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arXiv:2403.10501

Quantum States Seen by a Probe: Partial Trace Over a Region of Space

Quentin Ansel

Published Mar 15, 2024• 0 citations
arXivHTMLPDF

Abstract

The partial trace operation is usually considered in composite quantum systems, to reduce the state on a single subsystem. This operation has a key role in the decoherence effect and quantum measurements. However, partial trace operations can be defined in more generic situations. In particular, it can be used to restrict a quantum state (for a single or several quantum entities) on a specific region of space, the rest of the universe being treated as an environment. The reduced state is then interpreted as the state that can be detected by an ideal probe with a limited spatial extent. In this paper, such an operation is investigated for systems defined on a Fock Hilbert space. A generic expression of the reduced density matrix is computed, and it is applied to several case studies: eigenstates of the number operator, coherent states, and thermal states. These states admit very different behaviors. In particular, (i) a decoherence effect happens on eigenstates of the number operator (ii) coherent or thermal states remain coherent or thermal, but with an amplitude/temperature reduced non-trivially by the overlap between the field and the region of interest.

DOI
10.48550/arxiv.2403.10501

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The partial trace operation is usually considered in composite quantum systems, to reduce the state on a single subsystem. This operation has a key role in the decoherence effect and quantum measurements. However, partial trace operations can be defined in more generic situations. In particular, it can be used to restrict a quantum state (for a single or several quantum entities) on a specific region of space, the rest of the universe being treated as an environment. The reduced state is then interpreted as the state that can be detected by an ideal probe with a limited spatial extent. In this paper, such an operation is investigated for systems defined on a Fock Hilbert space. A generic expression of the reduced density matrix is computed, and it is applied to several case studies: eigenstates of the number operator, coherent states, and thermal states. These states admit very different behaviors. In particular, (i) a decoherence effect happens on eigenstates of the number operator (ii) coherent or thermal states remain coherent or thermal, but with an amplitude/temperature reduced non-trivially by the overlap between the field and the region of interest.
Partial trace is a very important mathematical operation in quantum mechanics. It is not only helpful in studying the subsystems of a composite quantum system but also used in computing a vast majority of quantum entanglement measures. Calculating partial trace becomes computationally very intensive with increasing number of qubits as the Hilbert space dimension increases exponentially. In this paper, we discuss about our new method of partial tracing which is based on set theory and it is more efficient. The proposed method of partial tracing overcomes all the limitations of the other well-known methods such as being computationally intensive and being limited to low dimensional Hilbert spaces. We give a detailed theoretical description of our method and also provide an explicit example of the computation. The merits of the new method and other key ideas are discussed.
The interpretation of the concept of reduced state is a subtle issue that has relevant consequences when the task is the interpretation of quantum mechanics itself. The aim of this paper is to argue that reduced states are not the quantum states of subsystems in the same sense as quantum states are states of the whole composite system. After clearly stating the problem, our argument is developed in three stages. First, we consider the phenomenon of environment-induced decoherence as an example of the case in which the subsystems interact with each other; we show that decoherence does not solve the measurement problem precisely because the reduced state of the measuring apparatus is not its quantum state. Second, the non-interacting case is illustrated in the context of no-collapse interpretations, in which we show that certain well-known experimental results cannot be accounted for due to the fact that the reduced states of the measured system and the measuring apparatus are conceived as their quantum states. Finally, we prove that reduced states are a kind of coarse-grained states, and for this reason they cancel the correlations of the subsystem with other subsystems with which it interacts or is entangled.
We consider a quantum system A∪B made up of degrees of freedom that can be partitioned into spatially disjoint regions A and B. When the full system is in a pure state in which regions A and B are entangled, the quantum mechanics of region A described without reference to its complement is traditionally assumed to require a reduced density matrix on A. While this is certainly true as an exact matter, we argue that under many interesting circumstances expectation values of typical operators anywhere inside A can be computed from a suitable pure state on A alone, with a controlled error. We use insights from quantum statistical mechanics-specifically the eigenstate thermalization hypothesis (ETH)-to argue for the existence of such "representative states."
We present a reduced density operator for electronically open molecules by explicitly averaging over the environmental degrees of freedom of the composite Hamiltonian. Specifically, we include the particle-number nonconserving (particle-breaking) interactions responsible for the sharing of electrons between the molecule and the environment, which are neglected in standard formulations of quantum statistical mechanics. We propose an unambiguous definition of the partial trace operation in the composite fermionic Fock space based on composite states in a second quantization framework built from a common orthonormal set of orbitals. Thereby, we resolve the fermionic partial trace ambiguity. The common orbital basis is constructed by spatial localization of the full orbital space, in which the full composite Hamiltonian naturally partitions into a molecule Hamiltonian, an environment Hamiltonian, and an interaction Hamiltonian. The new reduced density operator is based on the approximation of commutativity between the subsystem Hamiltonians (i.e., molecule and environment Hamiltonians) and the interaction Hamiltonian, but our methodology provides a hierarchical approach for improving this approximation. The reduced density operator can be viewed as a generalization of the grand canonical density operator. We are prompted to define the generalized chemical potential, which aligns with the standard interpretation of the chemical potential, apart from the possibility of fractional rather than strictly integer electron transfer in our framework. In contrast to standard approaches, our framework enables an explicit consideration of the electron occupancy in the environment at any level of theory, irrespective of the model used to describe the molecule. Specifically, our reduced density operator is fully compatible with all possible level-of-theory treatments of the environment. The approximations that render our reduced density operator identical to the grand canonical density operator are (i) restriction of excitations to occur within the same orbitals and (ii) assumption of equal interaction with the environment for all molecule spin orbitals (i.e., the wide-band approximation).
The coupling of a mesoscopic system with its environment usually causes total decoherence: at long times the reduced density matrix of the system evolves in time to a limit which is independent of its initial value, losing all the quantum information stored in its initial state. Under special circumstances, a subspace of the system's Hilbert space remains coherent, or "decoherence free", and the reduced density matrix approaches a non-trivial limit which contains information on its initial quantum state, despite the coupling to the environment. This situation is called "partial decoherence". Here we find the conditions for partial decoherence for a mesoscopic system (with NNN quantum states) which is coupled to an environment. When the Hamiltonian of the system commutes with the total Hamiltonian, one has "adiabatic decoherence", which yields N-1 time-independent combinations of the reduced density matrix elements. In the presence of a magnetic flux, one can measure circulating currents around loops in the system even at long times, and use them to retrieve information on the initial state. For N=2, we demonstrate that partial decoherence can happen only under adiabatic decoherence conditions. However, for N>2N>2N>2 we find partial decoherence even when the Hamiltonian of the system does not commute with the total Hamiltonian, and we obtain the general conditions for such non-adiabatic partial decoherence. For an electron moving on a ring, with N>2N>2N>2 single-level quantum dots, non-adiabatic partial decoherence can arise only when the total flux through the ring vanishes (or equals an integer number of flux quanta), and therefore there is no asymptotic circulating current.
Temperature is usually defined for physical systems at thermal equilibrium. Nevertheless one may wonder if it would be possible to attribute a meaningful notion of temperature to an arbitrary quantum state, beyond simply the thermal (Gibbs) state. In this Letter, we propose such a notion of temperature considering an operational task, inspired by the zeroth law of thermodynamics. Specifically, we define two effective temperatures for quantifying the ability of a quantum system to cool down or heat up a thermal environment. In this way we can associate an operationally meaningful notion of temperature to any quantum density matrix. We provide general expressions for these effective temperatures, for both single- and many-copy systems, establishing connections to concepts previously discussed in the literature. Finally, we consider a more sophisticated scenario where the heat exchange between the system and the thermal environment is assisted by a quantum reference frame. This leads to an effect of "coherent quantum catalysis," where the use of a coherent catalyst allows for exploiting quantum energetic coherences in the system, now leading to much colder or hotter effective temperatures. We demonstrate our findings using a two-level atom coupled to a single mode of the electromagnetic field.