Canonical ensemble theory is used to obtain simple exact scaling relationships (homology transformations) which establish equivalences between certain thermodynamic quantities for two state weak valued and mean valued energy quantum systems that are described by the same Hamiltonian. These equivalences define how a thermodynamic quantity for a weak (mean) valued energy system can be represented by an appropriately scaled version of the same quantity for a mean (weak) valued energy system. This provides: (i) a method for quantifying the thermodynamic properties of the typically more complicated (and difficult to prepare) weak valued energy systems in terms of those of the generally less complicated mean valued energy systems; and (ii) a systematic way to compare the properties of weak and mean valued energy systems. The results are validated and illustrated numerically by applying them to a spin ½ system in a uniform magnetic field. Several apparent thermodynamic effects induced by the weak measurement and state post-selection process required to prepare a weak valued energy system are also identified for this system.
Canonical ensemble theory is used to obtain simple exact scaling relationships (homology transformations) which establish equivalences between certain thermodynamic quantities for two state weak valued and mean valued energy quantum systems that are described by the same Hamiltonian. These equivalences define how a thermodynamic quantity for a weak (mean) valued energy system can be represented by an appropriately scaled version of the same quantity for a mean (weak) valued energy system. This provides: (i) a method for quantifying the thermodynamic properties of the typically more complicated (and difficult to prepare) weak valued energy systems in terms of those of the generally less complicated mean valued energy systems; and (ii) a systematic way to compare the properties of weak and mean valued energy systems. The results are validated and illustrated numerically by applying them to a spin ½ system in a uniform magnetic field. Several apparent thermodynamic effects induced by the weak measurement and state post-selection process required to prepare a weak valued energy system are also identified for this system.
A system composed of identical spins and described by a quantum mechanical pure state is analyzed within the statistical framework presented in Part I of this work. We explicitly derive the typical values of the entropy, of the energy, and of the equilibrium reduced density matrix of a subsystem for the two different statistics introduced in Part I. In order to analyze their consistency with thermodynamics, these quantities of interest are evaluated in the limit of large number of components of the isolated system. The main results can be summarized as follows: typical values of the entropy and of the equilibrium reduced density matrix as functions of the internal energy in the fixed expectation energy ensemble do not satisfy the requirement of thermodynamics. On the contrary, the thermodynamical description is recovered from the random pure state ensemble (RPSE), provided that one considers systems large enough. The thermodynamic limit of the considered properties for the spin system reveals a number of important features. First canonical statistics (and thus, canonical typicality as long as the fluctuations around the average value are small) emerges without the need of assuming the microcanonical space for the global pure state. Moreover, we rigorously prove (i) the equivalence of the "global temperature," derived from the entropy equation of state, with the "local temperature" determining the canonical state of the subsystems; and (ii) the equivalence between the RPSE typical entropy and the canonical entropy for the overall system.
We generalize the method introduced in J. Phys. A: Math. Gen. 35, 7255 (2002) based on the concept of thermodynamic equivalence and we transform a Fermi system of general density of states into a thermodynamically equivalent Bose system. This consists of mapping configurations of fermions from the original system onto configurations of bosons, the initial and final configurations having the same energy above the many-body ground state energy. In this way we obtain two systems of particles of different exclusion statistics, but which have the same entropies--and therefore identical canonical thermodynamic properties. This method enables one in general to calculate the system properties in either of the bosonic and fermionic representations. We check the method here in microscopic detail by calculating the equilibrium particle distributions in the two representations using the entropy maximization at fixed particle number and fixed ``fermionic'' and ``bosonic'' energies, respectively. Analytical calculations seem difficult to do, but we check the results numerically and we find them equal within the numerical accuracy.
Two approaches to describing the thermodynamics of a subsystem that interacts with a thermal bath are considered. Within the first approach, the mean system energy E_{S} is identified with the expectation value of the system Hamiltonian, which is evaluated with respect to the overall (system+bath) equilibrium distribution. Within the second approach, the system partition function Z_{S} is considered as the fundamental quantity, which is postulated to be the ratio of the overall (system+bath) and the bath partition functions, and the standard thermodynamic relation E_{S}=-d(ln Z_{S})/dbeta is used to obtain the mean system energy. Employing both classical and quantum-mechanical treatments, the advantages and shortcomings of the two approaches are analyzed in detail for various different systems. It is shown that already within classical mechanics both approaches predict significantly different results for thermodynamic quantities provided the system-bath interaction is not bilinear or the system of interest consists of more than a single particle. Based on the results, it is concluded that the first approach is superior.
Quantum weak measurement, measuring some observable quantities within the selected subensemble of the entire quantum ensemble, can produce many interesting results such as the superluminal phenomena. An outcome of such a measurement is the weak value which has been applied to amplify some weak signals of quantum interactions in lots of previous references. Here, we apply the weak measurement to the system of relativistic cold atoms. According to mass-energy equivalence, the internal energy of an atom will contribute its rest mass and consequently the external momentum of center of mass. This implies a weak coupling between the internal and external degrees of freedom of atoms moving in the free space. After a duration of this coupling, a weak value can be obtained by post-selecting an internal state of atoms. We show that the weak value can change the momentum uncertainty of atoms and consequently help us to experimentally measure the weak effects arising from mass-energy equivalence.
Preparing thermal equilibrium states is an essential task for finite-temperature quantum simulations. In statistical mechanics, microstates in thermal equilibrium can be obtained from statistical ensembles. To date, numerous ensembles have been devised, ranging from Gibbs ensembles such as the canonical and microcanonical ensembles to a variety of generalized ensembles. Since these ensembles yield equivalent thermodynamic predictions, one can freely choose an ensemble for computational convenience. In this paper, we exploit this flexibility to develop an efficient quantum algorithm for preparing thermal equilibrium states. We first present a quantum algorithm for implementing generalized ensembles within the framework of quantum singular value transformation. We then perform a detailed analysis of the computational cost and elucidate its dependence on the choice of the ensemble. Our analysis shows that employing an appropriate ensemble can significantly mitigate ensemble-dependent overhead and yield improved scaling of the computational cost with system size compared to existing methods based on the canonical ensemble. We also numerically demonstrate that our approach achieves a significant reduction in the computational cost even for small finite-size systems. Our algorithm applies to arbitrary thermodynamic systems at any temperature and is thus expected to offer a practical and versatile method for computing finite-temperature properties of quantum many-body systems. These results highlight the potential of ensemble design as a powerful tool for enhancing the efficiency of a broad class of quantum algorithms.
We investigate further the relationship between the entanglement spectrum of a composite many-body system and the energy spectrum of a subsystem making use of concepts of canonical thermodynamics. In many important cases the entanglement Hamiltonian is, in the limit of strong coupling between subsystems, proportional to the energy Hamiltonian of the subsystem. The proportionality factor is an appropriately defined coupling parameter, suggesting to interpret the latter as a inverse temperature. We identify a condition on the entanglement Hamiltonian which rigorously guarantees this interpretation to hold and removes any ambiguity in the definition of the entanglement Hamiltonian regarding contributions proportional to the unit operator. Illustrations of our findings are provided by spin ladders of arbitrary spin length, and by bilayer quantum Hall systems at total filling factor ν = 2. Within mean-field description, the latter system realizes an entanglement spectrum of free fermions with just two levels of equal modulus where the analogies to canonical thermodynamics are particularly close.