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Yuanzihe wuli pinglun

Density Matrix in Quantum Mechanics and the Distinction of Ensembles Having the Same Density Matrix

Gui‐Lu Long

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

Density matrix is one important tool in quantum mechanics,and it has very broad applications.However there are different definitions about the density matrix,and they describe quite different systems.There has been some misunderstanding about the density matrix in the community,and these misunderstandings hinder the right application of the density matrix.In this article,we discuss the different definitions of density matrix. We suggest to use the full density matrix,compressed density matrix and the reduced density matrix to describe the state of a complete quantum system,the state of an averaged particle in an ensemble and the state of part of a composite system.We stress that contrary to the wide accepted understanding that ensembles with the same compressed density matrix are physically indistinguishable, they are distinguishable through the so-called ensemble measurement.As an application,we suggest that the present conclusion that the present-day nuclear magnetic resonance quantum computation does not have quantum entanglement is groundless.

Related papers

Density Matrix in Quantum Mechanics and Distinctness of Ensembles Having the Same Compressed Density Matrix

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We clarify different definitions of the density matrix by proposing the use of different names, the full density matrix for a single-closed quantum system, the compressed density matrix for the averaged single molecule state from an ensemble of molecules, and the reduced density matrix for a part of an entangled quantum system, respectively. We show that ensembles with the same compressed density matrix can be physically distinguished by observing fluctuations of various observables. This is in contrast to a general belief that ensembles with the same compressed density matrix are identical. Explicit expression for the fluctuation of an observable in a specified ensemble is given. We have discussed the nature of nuclear magnetic resonance quantum computing. We show that the conclusion that there is no quantum entanglement in the current nuclear magnetic resonance quantum computing experiment is based on the unjustified belief that ensembles having the same compressed density matrix are identical physically. Related issues in quantum communication are also discussed.

•Jan 1, 2006•20 citations

Density Matrix in Quantum Mechanics and Distinctness of Ensembles of Fixed Particle Number Having the Same Compressed Density Matrix

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We clarify different definitions of the density matrix by proposing the use of different names, the full density matrix for a single-closed quantum system, the compressed density matrix for the averaged single molecule state from an ensemble of molecules, and the reduced density matrix for a part of an entangled quantum system, respectively. We show that ensembles with the same compressed density matrix can be physically distinguished by observing fluctuations of various observables. This is in contrast to a general belief that ensembles with the same compressed density matrix are identical. Explicit expression for the fluctuation of an observable in a specified ensemble is given. We have discussed the nature of nuclear magnetic resonance quantum computing. We show that the conclusion that there is no quantum entanglement in the current nuclear magnetic resonance quantum computing experiment is based on the unjustified belief that ensembles having the same compressed density matrix are identical physically. Related issues in quantum communication are also discussed.

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On the quantum mechanical nature in liquid NMR quantum computing

Gui‐Lu Long, Haoxiong Yan, Y s Li +6 more

The quantum nature of bulk ensemble NMR quantum computing --the center of recent heated debate, is addressed. Concepts of the mixed state and entanglement are examined, and the data in a 2 qubit liquid NMR quantum computation are analyzed. It is pointed out that the key problem in the current debate is the understanding of entanglement in a mixed state system. The following points are concluded in this Letter: 1)Density matrix describes the "state" of an average particle in an ensemble. It can not describe the state of an individual particle in an ensemble in detail; 2) Entanglement is a property of the wave function of a quantum particle(such as an molecule in a liquid NMR sample). Separability of the density matrix can not be used to measure the entanglement of mixed ensemble; 3)The evolution of states in bulk-ensemble NMR quantum computation is quantum mechanical; 4) The coefficient before the effective pure state density matrix, $ε$, is an measure of the simultaneity of the molecules in an ensemble. It reflects the intensity of the NMR signal and has no significance in quantifying the entanglement in the bulk ensemble NMR system. We conclude that the liquid NMR quantum computation is genuine, not just classical simulations.

arXiv:quant-ph/0007077•Jul 20, 2000•0 citations

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While the dynamic state of a single particle can be specified quantum mechanically in terms of its state function, any rigorous description of the state of a many-particle system would require the complete knowledge of the dynamic state functions of all the particles. That is not always possible. On the other hand, for a large number of particles in, or near, thermal equilibrium in a uniform sample, the principles of statistical mechanics may be invoked to describe the averaged expectation values of the physically observable properties of such a many-particle system. The basic concepts of the density-matrix formalism and the quantum mechanic analog of the classical Boltzmann equation commonly used for optical and magnetic resonance problems of many-particle quantum systems are introduced in this chapter. Applications of this approach to such specific problems as the resonant interaction of electromagnetic radiation with optical media of two-level atoms, nonlinear optics, and the laser rate equations and transient dynamics are discussed in this chapter. Definitions of the density operator and the density matrix Up to this point, in studying the dynamics of quantum mechanic systems, we have assumed that the state of the system can be specified in terms of a precisely known state function |Ψ〉. On the other hand, for a macroscopic medium containing many particles, it is not always possible to know the exact dynamic states of all the particles in the medium, even for physically identical particles.

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Quantum Mechanical Nature in Liquid NMR Quantum Computing

Long, Gui-Lu, Yan +15 more

The quantum nature of bulk ensemble NMR quantum computing the center of recent heated debate,is addressed. Concepts of the mixed and entanglement are examined, and the data in a two-qubit liquid NMRquantum computation are analyzed. The main points in this paper are: i) Density matrix describes the state of anaverage particle in an ensemble. It does not describe the of an individual particle in an ensemble; ii) Entanglementis a property of the wave function of a microscopic particle (such as a molecule in a liquid NMR sample), and separabilityof the density matrix cannot be used to measure the entanglement of mixed ensemble; iii) The evolution in bulk-ensemble NMRquantum computation is quantum-mechanical; iv) The coefficient before the effective pure densitymatrix, e, is a measure of the simultaneity of the molecules in an ensemble. It reflects the intensity of the NMR signaland has no significance in quantifying the entanglement in the bulk ensemble NMR system. The decomposition of thedensity matrix into product states is only an indication that the ensemble can be prepared by an ensemble with theparticles unentangled. We conclude that effective-pure-state NMR quantum computation is genuine, not just classicalsimulations.

理论物理通讯：英文版•Jan 1, 2002•4 citations

The Density Matrix Formalism

Bernard d’Espagnat

Along with an elementary description of the well-known density matrix formalism and of its application to the description of quantum-statistical ensembles, this chapter includes the proofs of a few propositions that are especially useful for understanding most of the developments. In the new formulation, ensembles are directly described by density matrices, observables are still described by Hermitean operators, and the rules giving the statistical distribution and the mean values of the predicted measurement results are, respectively, expressions. When stated in this manner, the rules of quantum mechanics apply directly not only to pure states but also to mixtures. This is a distinct advantage. In the case of mixtures defined by their density matrices, the new formulation also has another merit. If two proper mixtures are described by the same density matrix, any state vector representing a subensemble of one of these mixtures can be expressed as a linear combination of the state vectors representing the subensembles of the other mixture.

CRC Press eBooks•Mar 5, 2018•1 citation
Density matrix is one important tool in quantum mechanics,and it has very broad applications.However there are different definitions about the density matrix,and they describe quite different systems.There has been some misunderstanding about the density matrix in the community,and these misunderstandings hinder the right application of the density matrix.In this article,we discuss the different definitions of density matrix. We suggest to use the full density matrix,compressed density matrix and the reduced density matrix to describe the state of a complete quantum system,the state of an averaged particle in an ensemble and the state of part of a composite system.We stress that contrary to the wide accepted understanding that ensembles with the same compressed density matrix are physically indistinguishable, they are distinguishable through the so-called ensemble measurement.As an application,we suggest that the present conclusion that the present-day nuclear magnetic resonance quantum computation does not have quantum entanglement is groundless.
We clarify different definitions of the density matrix by proposing the use of different names, the full density matrix for a single-closed quantum system, the compressed density matrix for the averaged single molecule state from an ensemble of molecules, and the reduced density matrix for a part of an entangled quantum system, respectively. We show that ensembles with the same compressed density matrix can be physically distinguished by observing fluctuations of various observables. This is in contrast to a general belief that ensembles with the same compressed density matrix are identical. Explicit expression for the fluctuation of an observable in a specified ensemble is given. We have discussed the nature of nuclear magnetic resonance quantum computing. We show that the conclusion that there is no quantum entanglement in the current nuclear magnetic resonance quantum computing experiment is based on the unjustified belief that ensembles having the same compressed density matrix are identical physically. Related issues in quantum communication are also discussed.
We clarify different definitions of the density matrix by proposing the use of different names, the full density matrix for a single-closed quantum system, the compressed density matrix for the averaged single molecule state from an ensemble of molecules, and the reduced density matrix for a part of an entangled quantum system, respectively. We show that ensembles with the same compressed density matrix can be physically distinguished by observing fluctuations of various observables. This is in contrast to a general belief that ensembles with the same compressed density matrix are identical. Explicit expression for the fluctuation of an observable in a specified ensemble is given. We have discussed the nature of nuclear magnetic resonance quantum computing. We show that the conclusion that there is no quantum entanglement in the current nuclear magnetic resonance quantum computing experiment is based on the unjustified belief that ensembles having the same compressed density matrix are identical physically. Related issues in quantum communication are also discussed.
The quantum nature of bulk ensemble NMR quantum computing --the center of recent heated debate, is addressed. Concepts of the mixed state and entanglement are examined, and the data in a 2 qubit liquid NMR quantum computation are analyzed. It is pointed out that the key problem in the current debate is the understanding of entanglement in a mixed state system. The following points are concluded in this Letter: 1)Density matrix describes the "state" of an average particle in an ensemble. It can not describe the state of an individual particle in an ensemble in detail; 2) Entanglement is a property of the wave function of a quantum particle(such as an molecule in a liquid NMR sample). Separability of the density matrix can not be used to measure the entanglement of mixed ensemble; 3)The evolution of states in bulk-ensemble NMR quantum computation is quantum mechanical; 4) The coefficient before the effective pure state density matrix, εεε, is an measure of the simultaneity of the molecules in an ensemble. It reflects the intensity of the NMR signal and has no significance in quantifying the entanglement in the bulk ensemble NMR system. We conclude that the liquid NMR quantum computation is genuine, not just classical simulations.
While the dynamic state of a single particle can be specified quantum mechanically in terms of its state function, any rigorous description of the state of a many-particle system would require the complete knowledge of the dynamic state functions of all the particles. That is not always possible. On the other hand, for a large number of particles in, or near, thermal equilibrium in a uniform sample, the principles of statistical mechanics may be invoked to describe the averaged expectation values of the physically observable properties of such a many-particle system. The basic concepts of the density-matrix formalism and the quantum mechanic analog of the classical Boltzmann equation commonly used for optical and magnetic resonance problems of many-particle quantum systems are introduced in this chapter. Applications of this approach to such specific problems as the resonant interaction of electromagnetic radiation with optical media of two-level atoms, nonlinear optics, and the laser rate equations and transient dynamics are discussed in this chapter. Definitions of the density operator and the density matrix Up to this point, in studying the dynamics of quantum mechanic systems, we have assumed that the state of the system can be specified in terms of a precisely known state function |Ψ〉. On the other hand, for a macroscopic medium containing many particles, it is not always possible to know the exact dynamic states of all the particles in the medium, even for physically identical particles.
The quantum nature of bulk ensemble NMR quantum computing the center of recent heated debate,is addressed. Concepts of the mixed and entanglement are examined, and the data in a two-qubit liquid NMRquantum computation are analyzed. The main points in this paper are: i) Density matrix describes the state of anaverage particle in an ensemble. It does not describe the of an individual particle in an ensemble; ii) Entanglementis a property of the wave function of a microscopic particle (such as a molecule in a liquid NMR sample), and separabilityof the density matrix cannot be used to measure the entanglement of mixed ensemble; iii) The evolution in bulk-ensemble NMRquantum computation is quantum-mechanical; iv) The coefficient before the effective pure densitymatrix, e, is a measure of the simultaneity of the molecules in an ensemble. It reflects the intensity of the NMR signaland has no significance in quantifying the entanglement in the bulk ensemble NMR system. The decomposition of thedensity matrix into product states is only an indication that the ensemble can be prepared by an ensemble with theparticles unentangled. We conclude that effective-pure-state NMR quantum computation is genuine, not just classicalsimulations.
Along with an elementary description of the well-known density matrix formalism and of its application to the description of quantum-statistical ensembles, this chapter includes the proofs of a few propositions that are especially useful for understanding most of the developments. In the new formulation, ensembles are directly described by density matrices, observables are still described by Hermitean operators, and the rules giving the statistical distribution and the mean values of the predicted measurement results are, respectively, expressions. When stated in this manner, the rules of quantum mechanics apply directly not only to pure states but also to mixtures. This is a distinct advantage. In the case of mixtures defined by their density matrices, the new formulation also has another merit. If two proper mixtures are described by the same density matrix, any state vector representing a subensemble of one of these mixtures can be expressed as a linear combination of the state vectors representing the subensembles of the other mixture.