# Fluctuation theorem for quantum-state statistics

###### Abstract

We derive the fluctuation theorem for quantum-state statistics that can be obtained when we initially measure the total energy of a quantum system at thermal equilibrium, let the system evolve unitarily, and record the quantum-state data reconstructed at the end of the process. The obtained theorem shows that the quantum-state statistics for the forward and backward processes is related to the equilibrium free-energy difference through an infinite series of independent relations, which gives the quantum work fluctuation theorem as a special case, and reproduces the out-of-time-order fluctuation-dissipation theorem near thermal equilibrium. The quantum-state statistics exhibits a system-size scaling behavior that differs between integrable and non-integrable (quantum chaotic) systems as demonstrated numerically for one-dimensional quantum lattice models.

Fluctuation theorems (FTs) have played a central role in our understanding of how macroscopic irreversibility arises from microscopically reversible equation of motion Boc (); Evans et al. (1993); Gallavotti and Cohen (1995); Jarzynski (1997); Crooks (1999); Esposito et al. (2009); Campisi et al. (2011). The FTs lead to many fundamental relations in thermodynamics and statistical mechanics, including the second law of thermodynamics, the fluctuation-dissipation theorem (FDT) Callen and Welton (1951); Kubo (1957); Marconi et al. (2008), and Onsager’s reciprocity relation Onsager (1931); Casimir (1945).

The conventional approach to FTs in isolated quantum systems is based on the two-point measurement for work Kur (); Tas (); Esposito et al. (2009): one initially measures the total energy, let the system evolve according to a time-dependent Hamiltonian, and again measures the total energy at the end of the process. From the difference between the initial and final total energies, one can extract the work done on the system by an external force. The obtained work probability distributions for the forward and time-reversed processes are related to the equilibrium free-energy difference between the initial and final configurations (the quantum work FT). In this approach, one makes a projective energy measurement (with the outcome being the th eigenenergy of the final Hamiltonian) on the final state , so that one obtains limited information on the quantum state itself, i.e., only the diagonal information is available, where is the energy eigenstate.

How does the quantum state realized after the time evolution (including information on the off-diagonal elements , ) fluctuate? Here, by fluctuations of the quantum state we mean that the state fluctuates depending on the result of the initial energy measurement. If we repeat the procedure to (i) prepare the initial thermal equilibrium state, (ii) measure the total energy, (iii) perform a unitary time evolution, and (iv) reconstruct the quantum state , we can operationally determine the statistics of quantum states (Fig. 1). When the above procedure is repeated sufficiently many times, we obtain duplicated copies of quantum states, with which we can in principle reconstruct the quantum state using the technique of the quantum-state tomography Par (); Lvovsky and Raymer (2009).

The statistics of quantum states is closely related to quantum chaos, or non-integrability, of the system, the characterization of which has been a long-standing issue in statistical mechanics Berry (1987); Haake (2010). Suppose that after the first measurement the quantum state is projected to a certain eigenstate of the initial Hamiltonian. Then the state evolves within a subspace of the total Hilbert space due to the presence of conserved quantities. For integrable systems, the number of conserved quantities is extensive, so that the size of the subspace is highly constrained. Hence we expect that the resulting behavior of the quantum-state statistics is different between integrable and non-integrable systems.

Another motivation to study the quantum-state statistics is the recent finding of the out-of-time-order FDT Tsuji et al. (2018a), which relates chaotic properties of the system and a nonlinear response function involving a time-reversed process, and can be viewed as a higher-order extension of the conventional FDT. Provided that the conventional FDT can be derived from the quantum work FT near equilibrium, it is thus a natural question what is the underlying law that leads to the out-of-time-order FDT if applied near equilibrium.

In this paper, we show that the quantum-state statistics accumulated under a certain condition for the forward and time-reversed processes satisfies an infinite series of exact relations that are expressed in terms of the equilibrium free-energy difference between the initial and final configurations. The relations include the quantum work (Crooks) FT as a special case, and allow further extensions. Near equilibrium, the out-of-time-order FDT Tsuji et al. (2018a) is reproduced. We argue that the fluctuation of the quantum-state statistics shows a different system-size scaling between integrable and non-integrable systems, which can be used as a diagnosis of quantum chaos. This is demonstrated numerically for one-dimensional quantum lattice models.

Let us suppose that an isolated quantum system evolves in time according to the time-dependent Hamiltonian () (forward process). The initial and final Hamiltonians are denoted by and . The unitary evolution operator is given by , where represents the time-ordered product. We assume that the initial state is in thermal equilibrium with temperature , and is described by the canonical ensemble with the density matrix , where is the partition function. We denote the eigenvalues and orthonormal eigenvectors of () by () and (), respectively.

Suppose that we perform a projective energy measurement and obtain the measurement outcome with the probability , where the quantum state is projected from to . After the unitary time evolution, the quantum state becomes . At the end of the process, we record the quantum state reconstructed in the eigenbasis of the final Hamiltonian as . We here address the question of whether there is any law that governs the statistics of these quantum-state data when we repeat the above procedure. We show that it emerges when we accumulate the quantum-state data under a certain energy constraint given by . After taking the average, we obtain

(1) |

where the overline represents the average over the repeated processes, and is the Dirac delta function. For , corresponds precisely to the difference between the initial and final energies, which is equivalent to the work performed on the system. However, for off-diagonal elements, does not, in general, correspond to the work, but only has a formal meaning of the difference between the initial energy and the averaged final energy .

We also consider the time-reversed process with the Hamiltonian (), where represents the antiunitary time-reversal operator. The corresponding initial and final Hamiltonians are and , and the unitary evolution is given by . The initial state for the time-reversed process is assumed to be , where . In the same way as the forward process, we define

(2) |

where () and () are the eigenvalues and orthonormal eigenvectors of (), respectively, , , and .

Since is an operator acting on the Hilbert space, there are various ways to retrieve information from this object. Let us define distribution functions for the quantum-state statistics by taking the trace of the th moment of (),

(3) |

Here is a normalization constant determined by

(4) |

and is defined by the th power of with the symbol denoting the matrix multiplication and energy convolution simultaneously, i.e.,

(5) |

For the time-reversed process, the corresponding distribution function is defined by with the normalization condition and being the normalization constant for .

At , is identical to the work probability distribution: . For arbitrary , can be proven to take a real value (Appendix A). However, for , is not necessarily positive semidefinite. This prevents us from interpreting () as a probability distribution, though satisfies the normalization condition (4). Hence () should be regarded as a quasiprobability.

The main result of this paper is that the following relation holds between and its time-reversed partner :

(6) |

Here [] is the difference of the equilibrium free energies for the initial and final Hamiltonians at temperature . Note that the inverse temperature appearing in the free-energy argument is multiplied by in Eq. (6). For , the relation (6) reduces to the quantum work FT, . For , the relation (6) gives an extension of the FT to the quantum-state statistics. A remarkable feature of Eq. (6) is that it is valid for arbitrary unitary evolution , no matter how the system is driven away from equilibrium. Note that on the left-hand side of Eq. (6) each and strongly depends on , while the right-hand side is written in terms of the equilibrium quantities.

The relation (6) can be derived using the method of characteristic functions Talkner et al. (2007). Here we define a characteristic function for as the Fourier transform of ,

(7) |

which can be written as , where and are the Heisenberg representation of operators and , respectively (Appendix A). Hence () is classified into an out-of-time-ordered correlation function Lar (). By using the time-reversal property of , we find a symmetry relation , where is the characteristic function for . After Fourier transformation, we arrive at Eq. (6). The details of the proof is described in Appendix A.

By multiplying on both sides of Eq. (6) and using the normalization condition (4), we obtain the integral FT for the quantum-state statistics,

(8) |

where . For , the relation (8) is nothing but the Jarzynski equality, , while for it provides an extension of the Jarzynski equality. If one knows the distribution function , one can extract the equilibrium free-energy difference at temperature . Since is generated by the characteristic function , one can measure through the measurement of the out-of-time-ordered correlation function, for which various protocols have been proposed Swingle et al. (2016); Campisi and Goold (2017); Yao (); Zhu et al. (2016); Tsuji et al. (2017, 2018a); Yunger Halpern (2017); Yunger Halpern et al. (2018).

Applying Jensen’s inequality to the Jarzynski equality, one arrives at the second law of thermodynamics,

(9) |

One may wonder if one could derive a similar inequality

(10) |

from Eq. (8). This is, however, possible only if is positive semidefinite, since one cannot use Jensen’s inequality for non-positive-semidefinite distributions. We note that becomes positive semidefinite in the zero-temperature limit (). Let us assume that the ground state of the initial system (denoted by with the eigenenergy ) is unique. Then, in the zero-temperature limit,

(11) |

with . Thus, at zero temperature the inequality (10) holds. Of course, this does not mean that we have a new second law in addition to the existing one (9). At zero temperature is related to through , from which one obtains . Therefore, the inequality (10) reduces to the second law (9) at zero temperature [where ], and (10) does not provide new information in this case. In fact, the relation (6) reduces to the quantum work FT [Eq. (6) with ] in the zero-temperature limit. To obtain new information beyond the quantum work FT, one has to consider finite-temperature states.

If the relation (6) is applied near equilibrium, one can reproduce the out-of-time-order FDT Tsuji et al. (2018a) around zero frequency. This can be seen from the expansion of the integral FT (8) for and up to the third cumulants with respect to . If the Hamiltonian is split into the time-independent part and the rest as , where is an external field and is the coupled operator, then the second-order functional derivative on both sides of the cumulant expansions around (near equilibrium) leads to the near-zero-frequency part of the out-of-time-order FDT. Details of the derivation are given in Appendix B.

We have examined two aspects of : the distribution function for the quantum-state statistics and out-of-time-ordered correlation functions. For the latter, there have been various discussions in relation to chaotic properties of quantum systems Kit (); Mal (); Swingle et al. (2016); Rozenbaum et al. (2017); Aleiner et al. (2016); Fan et al. (2017); Tsuji et al. (2018b). Here we argue that there is a strong connection between the fluctuation in () and quantum chaotic nature (non-integrability) of the system. The crucial difference of () from the work probability distribution is that the former can take a negative value. In the following, we focus on the case of . We quantify the fluctuation in by the norm (),

(12) |

counts the negative portion of since (note that satisfies the normalization condition (4)).

As an illustration, let us consider the case that the Hamiltonian is suddenly quenched (i.e., ) and the initial temperature is . If we assume a non-degeneracy condition (Appendix C), is written for real Hamiltonians as . Using conserved quantities inherent in the system, the unitary transition matrix can be block-diagonalized as . If we define an entrywise-absolute-value matrix, , then , where denotes the Frobenius norm. Since the Frobenius norm is submultiplicative, satisfies an inequality, . By using the relation ( is the dimension of the th block Hilbert space) and ( is the dimension of the total Hilbert space), we obtain

(13) |

The right-hand side of this inequality strongly depends on the number of conserved quantities. As an estimate, let’s suppose that each block Hilbert space has approximately the same dimension (i.e., is independent of ). Then , i.e., the fluctuation in is constrained by the dimension of the block Hilbert space as compared to the dimension of the total Hilbert space. In integrable systems, the number of conserved quantities typically grows in proportion to the system size, so that is expected to decay exponentially in the large system-size limit. On the other hand, in non-integrable systems there is a finite number of conserved quantities, so that remains constant (or decays at most algebraically) as the system size increases. One can thus distinguish integrable and non-integrable systems by examining the system-size scaling behavior of .

We numerically demonstrate the relation (6) for the quantum-state statistics and the behavior of (12) for the one-dimensional model of hard-core bosons with the Hamiltonian,

(14) |

where () and () are the (next-)nearest-neighbor hopping and the strength of the interaction, respectively, and is the creation operator for hard-core bosons at site . We use as the unit of energy, and assumes the periodic boundary condition. The results are shown for the filling , where and are the number of particles and lattice sites, respectively. For other fillings, we obtain qualitatively similar results (Appendix C). To drive the system out of equilibrium, we perform an interaction quench at time . In this setup, (3) does not depend on and . We numerically solve the model by exact diagonalization (for details, see Appendix C).

The model (14) has been well studied in the context of quantum chaos Rigol (2009a); Santos and Rigol (2010). At , the model is known to be integrable. In the non-integrable case ( or ), the level-spacing statistics shows the Wigner-Dyson distribution, which is the universal property of quantum chaotic systems as expected from random matrix theory. The non-integrable model satisfies the eigenstate thermalization hypothesis Deutsch (1991); Srednicki (1994); Rigol et al. (2008), which is a sufficient condition for an isolated quantum system to be thermalized.

In the top and middle panels in Fig. 2, we plot the distribution functions for the forward process and for the time-reversed processes with , where we take a finite grid size to broaden the delta function (Appendix C). We clearly see that both and have negative parts. In the bottom panel of Fig. 2, we plot . The value of stays close to over the whole region of , which confirms the validity of the FT (6) for the quantum-state statistics. Small derivations are due to the finite grid used to plot and .

We numerically evaluate (12), which quantifies the negative portion of the distribution , for the one-dimensional hardcore boson model (14) in the limit of while keeping fixed (Appendix C). At zero temperature, is positive semidefinite (i.e., ) as explained earlier, and grows monotonically as temperature increases. In Fig. 3, we plot multiplied by the system size as a function of at for the quench . Clearly, shows a different scaling behavior between the integrable () and non-integrable () cases. For the integrable case, tends to decay exponentially (within one can still see slight bending of the curve in the log plot in Fig. 3), while for the non-integrable cases decays algebraically () and converges to the single universal curve. Even a tiny violation of integrability () causes a big difference in the behavior of . These results are consistent with the inequality (13). For the one-dimensional hardcore boson model (14), in the non-integrable case and due to the parity and translational symmetries. From (13), is roughly bounded by . If decays as a power law, , then the exponent must satisfy . The results shown in Fig. 3 indicate that the inequality for the exponent is saturated (i.e., ). In the integrable case shown in Fig. 3, the numerical estimate within suggests that with , the value of which is, however, non-universal and depends on the model parameters. We also simulate the same quantity for the one-dimensional spinless fermion model with nearest and next nearest neighbor hopping and interaction Rigol (2009b); Santos and Rigol (2010), and obtain similar results (Appendix C).

To summarize, we have studied the statistics of quantum states that can be obtained by the projective energy measurement followed by unitary evolution and quantum-state reconstruction in the energy basis. By accumulating the data of quantum states under a certain energy condition [Eq. (1)], we obtain the distribution function [Eq. (3)] which satisfies an infinite series of exact relations [Eq. (6)] (fluctuation theorem for the quantum-state statistics). It contains the quantum work fluctuation theorem as a special case, and if applied near equilibrium it reproduces the out-of-time-order fluctuation-dissipation theorem Tsuji et al. (2018a), which connects chaotic properties of the system and a nonlinear response function. We have discussed various aspects of the distribution function for the quantum-state statistics. In particular, the negativity of the distribution is closely related to the quantum chaotic nature (non-integrability) of the underlying model Hamiltonian. We have numerically demonstrated this for one-dimensional integrable and non-integrable quantum lattice models. The implications of the obtained relations to thermodynamics and thermalization in isolated quantum systems merit further study.

###### Acknowledgements.

N.T. is supported by JSPS KAKENHI Grant No. JP16K17729. M.U. acknowledges support by KAKENHI Grant No. JP26287088 and KAKENHI Grant No. JP15H05855.## Appendix A Proof of the fluctuation theorem for the quantum-state statistics

In this section, we prove the fluctuation theorem for the quantum-state statistics [Eq. (6)],

(15) |

The proof is actually similar to that for the ordinary quantum work fluctuation theorem using the method of characteristic functions Talkner et al. (2007).

Let us first recursively evaluate the product in the definition of in the energy eigenbasis,

(16) |

The normalization constant is determined by the direct calculation of the integral of ,

(17) |

Hence is given by the equilibrium partition function as

(18) |

In particular, is real (). is also real () as confirmed by taking the complex conjugate of ,

(19) |

By changing the summation labels as and and subsequently permuting the labels cyclicly, , one can see that (19) becomes identical to (16), proving the realness of .

After Fourier transformation, the characteristic function [Eq. (7)] is given by

(20) |

Here we define an operator

(21) | ||||

(22) |

With this, the characteristic function can be expressed in a compact form of