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.

Two Energy Measurements

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. In general it cannot be reproduced by measuring one fixed Hermitian work operator for all initial states and drives.

Nondegenerate Shortcut

If both energy measurements are nondegenerate, write the energy eigenvectors as $|n^0\rangle$ and $|m^\tau\rangle$. For any initial state, set $p_n^0=\langle n^0|\rho_0|n^0\rangle$. The first measurement removes coherence between these vectors, giving

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

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

With $u$ having units of inverse energy, 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 leaves the unread ensemble density operator unchanged. Conditioning on a particular outcome still changes the state. 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. Its proof uses initial Gibbs weights and unitarity; a reverse experiment is needed for the Crooks comparison, not for this equality. See Jarzynski Equality and Crooks Relation.

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.

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

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

Work Fluctuations After Dephasing

For the coherence-backaction example above, find the full TPM distribution, its variance, and its characteristic function.

Solution

Each of the four initial/final branches has probability $1/4$. Two branches have $W=0$, and the others have $W=\pm\epsilon$. Thus

$$P(W)=\tfrac12\delta(W)+\tfrac14\delta(W-\epsilon)+\tfrac14\delta(W+\epsilon),$$ $$\langle W\rangle=0,\qquad \operatorname{Var}(W)=\epsilon^2/2,\qquad G(u)=\frac{1+\cos(u\epsilon)}{2}.$$

A vanishing mean does not mean that every run transfers zero energy.

References

• M. Campisi, P. Hänggi, and P. Talkner, “Colloquium: Quantum fluctuation relations: Foundations and applications,” Reviews of Modern Physics 83, 771–791 (2011). arXiv:1012.2268.
• M. Esposito, U. Harbola, and S. Mukamel, “Nonequilibrium fluctuations, fluctuation theorems, and counting statistics in quantum systems,” Reviews of Modern Physics 81, 1665–1702 (2009).
• J. Kurchan, “A quantum fluctuation theorem,” arXiv:cond-mat/0007360 (2000).
• P. Talkner, E. Lutz, and P. Hänggi, “Fluctuation theorems: Work is not an observable,” Physical Review E 75, 050102(R) (2007).
• H. Tasaki, “Jarzynski relations for quantum systems and some applications,” arXiv:cond-mat/0009244 (2000).
• H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control, Cambridge University Press, 2010.