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### Abstract

A simple quantum mechanical model is used to investigate irreversibility. This model exhibits some features that do not appear in classical mechanics. The model is employed to investigate several definitions of entropy, the nature of microstates and macrostates, and the effects of a measurement and of environmental interactions on the reversibility of the system. The model illustrates the utility of the density operator in quantum mechanics.

I. INTRODUCTION

II. THE MODEL

III. PURE AND MIXED STATES: THE DENSITY OPERATOR

IV. MEASUREMENT

V. ENTROPY

VI. MACROSCOPIC MEASUREMENT

VII. INTERACTION WITH THE ENVIRONMENT

VIII. CONCLUSIONS

### Key Topics

- Entropy
- 45.0
- Quantum measurement theory
- 9.0
- Magnetization dynamics
- 8.0
- Magnetization measurement
- 6.0
- Quantum information
- 5.0

## Figures

Evolution of the spin expectation values under unitary evolution of the model system. The system starts in the state with all spins down. The dynamics are reversed after time step 199, causing the system to retrace its evolution back to the starting state.

Evolution of the spin expectation values under unitary evolution of the model system. The system starts in the state with all spins down. The dynamics are reversed after time step 199, causing the system to retrace its evolution back to the starting state.

Expectation value of the total magnetization. The total magnetization is zero in the initial state and rises rapidly to fluctuate around the value of 4, which corresponds to half the spins up and half down. The dynamics are reversed after time step 199, after which the magnetization retraces its evolution back to zero.

Expectation value of the total magnetization. The total magnetization is zero in the initial state and rises rapidly to fluctuate around the value of 4, which corresponds to half the spins up and half down. The dynamics are reversed after time step 199, after which the magnetization retraces its evolution back to zero.

Binomial distribution of the magnetization states for spins.

Binomial distribution of the magnetization states for spins.

Evolution of the spin expectation values with complete spin measurement. The evolution is unitary until time step 199. Then the complete measurement is performed. The dynamics are reversed after time step 199, but the measurement has wiped out the correlations encoded in the quantum state of the system. After measurement, the expectation values remain very close to 0.5.

Evolution of the spin expectation values with complete spin measurement. The evolution is unitary until time step 199. Then the complete measurement is performed. The dynamics are reversed after time step 199, but the measurement has wiped out the correlations encoded in the quantum state of the system. After measurement, the expectation values remain very close to 0.5.

Expectation value of total magnetization for time steps 199–400. The solid line is the same as in Fig. 2 and shows the reversal of the total magnetization under reversed dynamics. After a complete measurement (dashed line), the magnetization remains nearly constant around the value of 4, corresponding to half the spins up. After a macroscopic measurement (dotted line), the magnetization retains some of the information contained in the pre-measurement state, as seen in the dip in the magnetization near time step 400. After an interaction with the environment (dash-dotted line), much more of the information is retained, as seen in the strong decrease in the magnetization near time step 400.

Expectation value of total magnetization for time steps 199–400. The solid line is the same as in Fig. 2 and shows the reversal of the total magnetization under reversed dynamics. After a complete measurement (dashed line), the magnetization remains nearly constant around the value of 4, corresponding to half the spins up. After a macroscopic measurement (dotted line), the magnetization retains some of the information contained in the pre-measurement state, as seen in the dip in the magnetization near time step 400. After an interaction with the environment (dash-dotted line), much more of the information is retained, as seen in the strong decrease in the magnetization near time step 400.

The solid line shows the reversal of the macroscopic entropy under reversed dynamics. After a complete measurement (dashed line), the macroscopic entropy remains very close to the maximum entropy. After a macroscopic measurement (dotted line), the macroscopic entropy remains high for some time but dips toward zero near time step 400. After an interaction with the environment (dash-dotted line), the macroscopic entropy declines sharply near time step 400 but does not return all the way to zero.

The solid line shows the reversal of the macroscopic entropy under reversed dynamics. After a complete measurement (dashed line), the macroscopic entropy remains very close to the maximum entropy. After a macroscopic measurement (dotted line), the macroscopic entropy remains high for some time but dips toward zero near time step 400. After an interaction with the environment (dash-dotted line), the macroscopic entropy declines sharply near time step 400 but does not return all the way to zero.

Schematic representation of the effects on the density operator of complete versus macroscopic measurement. The complete measurement zeros out all of the off-diagonal matrix elements. If the states were rearranged according to the macrostate to which they belong (that is, grouped by total magnetization), then the macroscopic measurement would zero out all the off-diagonal blocks of the matrix.

Schematic representation of the effects on the density operator of complete versus macroscopic measurement. The complete measurement zeros out all of the off-diagonal matrix elements. If the states were rearranged according to the macrostate to which they belong (that is, grouped by total magnetization), then the macroscopic measurement would zero out all the off-diagonal blocks of the matrix.

Evolution of the spin expectation values with a macroscopic spin measurement. The evolution is unitary until time step 199. Then the macroscopic measurement is performed as in Fig. 7. The dynamics are reversed after time step 199.

Evolution of the spin expectation values with a macroscopic spin measurement. The evolution is unitary until time step 199. Then the macroscopic measurement is performed as in Fig. 7. The dynamics are reversed after time step 199.

The partial trace procedure sums the diagonal blocks of the expanded density operator.

The partial trace procedure sums the diagonal blocks of the expanded density operator.

Evolution of the spin expectation values with environmental interaction and partial trace after time step 199. The dynamics are reversed after time step 199.

Evolution of the spin expectation values with environmental interaction and partial trace after time step 199. The dynamics are reversed after time step 199.

## Tables

Microscopic entropy of the model system following various interventions. The microscopic entropy starts at zero in each case and remains constant under unitary evolution of the system.

Microscopic entropy of the model system following various interventions. The microscopic entropy starts at zero in each case and remains constant under unitary evolution of the system.

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