In this paper we establish a unique factorization theorem for pure quantum states expressed in computational basis. We show that there always exists unique factorization for any given N-qubit pure quantum state in terms of the tensor product of non-factorable or ``prime'' pure quantum states. This result is based on a simple criterion: Given N-qubit pure quantum state in computational basis can be factorized as the tensor product of an m-qubit pure quantum state and an n-qubit pure quantum state, where (m + n) = N, if and only if the rank of the certain associated matrix is equal to one. This simple criterion leads to a factorization algorithm which when applied to an N-qubit pure quantum state factorizes that state into the tensor product of non-factorable or ``prime'' pure quantum states. This paper shows that for any given N-qubit pure quantum state the said factorization always ``exists'' and is ``unique''. We demonstrated our work here on a computational basis.PACS Number: 03.67.Mn, 03.65.Ca, 03.65.UdIn this paper we establish a unique factorization theorem for pure quantum states expressed in computational basis. We show that there always exists unique factorization for any given N-qubit pure quantum state in terms of the tensor product of non-factorable or ``prime'' pure quantum states. This result is based on a simple criterion: Given N-qubit pure quantum state in computational basis can be factorized as the tensor product of an m-qubit pure quantum state and an n-qubit pure quantum state, where (m + n) = N, if and only if the rank of the certain associated matrix is equal to one. This simple criterion leads to a factorization algorithm which when applied to an N-qubit pure quantum state factorizes that state into the tensor product of non-factorable or ``prime'' pure quantum states. This paper shows that for any given N-qubit pure quantum state the said factorization always ``exists'' and is ``unique''. We demonstrated our work here on a computational basis.PACS Number: 03.67.Mn, 03.65.Ca, 03.65.Ud

Annals of Mathematics and PhysicsSep 15, 20231 citation

Pure and Mixed States

J. C. A. Barata, Marcos Brum, Victor Chabu +1 more

arXiv:1909.06232Brazilian Journal of PhysicsNov 9, 20207 citations

Computing the geometric measure of entanglement of multipartite pure states by means of non-negative tensors

Shenglong Hu, Liqun Qi, Guofeng Zhang

The geometric measure of entanglement for pure states has attracted much attention. On the other hand, the spectral theory of non-negative tensors (hypermatrices) has been developed rapidly. In this paper, we show how the spectral theory of non-negative tensors can be applied to the study of the geometric measure of entanglement for pure states. For symmetric pure multipartite qubit or qutrit states an elimination method is given. For symmetric pure multipartite qudit states, a numerical algorithm with randomization is presented. We also show that a nonsymmetric pure state can be augmented to a symmetric one whose amplitudes can be encoded in a non-negative symmetric tensor, so the geometric measure of entanglement can be calculated. Several examples, such as $m\mathrm{GHZ}$ states, $W$ states, inverted $W$ states, qudits, and nonsymmetric states, are used to demonstrate the power of the proposed methods. Given a pure state, one can always find a change of basis (a unitary transformation) so that all the probability amplitudes of the pure state are non-negative under the new basis. Therefore, the methods proposed here can be applied to a very wide class of multipartite pure states.The geometric measure of entanglement for pure states has attracted much attention. On the other hand, the spectral theory of non-negative tensors (hypermatrices) has been developed rapidly. In this paper, we show how the spectral theory of non-negative tensors can be applied to the study of the geometric measure of entanglement for pure states. For symmetric pure multipartite qubit or qutrit states an elimination method is given. For symmetric pure multipartite qudit states, a numerical algorithm with randomization is presented. We also show that a nonsymmetric pure state can be augmented to a symmetric one whose amplitudes can be encoded in a non-negative symmetric tensor, so the geometric measure of entanglement can be calculated. Several examples, such as $m\mathrm{GHZ}$ states, $W$ states, inverted $W$ states, qudits, and nonsymmetric states, are used to demonstrate the power of the proposed methods. Given a pure state, one can always find a change of basis (a unitary transformation) so that all the probability amplitudes of the pure state are non-negative under the new basis. Therefore, the methods proposed here can be applied to a very wide class of multipartite pure states.

Physical Review AJan 6, 201652 citations

Pure and mixed state

Dipankar Bhattacharyya, Jyotirmoy Guha

Institute of Physics eBooksJan 1, 20220 citations

Preparation of 3-qubit and 4-qubit pesudo-pure states in NMR quantum computation

Fei Xue, Juan Du, Fan Yang +4 more

Basing on the temporal averaging, schedules to prepare 3-qubit and 4-qubit pesudo-pure states in nuclear magnetic resonance (NMR) quantum computation are presented in this paper. Utilizing Lambda(n) (not) gate, our schedules reduced the experiment number that is needed to prepare pesudo-pure state in a homonuclear spin system: only three experiments are needed to prepare 3-qubit peudo-pure state; and only five experiments are needed to prepare 4-qubit peudo-pure state. Then we implemented the schedule for 3-qubit psudo-pure state in a chain-coupled spin system to confirm the schedule, and gave the simulation result of the schedule for the 4-qubit peudo-pure state.Basing on the temporal averaging, schedules to prepare 3-qubit and 4-qubit pesudo-pure states in nuclear magnetic resonance (NMR) quantum computation are presented in this paper. Utilizing Lambda(n) (not) gate, our schedules reduced the experiment number that is needed to prepare pesudo-pure state in a homonuclear spin system: only three experiments are needed to prepare 3-qubit peudo-pure state; and only five experiments are needed to prepare 4-qubit peudo-pure state. Then we implemented the schedule for 3-qubit psudo-pure state in a chain-coupled spin system to confirm the schedule, and gave the simulation result of the schedule for the 4-qubit peudo-pure state.

Jan 1, 20020 citations

Identifying Pure States

Giacomo Mauro D’Ariano, Giulio Chiribella, Paolo Perinotti

Cambridge University Press eBooksNov 2, 20160 citations

m\mathrm{GHZ}

states,

W

states, inverted

W

states, qudits, and nonsymmetric states, are used to demonstrate the power of the proposed methods. Given a pure state, one can always find a change of basis (a unitary transformation) so that all the probability amplitudes of the pure state are non-negative under the new basis. Therefore, the methods proposed here can be applied to a very wide class of multipartite pure states.