Two-Point Measurement Scheme

The two-point measurement scheme is the standard operational protocol for assigning a work distribution to a closed driven quantum system. It measures the system energy at the beginning, applies the drive, measures the system energy at the end, and defines work as the difference between the two recorded energies.

The scheme is important because it gives a positive probability distribution and leads directly to the usual quantum Jarzynski and Crooks relations for initially thermal states. It is also subtle: the first energy measurement is a real projective measurement, so it can erase energy-basis coherence. The scheme defines a protocol-dependent work distribution, not a universal work observable.

For the underlying heat/work convention, see Energy, Heat, and Work. For the fluctuation theorems that use this scheme, see Fluctuation Theorems.

Protocol

Consider a finite-dimensional closed system driven from Hamiltonian $H_0$ at time $0$ to Hamiltonian $H_\tau$ at time $\tau$. Let

$$H_0 = \sum_n E_n^0\Pi_n^0, \qquad H_\tau = \sum_m E_m^\tau\Pi_m^\tau,$$

where $\Pi_n^0$ and $\Pi_m^\tau$ are energy projectors. The projectors may have rank greater than one, so this notation includes degeneracies.

The protocol is:

1. Prepare the initial state $\rho_0$.

2. Measure $H_0$ projectively. Outcome $n$ has probability

   $$p_n = \operatorname{Tr} \left[ \Pi_n^0\rho_0 \right].$$

3. After outcome $n$, use the Lüders energy update

   $$\rho_{0|n} = \frac{\Pi_n^0\rho_0\Pi_n^0} {\operatorname{Tr}(\Pi_n^0\rho_0)}$$

   when $p_n$ is nonzero.

4. Evolve the system with the driven unitary $U_{\tau,0}$.

5. Measure $H_\tau$ projectively. Outcome $m$ has conditional probability

   $$p(m|n) = \operatorname{Tr} \left[ \Pi_m^\tau U_{\tau,0}\rho_{0|n}U_{\tau,0}^\dagger \right].$$

6. Assign the work value

   $$W_{mn} = E_m^\tau-E_n^0.$$

The joint probability of the two outcomes is

$$p(m,n) = \operatorname{Tr} \left[ \Pi_m^\tau U_{\tau,0} \Pi_n^0\rho_0\Pi_n^0 U_{\tau,0}^\dagger \Pi_m^\tau \right].$$

The work distribution is therefore

$$P(W) = \sum_{m,n} p(m,n)\, \delta\!\left( W-W_{mn} \right).$$

This is a probability distribution over recorded energy differences. It is not the spectral distribution of a single operator called work.

Nondegenerate Shortcut

If both energy measurements are nondegenerate, write the energy eigenvectors as $|n^0\rangle$ and $|m^\tau\rangle$. If the initial state is diagonal in the initial energy basis,

$$\rho_0 = \sum_n p_n^0 |n^0\rangle\langle n^0|,$$

then

$$p(m,n) = p_n^0 \left| \langle m^\tau|U_{\tau,0}|n^0\rangle \right|^2,$$

and

$$P(W) = \sum_{m,n} p_n^0 \left| \langle m^\tau|U_{\tau,0}|n^0\rangle \right|^2 \delta\!\left( W-E_m^\tau+E_n^0 \right).$$

This is the formula most often quoted in introductory discussions. The projector formula above is safer because it handles degeneracy and arbitrary initial states.

Characteristic Function

The characteristic function of the work distribution is

$$G(u) = \int dW\,e^{iuW}P(W).$$

Using the two-point probabilities gives

$$G(u) = \operatorname{Tr} \left[ e^{iuH_\tau} U_{\tau,0} e^{-iuH_0} \rho_0^{D} U_{\tau,0}^\dagger \right],$$

where

$$\rho_0^{D} = \sum_n \Pi_n^0\rho_0\Pi_n^0$$

is the initial state dephased in the energy eigenspaces measured by the first energy measurement.

If $[\rho_0,H_0]=0$, then $\rho_0^{D}=\rho_0$. If $\rho_0$ has coherence between different initial energy eigenspaces, then the two-point measurement characteristic function is built from the dephased state, not from the original coherent state.

Average Work

The mean work assigned by the scheme is

$$\langle W\rangle_{\mathrm{TPM}} = \sum_{m,n} p(m,n) \left( E_m^\tau-E_n^0 \right).$$

Equivalently,

$$\langle W\rangle_{\mathrm{TPM}} = \operatorname{Tr} \left[ H_\tau U_{\tau,0}\rho_0^{D}U_{\tau,0}^\dagger \right] - \operatorname{Tr} \left[ H_0\rho_0^{D} \right].$$

When $[\rho_0,H_0]=0$, this is the actual average energy change of the closed driven system:

$$\langle W\rangle_{\mathrm{TPM}} = \operatorname{Tr} \left[ H_\tau U_{\tau,0}\rho_0U_{\tau,0}^\dagger \right] - \operatorname{Tr} \left[ H_0\rho_0 \right].$$

When $\rho_0$ has energy coherence, the scheme first dephases the state. The TPM average can then differ from the energy change that would occur under the drive without the initial measurement.

Initial Gibbs States

The scheme is especially natural for initial Gibbs states,

$$\rho_0 = \frac{e^{-\beta H_0}}{Z_0}.$$

Such states commute with $H_0$, so the first energy measurement does not change the density operator. In this setting the TPM distribution is compatible with the usual quantum work fluctuation relations.

For example, the exponential work average is

$$\left\langle e^{-\beta W}\right\rangle = \sum_{m,n} p(m,n) e^{-\beta(E_m^\tau-E_n^0)}.$$

With unitary evolution and an initial Gibbs state, this reduces to

$$\left\langle e^{-\beta W}\right\rangle = \frac{Z_\tau}{Z_0} = e^{-\beta\Delta F}.$$

This is the Jarzynski equality in the closed quantum TPM setting. The detailed assumptions and the Crooks relation are treated on Fluctuation Theorems.

Degeneracies

Energy degeneracies should be handled with energy projectors, not with arbitrarily chosen basis vectors. If several states share the same energy $E_n^0$, the energy measurement reports the eigenspace projector $\Pi_n^0$.

The standard Lüders update

$$\rho \mapsto \frac{\Pi_n^0\rho\Pi_n^0} {\operatorname{Tr}(\Pi_n^0\rho)}$$

preserves coherence inside the degenerate energy eigenspace while removing coherence between different energy eigenspaces.

A refined measurement that resolves extra labels inside the degenerate subspace is a different apparatus. It may produce the same reported energy value but a different post-measurement state. That difference can change the later work statistics if the drive is sensitive to coherence inside the degenerate subspace. The work protocol should therefore state whether the energy measurement is Lüders-degenerate or refined.

For the measurement-theory background, see Projective Measurements and State Update Rules.

Simple Commuting Quench

Let

$$H_0 = \epsilon_0|e\rangle\langle e|, \qquad H_\tau = \epsilon_\tau|e\rangle\langle e|,$$

and suppose the sudden quench does not rotate the eigenvectors, so $U_{\tau,0}=I$ during the idealized instantaneous change. If the initial excited-state population is $p_e$, then the TPM distribution is

$$P(W) = (1-p_e)\delta(W) + p_e\, \delta\!\left( W-(\epsilon_\tau-\epsilon_0) \right).$$

The ground-state branch has no energy change. The excited-state branch changes by the level-spacing difference. The mean work is

$$\langle W\rangle = p_e(\epsilon_\tau-\epsilon_0).$$

This is the same split used in Energy, Heat, and Work: changing the level spacing at fixed population is work.

Coherence Backaction Example

The effect of the first measurement can be seen with a two-level Hamiltonian

$$H_0=H_\tau = \epsilon |e\rangle\langle e|.$$

Let the initial state be

$$|+\rangle = \frac{|g\rangle+|e\rangle}{\sqrt2}, \qquad \rho_0=|+\rangle\langle+|.$$

Let the drive be the Hadamard-like unitary

$$U|g\rangle=|+\rangle, \qquad U|e\rangle=|-\rangle,$$

where $|-\rangle=(|g\rangle-|e\rangle)/\sqrt2$. Without an initial measurement, $U|+\rangle=|g\rangle$, so the actual average energy change is

$$\Delta E_{\mathrm{unmeasured}} = 0-\frac{\epsilon}{2} = - \frac{\epsilon}{2}.$$

In the TPM scheme, the first energy measurement dephases the state:

$$\rho_0^D = \frac12|g\rangle\langle g| + \frac12|e\rangle\langle e|.$$

After $U$, both branches have final excited-state probability $1/2$. The final average energy is $\epsilon/2$, the initial measured average energy is also $\epsilon/2$, and therefore

$$\langle W\rangle_{\mathrm{TPM}} = 0.$$

The discrepancy is not an error. It means the TPM scheme describes the experiment in which the first energy measurement was actually performed.

Open-System Variants

The simplest TPM scheme is a closed-system protocol. For an open system, several variants exist:

• Measure only the reduced system energy at the beginning and end. This gives a distribution of system energy changes, not a clean separation of heat and work.
• Measure the total system-plus-bath energy. This is closer to an inclusive energy budget but may be experimentally inaccessible.
• Introduce counting fields for bath energy changes or particle transfers.
• Use quantum-jump trajectories when jumps correspond to resolved reservoir exchanges.
• Use weak measurements or interferometric schemes to access characteristic functions.

These variants answer different questions. A reduced-system energy difference may be useful, but it should not automatically be called work or heat without specifying the drive, reservoir, and measurement record.

Common Mistakes

1. Treating $W_{mn}=E_m^\tau-E_n^0$ as eigenvalues of a universal work operator.

2. Applying the TPM formula to a coherent initial state without noting that the first measurement dephases the state.

3. Replacing degenerate energy projectors by arbitrary eigenvectors without stating the refinement.

4. Using the closed-system TPM distribution for an open-system process without accounting for bath energy.

5. Assuming the Jarzynski equality follows from the TPM formula alone. It also requires the right initial equilibrium state and the correct reverse-process assumptions.

6. Forgetting that a noisy or weak energy measurement defines a different protocol from an ideal projective energy measurement.

Exercises

Normalization

Show that the joint TPM probabilities

$$p(m,n) = \operatorname{Tr} \left[ \Pi_m^\tau U \Pi_n^0\rho_0\Pi_n^0 U^\dagger \Pi_m^\tau \right]$$

sum to one.

Solution

Sum first over $m$ and use $\sum_m\Pi_m^\tau=I$:

$$\sum_m p(m,n) = \operatorname{Tr} \left[ U \Pi_n^0\rho_0\Pi_n^0 U^\dagger \right].$$

By cyclicity of the trace and unitarity,

$$\operatorname{Tr} \left[ U \Pi_n^0\rho_0\Pi_n^0 U^\dagger \right] = \operatorname{Tr} \left[ \Pi_n^0\rho_0\Pi_n^0 \right].$$

Now sum over $n$:

$$\sum_n \operatorname{Tr} \left[ \Pi_n^0\rho_0\Pi_n^0 \right] = \sum_n \operatorname{Tr} \left[ \Pi_n^0\rho_0 \right] = \operatorname{Tr}\rho_0 = 1.$$

Commuting Two-Level Quench

For $H_0=\epsilon_0|e\rangle\langle e|$, $H_\tau=\epsilon_\tau|e\rangle\langle e|$, and $U=I$, derive $P(W)$ for an arbitrary initial population $p_e$.

Solution

The energy eigenvectors do not change. The ground branch has energy $0$ both initially and finally, so $W=0$. The excited branch has work value

$$W=\epsilon_\tau-\epsilon_0.$$

The initial probabilities are $p_g=1-p_e$ and $p_e$. Therefore

$$P(W) = (1-p_e)\delta(W) + p_e \delta\!\left( W-(\epsilon_\tau-\epsilon_0) \right).$$

The mean is

$$\langle W\rangle = p_e(\epsilon_\tau-\epsilon_0).$$

Coherent Initial State

Use the coherence backaction example above to verify that the TPM mean work is $0$ while the unmeasured average energy change is $-\epsilon/2$.

Solution

The initial coherent state is $|+\rangle=(|g\rangle+|e\rangle)/\sqrt2$, so its initial average energy is

$$\langle H_0\rangle = \frac{\epsilon}{2}.$$

The unitary sends $|+\rangle$ to $|g\rangle$, so without an initial energy measurement the final average energy is $0$. Therefore

$$\Delta E_{\mathrm{unmeasured}} = 0-\frac{\epsilon}{2} = - \frac{\epsilon}{2}.$$

In the TPM protocol, the initial measurement produces $|g\rangle$ or $|e\rangle$ with probability $1/2$ each. The unitary maps these states to $|+\rangle$ and $|-\rangle$, each of which has final excited-state probability $1/2$. Thus the final average energy after the TPM first measurement is $\epsilon/2$. The initial measured average energy is also $\epsilon/2$, hence

$$\langle W\rangle_{\mathrm{TPM}} = 0.$$

References

• P. Talkner, E. Lutz, and P. Hänggi, “Fluctuation theorems: Work is not an observable,” Physical Review E 75, 050102, 2007.
• M. Campisi, P. Hänggi, and P. Talkner, “Colloquium: Quantum fluctuation relations: Foundations and applications,” Reviews of Modern Physics 83, 771, 2011.
• H. Tasaki, “Jarzynski relations for quantum systems and some applications,” arXiv:cond-mat/0009244, 2000.
• J. Kurchan, “A quantum fluctuation theorem,” arXiv:cond-mat/0007360, 2000.
• M. Esposito, U. Harbola, and S. Mukamel, “Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems,” Reviews of Modern Physics 81, 1665, 2009.
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press, 2010.