Decoherence-Free Subsystems (DFS) are a powerful means of protecting quantum information against noise with known symmetry properties. Although Hamiltonians theoretically exist that can implement a universal set of logic gates on DFS encoded qubits without ever leaving the protected subsystem, the natural Hamiltonians that are available in specific implementations do not necessarily have this property. Here we describe some of the principles that can be used in such cases to operate on encoded qubits without losing the protection offered by the DFS. In particular, we show how dynamical decoupling can be used to control decoherence during the unavoidable excursions outside of the DFS. By means of cumulant expansions, we show how the fidelity of quantum gates implemented by this method on a simple two-physical-qubit DFS depends on the correlation time of the noise responsible for decoherence. We further show by means of numerical simulations how our previously introduced "strongly modulating pulses" for NMR quantum information processing can permit high-fidelity operations on multiple DFS encoded qubits in practice, provided that the rate at which the system can be modulated is fast compared to the correlation time of the noise. The principles thereby illustrated are expected to be broadly applicable to many implementations of quantum information processors based on DFS encoded qubits.
Decoherence-free subsystems have been successfully developed as a tool to preserve fragile quantum information against noises. In this letter, we develop a structure theory for decoherence-free subsystems. Based on it, we present an effective algorithm to construct a set of maximal decoherence-free subsystems in the sense that any other such subsystem is a subspace of one of them. As an application of these techniques in quantum many body systems, we propose a simple and numerically robust method to determine if two irreducible tensors are repeated, a key step in deciding if they are equivalen…
We discuss the structure of decoherence-free subsystems for a bosonic channel affected by collective depolarization. A single use of the channel is defined as a transmission of a pair of bosonic modes. Collective depolarization consists in a random linear U(2) transformation of the respective mode operators, which is assumed to be identical for $N$ consecutive uses of the channel. We derive a recursion formula that characterizes the dimensionality of available decoherence-free subsystems in such a setting.
Decoherence is the phenomenon of non-unitary dynamics that arises as a consequence of coupling between a system and its environment. It has important harmful implications for quantum information processing, and various solutions to the problem have been proposed. Here we provide a detailed a review of the theory of decoherence-free subspaces and subsystems, focusing on their usefulness for preservation of quantum information.
We introduce a generalized theory of decoherence-free subspaces and subsystems (DFSs), which do not require accurate initialization. We derive a new set of conditions for the existence of DFSs within this generalized framework. By relaxing the initialization requirement we show that a DFS can tolerate arbitrarily large preparation errors. This has potentially significant implications for experiments involving DFSs, in particular for the experimental implementation, over DFSs, of the large class of quantum algorithms which can function with arbitrary input states.
The use of d-state systems, or qudits, in quantum information processing is discussed. Three-state and higher dimensional quantum systems are known to have very different properties from two-state systems, i.e., qubits. In particular there exist qudit states which are not equivalent under local unitary transformations unless a selection rule is violated. This observation is shown to be an important factor in the theory of decoherence-free, or noiseless, subsystems. Experimentally observable consequences and methods for distinguishing these states are also provided, including the explicit const…
When the environmental disturbace to a quantum system has a wavelength much larger than the system size, all qubits localized within a small area are under action of the same error operators. Noiseless subsystem and decoherence free subspace are known to correct such collective errors. We construct simple quantum circuits, which implement these collective error correction codes, for a small number $n$ of physical qubits. A single logical qubit is encoded with $n=3$ and $n=4$, while two logical qubits are encoded with $n=5$. The recursive relations among the subspaces employed in noiseless subs…
This paper provides a general theory for characterizing and constructing a decoherence-free (DF) subsystem for an infinite dimensional linear open quantum system. The main idea is that, based on the Heisenberg picture of the dynamics rather than the commonly-taken Schrodinger picture, the notions of controllability and observability in control theory are employed to characterize a DF subsystem. A particularly useful result is a general if and only if condition for a linear system to have a DF component; this condition is used to demonstrate how to actually construct a DF dynamics in some speci…
We show that private shared reference frames can be used to perform private quantum and private classical communication over a public quantum channel. Such frames constitute a novel type of private shared correlation (distinct from private classical keys or shared entanglement) useful for cryptography. We present optimally efficient schemes for private quantum and classical communication given a finite number of qubits transmitted over an insecure channel and given a private shared Cartesian frame and/or a private shared reference ordering of the qubits. We show that in this context, it is use…
We describe exchange-only universal quantum computation and leakage reduction in the 3-qubit decoherence free subsystem (DFS). We discuss the angular momentum structure of the DFS, the proper forms for the DFS CNOT and leakage reduction operators in the total angular momentum basis, and new exchange-only pulse sequences for the CNOT and leakage reduction operators. Our new DFS CNOT sequence requires 22 pulses in 13 time steps. The DFS leakage reduction sequence, the first explicit leakage reduction sequence of its kind, requires 30 pulses in 20 time steps. Although the search for sequences was…
The Uhrig dynamical decoupling sequence achieves high-order decoupling of a single system qubit from its dephasing bath through the use of bang-bang Pauli pulses at appropriately timed intervals. High-order decoupling of single and multiple qubit systems from baths causing both dephasing and relaxation can also be achieved through the nested application of Uhrig sequences, again using single-qubit Pauli pulses. For the 3-qubit decoherence free subsystem (DFS) and related subsystem encodings, Pauli pulses are not naturally available operations; instead, exchange interactions provide all require…
Quantum entanglement has been used as an experimental tool to control the state of a quantum subsystem by measurements in its twin subsystem by Kwiat and Chiao 21 year ago, already, with entangled subsystems linked by an auxiliary classical coincidence channel. We propose to extend the idea one significant step further consisting essentially in eliminating the coincidence channel in the reminded experiment. The current availability of bright biphoton sources makes such experiments feasible.
In order to analyze the effect of chaos or order on the rate of decoherence in a subsystem, we aim to distinguish effects of the two types of dynamics by choosing initial states as random product states from two factor spaces representing two subsystems. We introduce a random matrix model that permits to vary the coupling strength between the subsystems. The case of strong coupling is analyzed in detail, and we find no significant differences except for very low-dimensional spaces.
Since there are many examples in which no decoherence-free subsystems exist (among them all cases where the error generators act irreducibly on the system Hilbert space), it is of interest to search for novel mechanisms which suppress decoherence in these more general cases. Drawing on recent work (quant-ph/0502153) we present three results which indicate decoherence suppression without the need for noiseless subsystems. There is a certain trade-off; our results do not necessarily apply to an arbitrary initial density matrix, or for completely generic noise parameters. On the other hand, our c…
A generalized formal framework for decoherence, that can be used both in open and closed quantum systems, is sketched. In this context, the relationship between the decoherence of a closed system and the decoherence of its subsystems is studied, and the corresponding decoherence times are defined: for macroscopic systems, the decoherence time of the closed system is much greater than the decoherence time of its subsystems. Finally, it is shown that the application of the new formal framework to a well-known model leads to physically adequate results.
The dynamics of two coupled spins-1/2 coupled to a spin-bath is studied as an extended model of the Tessieri-Wilkie Hamiltonian \cite{TWmodel}. The pair of spins served as an open subsystem were prepared in one of the Bell states and the bath consisted of some spins-1/2 is in a thermal equilibrium state from the very beginning. It is found that with the increasing the coupling strength of the bath spins, the bath forms a resonant antiferromagnetic order. The polarization correlation between the two spins of the subsystem and the concurrence are recovered in some extent to the isolated subsyste…
The three-spin-$1/2$ decoherence-free subsystem defines a logical qubit protected from collective noise and supports exchange-only universal gates. Such logical qubits are well-suited for implementation with electrically-defined quantum dots. Exact exchange-only entangling logical gates exist but are challenging to construct and understand. We use a decoupling strategy to obtain straightforward approximate entangling gates. A benefit of the strategy is that if the physical spins are aligned, then it can implement evolution under entangling Hamiltonians. Hamiltonians expressible as linear combi…
We outline a proposal for a method of preparing an encoded two-state system (logical qubit) that is immune to collective noise acting on the Hilbert space of the states supporting it. The logical qubit is comprised of three photonic three-state systems (qutrits) and is generated by the process of spontaneous parametric down conversion. The states are constructed using linear optical elements along with three down-conversion sources, and are deemed successful by the simultaneous detection of six events. We also show how to select a maximally entangled state of two qutrits by similar methods. Fo…
This paper studies the combinatoric structure of the set of all representations, up to equivalence, of a finite-dimensional semisimple Lie algebra. This has intrinsic interest as a previously unsolved problem in representation theory, and also has applications to the understanding of quantum decoherence. We prove that for Hilbert spaces of sufficiently high dimension, decoherence-free subspaces exist for almost all representations of the error algebra. For decoherence-free subsystems, we plot the function $f_d(n)$ which is the fraction of all $d$-dimensional quantum systems which preserve $n$ …
We present a model of an N-qubit channel where consecutive qubits experience correlated random rotations. Our model is an extension to the standard decoherence-free subsystems approach (DFS) which assumes that all the qubits experience the same disturbance. The variation of rotations acting on consecutive qubits is modeled as diffusion on the SU(2) group. The model may be applied to spins traveling in a varying magnetic field, or to photons passing through a fiber whose birefringence fluctuates over the time separation between photons. We derive an explicit formula describing the action of the…
The bidirectional steerability between different-size subsystems is discussed for a single parameter accelerated qubit-qutrit system. The decoherence due to the mixing and acceleration parameters is investigated, where for the total system and the qutrit, it increases as the mixing parameter increases, while it decreases for the qubit. The non-classical correlations are quantified by using the local quantum uncertainty, where it increases at large values of the acceleration parameter. The possibility that each subsystem steers each other is studied, where the behavior of the steering inequalit…
Choosing a specific way of dividing a closed system into parts is a starting point for the decoherence program and for the quantum thermalization program. It is shown that one can always chose such way of partitioning that decoherence-assisted classicality does not emerge and thermalization does not occur. Implications of this result are discussed.
A critical step in experimental quantum information processing (QIP) is to implement control of quantum systems protected against decoherence via informational encodings, such as quantum error correcting codes, noiseless subsystems and decoherence free subspaces. These encodings lead to the promise of fault tolerant QIP, but they come at the expense of resource overheads. Part of the challenge in studying control over multiple logical qubits, is that QIP test-beds have not had sufficient resources to analyze encodings beyond the simplest ones. The most relevant resources are the number of avai…
Quantum information requires protection from the adverse affects of decoherence and noise. This review provides an introduction to the theory of decoherence-free subspaces, noiseless subsystems, and dynamical decoupling. It addresses quantum information preservation as well protected computation.
Quantum dissipation is studied within two model oscillators, the Caldirola-Kanai (CK) oscillator as an open system with one degree of freedom and the Bateman-Feshbach-Tikochinsky (BFT) oscillator as a closed system with two degrees of freedom. Though these oscillators describe the same classical damped motion, the CK oscillator retains the quantum coherence, whereas the damped subsystem of the BFT oscillator exhibits both quantum decoherence and classical correlation. Furthermore the amplified subsystem of the BFT oscillator shows the same degree of quantum decohernce and classical correlation…
We study the robustness of the GHZ (or ``cat'') class of multi-partite states under decoherence. The noise model is described by a general completely positive map for qubits independently coupled to the environment. In particular, the robustness of N-party entanglement is studied in the large N limit when (a) the number of spatially separated subsystems is fixed but the size of each subsystem becomes large (b) the size of the subsystems is fixed while their number becomes arbitrarily large. We obtain conditions for entanglement in these two cases. Among our other results, we show that the pari…
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…
It has been claimed that decoherence of open quantum systems explains the tendency of macroscopic systems to exhibit quasiclassical behavior. We show that quasiclassicality is in fact an unremarkable property, characterizing generic subsystems of environments even in the absence of dynamical decoherence. It is suggested that decoherence is best regarded as explaining the persistence of true classicality, rather than the emergence, rather than the emergence of quasiclassicality.
Hierarchy of mixed symmetry protected topological states in extended cluster states under subsystem decoherence
Graduate School of Engineering Science, Akita University（秋田大学工学研究科）；Department of Physics, College of Humanities and Sciences, Nihon University（日本大学文理学部物理学科）
We study the decoherence induced by the environment over a composite quantum system, comprising two coupled subsystems A and B, which may be a harmonic or an upside-down oscillators. We analyze the case in which the B-subsystem is in direct interaction with a thermal bath, while the other remains isolated from the huge reservoir. We compare the results concerning the decoherence suffered by the A-subsystem.
The dynamic of correlations in a system composed of a two-mode quantum field coupled with the environment is studied. The quantum field corresponds to two entangled coherent states whose amplitude we vary up to the mesoscopic regime. We show that under the onset of decoherence, correlations in the quantum field are not lost but transferred to the environment. We also found that sudden transitions in the decoherence regimes appear along with the dynamics depending on the coherent states' amplitude. Increasing the amplitude of the entangled coherence state results in the apparition of a metastab…
In this paper, we present a theoretical description to the quantum coherence and decoherence phenomena of energy transfer in photosynthesis observed in a recent experiment [see Science 316, 1462 (2007)]. As a successive two-color laser pulses with selected frequencies cast on a sample of the photosynthetic purple bacterium Rb. sphaeroides two resonant excitations of electrons in chromophores can be generated. However, this effective two-level subsystem will interact with its protein environment and decoherence is inevitable. We describe this subsystem coupled with its environment as a dynamica…
The different methods of reducing decoherence in quantum devices are discussed from the unified point of view based on the energy conservation principle and the concept of forbidden transitions. Minimal decoherence model, "bang-bang" techniques, Zeno effect and decoherence-free subspaces and subsystems are studied as particular examples.
A perturbative treatment of reduced density operators of quantum subsystems is implemented in the same spirit as Fermi Golden Rule for scattering. Analytic expressions for linear entropy (a measure of purity loss, and in some cases of coherence loss) and for subsystem's energy variations (dissipation) are given. Application to electromagnetic field superposition states in a dissipative cavity is performed. Evaluation of typical field and reservoir time scales ($τ_{F}$ and $τ_{R}$) show that even for small temperatures they are very different. We also indicate the condition on the cutoff temper…
The effect induced by an environment on a composite quantum system is studied. The model considers the composite system as comprised by a subsystem A coupled to a subsystem B which is also coupled to an external environment. We study all possible four combinations of subsystems A and B made up with a harmonic oscillator and an upside down oscillator. We analyzed the decoherence suffered by subsystem A due to an effective environment composed by subsystem B and the external reservoir. In all the cases we found that subsystem A decoheres even though it interacts with the environment only through…
A common misconception is that decoherence gives the eigenstates that we observe to be fairly definite about a subsystem (e.g., approximate eigenstates of position) as the elements of the Schmidt basis in which the density matrix of the subsystem is diagonal. Here I show that in simple examples of linear systems with gaussian states, the Schmidt basis states have as much mean uncertainty about position as the full density matrix with its combination of different possibilities.
This work addresses quantum adiabatic decoherence of many-body spin systems coupled with a boson field in the framework of open quantum systems theory. We generalize the traditional spin-boson model by considering a system-environment interaction Hamiltonian that represents a partition of non-interacting subsystems and highlights the collective correlation that appears exclusively due to the coupling with a common environment. Remarkably, this simple, exactly solvable model encompasses relevant aspects of a many-body open quantum system and features the subtle quantum effects that arise when t…
We show how to realize, by means of non-abelian quantum holonomies, a set of universal quantum gates acting on decoherence-free subspaces and subsystems. In this manner we bring together the quantum coherence stabilization virtues of decoherence-free subspaces and the fault-tolerance of all-geometric holonomic control. We discuss the implementation of this scheme in the context of quantum information processing using trapped ions and quantum dots.
We study decoherence in a circuit QED system consisting of a charge qubit and two superconducting transmission line resonators (TLRs). We show that in the dispersive regime of the circuit QED system one TLR can be used as an auxiliary subsystem to realize decoherence-free quantum dynamics of the bipartite target system consisting of the charge qubit and the other TLR conditioned on the auxiliary TLR initially being a proper number state. Our study gives new insight into control and manipulation of decoherence in quantum systems.
A state selected at random from the Hilbert space of a many-body system is overwhelmingly likely to exhibit highly non-classical correlations. For these typical states, half of the environment must be measured by an observer to determine the state of a given subsystem. The objectivity of classical reality-the fact that multiple observers can agree on the state of a subsystem after measuring just a small fraction of its environment-implies that the correlations found in nature between macroscopic systems and their environments are very exceptional. Building on previous studies of quantum Darwin…
We develop a structure theory for decoherence-free subspaces and noiseless subsystems that applies to arbitrary (not necessarily unital) quantum operations. The theory can be alternatively phrased in terms of the superoperator perspective, or the algebraic noise commutant formalism. As an application, we propose a method for finding all such subspaces and subsystems for arbitrary quantum operations. We suggest that this work brings the fundamental passive technique for error correction in quantum computing an important step closer to practical realization.
We discuss the quantum dynamics of an isolated composite system consisting of weakly interacting many-body subsystems. We focus on one of the subsystems, S, and study the dependence of its quantum correlations and decoherence rate on the state of the weakly-coupled complementary part E, which represents the environment. As a theoretical laboratory, we consider a composite system made of two stacked quantum Ising chains, locally and homogeneously weakly coupled. One of the chains is identified with the subsystem S under scrutiny, and the other one with the environment E. We investigate the beha…
Using the concept of von Neumann entropy, we quantify the information content of the various components of the quantum walk system, including the mutual information between its subsystems (coin and position) and use it to give a precise formulation of the measure of entanglement between subsystems.
We investigate the robustness of multiparty nonlocality under local decoherence, acting independently and equally on each subsystems. To be specific, we consider an N-qubit GHZ state under depolarization, dephasing, or dissipation channel, and tested the nonlocality by violation of Mermin-Klyshko inequality, which is one of Bell's inequalities for multi-qubit systems. The results show that the robustness of nonlocality increases with the number of qubits, and that the nonlocality of an N-qubit GHZ state with even N is extremely persistent against dephasing.
We have studied the emergence of classical states in the perturbative interaction model. The states which interact with many other degrees of freedom, such as the center of mass of a macro-object, play important role. Although the random phase mechanism is effective as same as Zurek's strong correlation model, there are enormous states, each of which independently developes due to the orthogonality of the environmental states. In these privileged states, the subsystem picture with the separability is stable.
For the standard Quantum Brownian Motion (QBM) model, we point out the occurrence of simultaneous (parallel), mutually irreducible and autonomous decoherence processes. Besides the standard, one Brownian particle, we show there is at least another one system undergoing the dynamics described by the QBM effect. We do this by selecting the two mutually irreducible, global structures (decompositions into subsystems) of the composite system of the QBM model. A generalization of this observation is a new, challenging task in the foundations of the decoherence theory. We do not place our findings in…
The interest in decoherence-free, or noiseless subsystems (DFS/NSs) of quantum systems is both of fundamental and practical interest. Understanding the invariance of a set of states under certain transformations is mutually associated with a better understanding of some fundamental aspects of quantum mechanics as well as the practical utility of invariant subsystems. For example, DFS/NSs are potentially useful for protecting quantum information in quantum cryptography and quantum computing as well as enabling universal computation. Here we discuss transformations which are compatible with a DF…
Dynamical irreversibility and local decoherence in quantum many-body chaos
Instituto de Física, Universidad Nacional Autónoma de México（墨西哥国立自治大学物理研究所）；Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México（墨西哥国立自治大学核科学研究所）；Center for Theoretical Physics of Complex Systems, Institute for Basic Science (IBS)（基础科学研究院复杂系统理论物理中心）
We consider the claim that decoherence explains the emergence of classicality in quantum systems, and conclude that it does not. We show that, given a randomly chosen universe composed of a variety of subsystems, some of which are macroscopic and subject to decoherence-inducing interactions, and some of which are microscopic, the macroscopic subsystems will not display any distinctively classical behavior. Therefore, a universe in which macroscopic and microscopic do display distinct behavior must be in a very special, highly nongeneric quantum state.
In this paper we investigate a open two-qubit model whose dynamics is not exactly solvable. When the initial state is the maximum entangled state, as the exactly solvable open two-qubit model [D. Tolkunov and V. Privman, Phys. Rev. A 71, 060308(R) (2005)], the decay of entanglement of formation of the model, expressed by concurrence is also governed by the product of suppression factors describing decoherence of the subsystems (qubits). However, if the initial state is not the maximum entangled state, its concurrence will decrease faster than the product of the suppression factors describing d…
In order to analyze the effect of chaos or order on the rate of decoherence in a subsystem we aim to distinguish effects of the two types of dynamics from those depending on the choice of the wave packet. To isolate the former we introduce a random matrix model that permits to vary the coupling strength between the subsystems. The case of strong coupling is analyzed in detail, and we find at intermediate times a weak effect of spectral correlations that is reminiscent of the correlation hole.
Universal quantum computation on decoherence-free subspaces and subsystems (DFSs) is examined with particular emphasis on using only physically relevant interactions. A necessary and sufficient condition for the existence of decoherence-free (noiseless) subsystems in the Markovian regime is derived here for the first time. A stabilizer formalism for DFSs is then developed which allows for the explicit understanding of these in their dual role as quantum error correcting codes. Conditions for the existence of Hamiltonians whose induced evolution always preserves a DFS are derived within this st…
Quantum non-Markovianity modifies the environmental decoherence of a system. This situation is enriched in complex systems owing to interactions among subsystems. We consider the problem of distinguishing the multiple sources of non-Markovianity using a simple power spectrum technique, applied to a qubit interacting with another qubit via a Jaynes-Cummings type Hamiltonian and simultaneously subjected to some well known noise channels, such as, the random telegraph noise and non-Markovian amplitude damping, which exhibit both Markovian as well as non-Markovian dynamics under different paramete…
The existence of a decoherence-free subspace/subsystem (DFS) requires that the noise possesses a symmetry. In this work we consider noise models in which perturbations break this symmetry, so that the DFS for the unperturbed model experiences noise. We ask whether in this case there exist subspaces/subsystems that have less noise than the original DFS. We develop a numerical method to search for such minimal noise subsystems and apply it to a number of examples. For the examples we examine, we find that if the perturbation is local noise then there is no better subspace/subsystem than the orig…
Quantum information processing requires a high degree of isolation from the detrimental effects of the environment as well as an extremely precise level of control on the way quantum dynamics unfolds in the information-processing system. In this paper, we show how these two goals can be ideally achieved by hybridizing the concepts of noiseless subsystems and of holonomic quantum computation. An all-geometric universal computation scheme based on non-adiabatic and non-Abelian quantum holonomies embedded in a four-qubit noiseless subsystem for general collective decoherence is proposed. The impl…
Consider a discrete-time quantum walk on the $N$-cycle governed by the following condition: at every time step of the walk, the option persists, with probability $p$, of exercising a projective measurement on the coin degree of freedom. For a bipartite quantum system of this kind, we prove that the von Neumann entropy of the total density operator converges to its maximum value. Thus, when influenced by decoherence, the mutual information between the two subsystems, corresponding respectively to the space of the coin and the space of the walker, eventually must diminish to zero. To put it plai…
