstly have on the average a high value of concurrence, secondly can be constructed by a universal unitary operator independent of the input states. We also show that these states are the only ones which can be produced with perfect fidelity by any quantum operation designed for intertwining two pure states with a relative phase. A probabilistic method is proposed for producing any pre-determined relative phase into the combination of any two arbitrary states.
In this work, a variation of the problem originally solved by Verstraete, Audenaert, and De Moor [Phys. Rev. A 64, 012316 (2001)] on what is the maximum entanglement that can be created in a two-qubit system by a global unitary transformation is considered and solved when permutation invariance in the state is imposed. The additional constraint of permutation symmetry appears naturally in the context of bosonic systems or spin states. We also characterise symmetric two-qubit states that remain separable after any global unitary transformation, called symmetric absolutely separable states (SAS), or absolutely classical for spin states. This allows us to determine the maximal radius of a ball of SAS states around the maximally mixed state in the symmetric sector, and the minimal radius of a ball that includes the set of SAS states. For three-qubit systems, a necessary condition for absolute separability of symmetric states is given, which leads us to upper bounds on the ball radii similar to those studied for the two-qubit system.
A general method in constructing a complete set of wave functions for multipartite identical qubits is presented based on the irreducible representations of the permutation group and the nth rank tensors. Particular examples for n =2, 3, and 4 are derived and the entanglement behavior for each state is examined from several criteria. It is found that the states so constructed are all bound entangled states. For the case of even n, all the states are found to have maximum "n-tangle". The symmetry in spin space is found to increase the n-tangle in general. The "n-tangle" for n = 4 is found not always representing 4-way entanglement. It measures the degree of spin-space symmetry instead. A useful relationship in the classification between systems containing different number of qubits is given in terms of the Young's Tableaux based on our analysis.
Abstract Dicke states are completely symmetric states of multiple qubits (2-level systems), and qudit Dicke states are their d -level generalization. We define here q -deformed qudit Dicke states using the quantum algebra s u q ( d ) . We show that these states can be compactly expressed as a weighted sum over permutations with q -factors involving the so-called inversion number, an important permutation statistic in Combinatorics. We use this result to compute the bipartite entanglement entropy of these states. We also discuss the preparation of these states on a quantum computer, and show that introducing a q -dependence does not change the circuit gate count.
In this work, we study permutation-invariance and bandwidth-tuning in quantum fidelity kernels for classifying vector boson scattering (VBS) against QCD multijet background using proton-proton collisions simulated at s=13.6\sqrt{s}=13.6s​=13.6 TeV, with detector simulation corresponding to Run-2 CMS detector. We employ partially and fully symmetrized hardware-efficient kernel variants that embed SnS_nSn​ symmetry of the final-state jets. In noiseless simulation, some symmetry-enhanced kernels outperform a non-invariant HEA kernel, a classical Gaussian kernel baseline, and a product state quantum kernel in terms of the area under the curve (AUC) of the receiver operator characteristic curve. The bandwidth γ\gammaγ crucially affects the classification performance, and its optimal value is noticeably affected by the introduction of finite-shot estimation (10 000 shots) leading to the vanishing of the AUC gains over the classical baseline. Finally, accompanying runs on VTT’s Q50 quantum computer exhibit increased noise challenges for the symmetrized entanglement pattern.