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Chinese Journal of Quantum Electronics

Measurement of mechanical quantity in quantum pure state and mixed state

Deng-Yu Zhang

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

The quantum pure state and mixed state are the most important quantum states in quantum information.They are the foundation of studying density operator,quantum entanglement and quantum decoherence.The two states' definition and innate differences are given in quantum theory.The observable effect for quantum pure states' mutually phase is the most important characteristics in quantum interfer- ence.The transformation from quantum pure state to mixed state is analyzed through measurement of mechanic quantity (?)_z,(?)_x and (?)_n by quantum mechanics methods.The result shows that the measurement of mechanical quantity is anisotropic in the quantum pure state but for quantum mixed state is isotropic.

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The quantum pure state and mixed state are the most important quantum states in quantum information.They are the foundation of studying density operator,quantum entanglement and quantum decoherence.The two states' definition and innate differences are given in quantum theory.The observable effect for quantum pure states' mutually phase is the most important characteristics in quantum interfer- ence.The transformation from quantum pure state to mixed state is analyzed through measurement of mechanic quantity (?)_z,(?)_x and (?)_n by quantum mechanics methods.The result shows that the measurement of mechanical quantity is anisotropic in the quantum pure state but for quantum mixed state is isotropic.
Irreversibility in quantum measurements is considered from the point of quantum information theory. For that purpose the information transfer between the measured object S and measuring system O is analyzed. It's found that due to the principal constraints of quantum-mechanical origin, the information about the purity of S state isn't transferred to O during the measurement of arbitraryS observable V. Consequently O can't discriminate the pure and mixed S ensembles with the same . As the result, the random outcomes should be detected by O in V measurement for S pure ensemble of V eigenstate superposition. It's shown that the outcome probabilties obey to Born rule. The influence of O decoherence by its environment is studied, however the account of its effects doesn't change these results principally.
Interaction with environment may lead to the transition of quantum system from pure state to the mixed one. In this case, the problem of definition of entanglement may arise. In particular, quantitative measure of entanglement concurrence is introduced differently for pure and mixed states. To solve this problem, alternative definition of concurrence of mixed states is introduced in present work. In particular, concurrence of mixed states is determined as an average of pure states concurrences. Averaging is carried out over the quantum ensemble, corresponding to all possible realization of random variable.
The paper is an elaborate comparative assessment of the most relevant entropy-based quantities to be applied in quantifying quantum entanglement in pure, mixed, bipartite, and multipartite quantum systems. The study fills the existing gap in the literature in which most of the previous literature examines individual measures, but a lot of knowledge is lacking on what these measures are regarding their strengths, weaknesses, and the operational usefulness against each other. The study can prove the behaviour of each measure under noise, decoherence, dimensional scaling, and variation of state-type by analysing von Neumann entropy, the Tsallis entropy, entropy of entanglement of formation, concurrence and logarithmic negativity. Numerical studies with pure two-qubit states, Werner states, GHZ states, W states and mixed random states demonstrate specific sensitivity properties and computational limitations which ensure that no single measure can be a universal quantifier. The paper goes on to say that entropy-only measures do not work in mixed regimes because of a dominance in classical correlations. The results have several important implications to quantum communication, quantum simulation, and quantum computing in the NISQ-era, where the choice of measurement has a direct impact on benchmarking, error-correction scheme, and protocol performance based on entanglement. The paper ends with a recommendation of contextual, purpose-based scheme of selecting entanglement measures and suggests a further study on scalable benchmarking, hybrid quantifiers, and operational performance-based entanglement measurements. These observations reinforce the theoretical premise of entanglement studies and enable the advancement of more effective quantum technologies in the field of practice.
The classification of the multipartite entanglement is an important problem in quantum information theory. We propose a class of two qubit mixed states σAB=p∣χ1⟩⟨χ1∣⊗ρ1+(1−p)∣χ2⟩⟨χ2∣⊗ρ2σ_{AB}= p|χ_{1}\rangle\langleχ_{1}|\otimesρ_{1}+(1-p)|χ_{2}\rangle\langleχ_{2}|\otimesρ_{2}σAB​=p∣χ1​⟩⟨χ1​∣⊗ρ1​+(1−p)∣χ2​⟩⟨χ2​∣⊗ρ2​, where ∣χ1⟩=α∣0⟩+β∣1⟩|χ_{1}\rangle=α|0\rangle+β|1\rangle∣χ1​⟩=α∣0⟩+β∣1⟩, ∣χ2⟩=β∣0⟩+(−1)nα∣1⟩|χ_{2}\rangle=β|0\rangle+(-1)^{n}α|1\rangle∣χ2​⟩=β∣0⟩+(−1)nα∣1⟩. We have shown that the state σABσ_{AB}σAB​ represent a classical state when nnn is odd while it represent a non-classical state when nnn is even. The purification of the state σABσ_{AB}σAB​ is studied and found that the purification is possible if the spectral decomposition of the density matrices ρ1ρ_{1}ρ1​ and ρ2ρ_{2}ρ2​ represent pure states. We have established a relationship between three tangle, which measures the amount of entanglement in three qubit system and the quantity ⟨χ1∣χ2⟩\langleχ_{1}|χ_{2}\rangle⟨χ1​∣χ2​⟩, which identifies whether the two qubit mixed state is classical or non-classical. The three qubit purified state is then classified as a separable or biseparable or W-type or GHZ-type state using the quantum correlation, measured by geometric discord, of its reduced two qubit density matrix.