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In this paper we return to the problem of reduced-state dynamics in the presence of an interacting environment. The question we investigate is how to appropriately model a particular system evolution given some knowledge of the system-environment interaction. When the experimenter takes into account certain known features of the interaction such as its invariant subspaces or its non-local content, it may not be possible to consistently model the system evolution over a certain time interval using a standard Stinespring dilation, which assumes the system and environment to be initially uncorrelated. Simple examples demonstrating how restrictions can emerge are presented below. When the system and environment are qubits, we completely characterize the set of unitaries that always generate reduced dynamics capable of being modeled using a consistent Stinespring dilation. Finally, we show how any initial correlations between the system and environment can be certified by observing the system transformation alone during certain joint evolutions.
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The optimal state determination (or tomography) is studied for a composite system of two qubits when measurements can be performed on one of the qubits and interactions of the two qubits can be implemented. The goal is to minimize the number of interactions to be used. The algebraic method applied in the paper leads to an extension of the concept of mutually unbiased measurements.
We use the Koashi-Imoto decomposition of the degrees of freedom of joint system-environment initial states to investigate the reduced dynamics. We show that a subset of joint system-environment initial states guarantees completely positive reduced dynamics, if and only if the system privately owns all quantum degrees of freedom and can locally access the classical degrees of freedom, without disturbing all joint initial states in the given subset. Furthermore, we show that the quantum mutual information for such kinds of states must be independent of the quantum degrees of freedom.
The quantum channels with memory, known as non-Markovian channels, are of crucial importance for a realistic description of a variety of physical systems, and pave ways for new methods of decoherence control by manipulating the properties of environment such as its frequency spectrum. In this work, the reduced dynamics of coin in a discrete-time quantum walk is characterized as a non-Markovian quantum channel. A general formalism is sketched to extract the Kraus operators for a $t$-step quantum walk. Non-Markovianity, in the sense of P-indivisibility of the reduced coin dynamics, is inferred from the non-monotonous behavior of distinguishably of two orthogonal states subjected to it. Further, we study various quantum information theoretic quantities of a qubit under the action of this channel, putting in perspective, the role such channels can play in various quantum information processing tasks.
Effective quantum computation relies upon making good use of the exponential information capacity of a quantum machine. A large barrier to designing quantum algorithms for execution on real quantum machines is that, in general, it is intractably difficult to construct an arbitrary quantum state to high precision. Many quantum algorithms rely instead upon initializing the machine in a simple state, and evolving the state through an efficient (i.e. at most polynomial-depth) quantum algorithm. In this work, we show that there exist families of quantum states that can be prepared to high precision with circuits of linear size and depth. We focus on real-valued, smooth, differentiable functions with bounded derivatives on a domain of interest, exemplified by commonly used probability distributions. We further develop an algorithm that requires only linear classical computation time to generate accurate linear-depth circuits to prepare these states, and apply this to well-known and heavily-utilized functions including Gaussian and lognormal distributions. Our procedure rests upon the quantum state representation tool known as the matrix product state (MPS). By efficiently and scalably encoding an explicit amplitude function into an MPS, a high fidelity, linear-depth circuit can directly be generated. These results enable the execution of many quantum algorithms that, aside from initialization, are otherwise depth-efficient.
Whereas it is easy to reduce the translational symmetry of a molecular system by using, e.g., Jacobi coordinates the situation is much more involved for the rotational symmetry. In this paper we address the latter problem using {it holonomy reduction}. We suggest that the configuration space may be considered as the reduced holonomy bundle with a connection induced by the mechanical connection. Using the fact that for the special case of the three-body problem, the holonomy group is SO(2) (as opposed to SO(3) like in systems with more than three bodies) we obtain a holonomy reduced configuration space of topology $ mathbf{R}_+^3 times S^1$. The dynamics then takes place on the cotangent bundle over the holonomy reduced configuration space. On this phase space there is an $S^1$ symmetry action coming from the conserved reduced angular momentum which can be reduced using the standard symplectic reduction method. Using a theorem by Arnold it follows that the resulting symmetry reduced phase space is again a natural mechanical phase space, i.e. a cotangent bundle. This is different from what is obtained from the usual approach where symplectic reduction is used from the outset. This difference is discussed in some detail, and a connection between the reduced dynamics of a triatomic molecule and the motion of a charged particle in a magnetic field is established.
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