Dependence of a quantum-mechanical system on its own initial state and the initial state of the environment it interacts with

Hutter, Adrian; Wehner, Stephanie

doi:10.1103/physreva.87.012121

Cited by 16 publications

(22 citation statements)

References 28 publications

(44 reference statements)

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“…We refer the reader to recent works that used decoupling theorems in order to derive general conditions under which thermal equilibrium is achieved [HW13,Hut11,dRHRW13]. As such, we believe that our results shed light on the speed at which thermal equilibrium is reached for generic two-body dynamics.…”

Section: Applicationsmentioning

confidence: 55%

“…For example, del Rio et al [dRÅR + 11] used the decoupling theorem of [DBWR10] to study the work cost of an erasure in a fully quantum context. Also, general conditions under which thermal equilibrium is reached are analyzed in [HW13,Hut11,dRHRW13]. In a different area, Hayden and Preskill [HP07] proved that an mqubit quantum state that was dropped into a black hole could be recovered with high fidelity from an amount of Hawking radiation containing slightly more than m qubits of quantum information, as long as the dynamics of the black hole approximates a unitary two-design sufficiently well.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Decoupling with Random Quantum Circuits

Brown

Fawzi

2015

Commun. Math. Phys.

105

132

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Decoupling has become a central concept in quantum information theory with applications including proving coding theorems, randomness extraction and the study of conditions for reaching thermal equilibrium. However, our understanding of the dynamics that lead to decoupling is limited. In fact, the only families of transformations that are known to lead to decoupling are (approximate) unitary two-designs, i.e., measures over the unitary group which behave like the Haar measure as far as the first two moments are concerned. Such families include for example random quantum circuits with O(n 2 ) gates, where n is the number of qubits in the system under consideration. In fact, all known constructions of decoupling circuits use Ω(n 2 ) gates. Here, we prove that random quantum circuits with O(n log 2 n) gates satisfy an essentially optimal decoupling theorem. In addition, these circuits can be implemented in depth O(log 3 n). This proves that decoupling can happen in a time that scales polylogarithmically in the number of particles in the system, provided all the particles are allowed to interact. Our proof does not proceed by showing that such circuits are approximate two-designs in the usual sense, but rather we directly analyze the decoupling property.

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Section: Applicationsmentioning

confidence: 55%

Section: Introductionmentioning

confidence: 99%

Decoupling with Random Quantum Circuits

Brown

Fawzi

2015

Commun. Math. Phys.

105

132

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“…When the initial state is spread over many energy levels, and we consider the set of observables for which this state is an eigenstate, most observables are initially out of equilibrium yet equilibrate rapidly. Moreover, all two-outcome measurements, where one of the projectors is of low rank, equilibrate rapidly.The topic of equilibration time scales has been of much interest lately [1][2][3][4][5][6][7][8][9]. Given that it has been shown that quantum systems equilibrate under rather general conditions [10][11][12], it is important to understand the time scale for the process.…”

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confidence: 99%

Quantum systems equilibrate rapidly for most observables

et al. 2014

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Considering any Hamiltonian, any initial state, and measurements with a small number of possible outcomes compared to the dimension, we show that most measurements are already equilibrated. To investigate non-trivial equilibration we therefore consider a restricted set of measurements. When the initial state is spread over many energy levels, and we consider the set of observables for which this state is an eigenstate, most observables are initially out of equilibrium yet equilibrate rapidly. Moreover, all two-outcome measurements, where one of the projectors is of low rank, equilibrate rapidly.The topic of equilibration time scales has been of much interest lately [1][2][3][4][5][6][7][8][9]. Given that it has been shown that quantum systems equilibrate under rather general conditions [10][11][12], it is important to understand the time scale for the process. However, attempts to derive an upper bound on equilibration time have resulted in very large time scales. Short and Farrelly [1], for instance, obtain a very general bound, of which we give an improved derivation in appendix A, which scales with the dimension of the system, typically exponentially in the number of particles.A quantum system is said to undergo equilibration when its quantum state spends most of its time almost indistinguishable from a fixed (time-invariant) steadystate. This is not the same as thermalization, in which the steady-state is a Gibbs state. Thus, thermalization is a special case of equilibration, and understanding equilibration times is a key step in understanding thermalization times.When one discusses quantum equilibration, it is common to refer to either subsystem equilibration [10,13], in which a small system equilibrates due to contact with a bath, or observable equilibration, in which a fully closed system appears to equilibrate due to the limited information offered by outcomes of a particular set of observables. The latter was initially shown by Reimann [11,14], as a statement that the expectation values of quantum observables stay predominantly close to a static value, and was later built-on by Short [15], who showed that these results apply even if one considers all the information that can be gathered from the observable, instead of just the expectation value.In this paper we consider any finite-dimensional system and any Hamiltonian, and show that most N -outcome observables are initially in equilibrium (for N small compared to the dimension). To investigate timescales we therefore turn to a natural class of observables which are initially typically out of equilibrium -those with a definite initial value (i.e. observables for which the initial state is an eigenstate). We show that, for pure initial states spread over many energy levels, most of these observables equilibrate in very short times -in fact, most equilibrate essentially as fast as possible. Moreover, in the case of two-outcome observables where one of the projectors is of low rank we show that all observables equilibrate fast (for any initial state spread...

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“…Another factor that determines how quickly S thermalizes is the structure of the Hamiltonian of SE: the systems must be fully interacting and it helps if their joint evolution drives them through many different states. There have also been converse results, on states that do not equilibrate [18,19]. The results of [5][6][7][8][9], originally derived through measure concentration techniques and from properties of the system's Hamiltonian, emerge here as direct consequence of our general approach, in the special case where the reference R is classical.…”

Section: Second Example: Noise Modelsmentioning

confidence: 63%

“…It was shown that the time scales of thermalization depend again on the size of the subsystem, as well as on the emergent speed of light [19].…”

Section: B Time Scalesmentioning

confidence: 99%

Relative thermalization

et al. 2016

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When studying thermalization of quantum systems, it is typical to ask whether a system interacting with an environment will evolve towards a local thermal state. Here, we show that a more general and relevant question is "when does a system thermalize relative to a particular reference?" By relative thermalization we mean that, as well as being in a local thermal state, the system is uncorrelated with the reference. We argue that this is necessary in order to apply standard statistical mechanics to the study of the interaction between a thermalized system and a reference. We then derive a condition for relative thermalization of quantum systems interacting with an arbitrary environment. This condition has two components: the first is state-independent, reflecting the structure of invariant subspaces, like energy shells, and the relative sizes of system and environment; the second depends on the initial correlations between reference, system and environment, measured in terms of conditional entropies. Intuitively, a small system interacting with a large environment is likely to thermalize relative to a reference, but only if, initially, the reference was not highly correlated with the system and environment. Our statement makes this intuition precise, and we show that in many natural settings this thermalization condition is approximately tight. Established results on thermalization, which usually ignore the reference, follow as special cases of our statements. I. THE CASE FOR RELATIVE THERMALIZATION A. Subjectivity in thermodynamicsThermodynamics was originally developed to study and improve the performance of steam engines: to turn the heat of a gas into work, as efficiently as possible. Today, it is also being applied to study heat and work flows in the micro and nano regimes. In fact, advances in the manipulation of small systems have allowed us to extract work from systems such as quantum dots and trapped ions [1,2]. Yet, thermodynamics as a science is still adapting to this new regime, and it still bears some of the traits of the gaseous systems for which it was first designed. For example, the information available about the state of a gas used to be limited and objective: we would measure the temperature, pressure and volume of a gas, but we could not keep track of each individual particle. Crucially, all conceivable observers had access to the same information about the state of the system, and could manipulate it in equivalent ways-like letting a gas expand to obtain work. And yet, since very early on, several thought experiments have challenged the idea that thermodynamics should be objective. In 1871 James Maxwell realized that a "demon" able to measure the position and velocity of the particles of a gas could extract more work from it than the typical observer implicit in standard thermodynamics [3]. Picking up on Maxwell's idea on the power of information, Leó Szilárd imagined a partitioned box with a single-particle gas on one side. Depending on their information on the location of the particle, tw...

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Dependence of a quantum-mechanical system on its own initial state and the initial state of the environment it interacts with

Cited by 16 publications

References 28 publications

Decoupling with Random Quantum Circuits

Decoupling with Random Quantum Circuits

Quantum systems equilibrate rapidly for most observables

Relative thermalization

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