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| A '''decoherence-free subspace''' (DFS) is a [[Linear subspace|subspace]] of a system's [[Hilbert space]] that is [[Invariant (mathematics)|invariant]] to non-[[Unitarity (physics)|unitary]] dynamics. Alternatively stated, they are a small section of the system Hilbert space where the system is [[Coupling (physics)|decoupled]] from the environment and thus its evolution is completely unitary. DFSs can also be characterized as a special class of [[Decoherence-free subspaces#Quantum error-correcting codes(QECCs)|quantum error correcting codes]]. In this representation they are ''passive'' error-preventing codes since these subspaces are encoded with information that (possibly) won't require any ''active'' stabilization methods. These subspaces prevent destructive environmental interactions by isolating [[quantum information]]. As such, they are an important subject in [[quantum computing]], where ([[Coherence (physics)|coherent]]) control of quantum systems is the desired goal. [[Quantum decoherence|Decoherence]] creates problems in this regard by causing loss of coherence between the [[quantum states]] of a system and therefore the decay of their [[interference]]{{Disambiguation needed|date=June 2011}} terms, thus leading to loss of information from the (open) quantum system to the surrounding environment. Since quantum computers cannot be isolated from their environment (i.e. we cannot have a truly isolated quantum system in the real world) and information can be lost, the study of DFSs is important for the implementation of quantum computers into the real world.
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| == Background ==
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| === Origins ===
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| The study of DFSs began with a search for structured methods to avoid decoherence in the subject of [[Quantum computing|quantum information processing]] (QIP). The methods involved attempted to identify particular states which have the potential of being unchanged by certain decohering processes (i.e. certain interactions with the environment). These studies started with observations made by G.M. Palma, K-A Suominen, and [[Artur Ekert|A.K. Ekert]], who studied the consequences of pure dephasing on two [[qubits]] that have the same interaction with the environment. They found that two such qubits do not decohere.<ref name="Lidar and Whaley">[http://arxiv.org/abs/quant-ph/0301032 Decoherence-free subspaces and subsystems] from [[arXiv]]</ref> Originally the term "sub-decoherence" was used by Palma to describe this situation. Noteworthy is also independent work by [[Martin Bodo Plenio|Martin Plenio]], [[Vlatko Vedral]] and [[Peter Knight]] who constructed an error correcting code with codewords that are invariant under a particular unitary time evolution in spontaneous emission.<ref name="Plenio, Vedral and Knight">[http://arxiv.org/abs/quant-ph/9603022 Quantum error correction in the presence of spontaneous emission] from [[arXiv]]</ref>
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| === Further development === | |
| Shortly afterwards, L-M Duan and G-C Guo also studied this phenomenon and reached the same conclusions as Palma, Suominen, and Ekert. However, Duan and Guo applied their own terminology, using "coherence preserving states" to describe states that do not decohere with dephasing. Duan and Guo furthered this idea of combining two qubits to preserve coherence against dephasing, to both collective dephasing and dissipation showing that decoherence is prevented in such a situation. This was shown by assuming knowledge of the system-environment [[coupling constant|coupling strength]]. However, such models were limited since they dealt with the decoherence processes of dephasing and dissipation solely. To deal with other types of decoherences, the previous models presented by Palma, Suominen, and Ekert, and Duan and Guo were cast into a more general setting by P. Zanardi and M. Rasetti. They expanded the existing mathematical framework to include more general system-environment interactions, such as collective decoherence-the same decoherence process acting on all the states of a quantum system and general [[Hamiltonian (quantum mechanics)|Hamiltonian]]s. Their analysis gave the first formal and general circumstances for the existence of decoherence-free (DF) states, which did not rely upon knowing the system-environment coupling strength. Zanardi and Rasetti called these DF states "error avoiding codes". Subsequently, [[Daniel Lidar|Daniel A. Lidar]] proposed the title "decoherence-free subspace" for the space in which these DF states exist. Lidar studied the strength of DF states against [[Perturbation theory (quantum mechanics)|perturbation]]s and discovered that the coherence prevalent in DF states can be upset by evolution of the system Hamiltonian. This observation discerned another prerequisite for the possible use of DF states for quantum computation. A thoroughly general requirement for the existence of DF states was obtained by Lidar, D. Bacon, and K.B. Whaley expressed in terms of the [[Quantum decoherence#Operator-sum representation|Kraus operator-sum representation]] (OSR).
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| === Recent research ===
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| A subsequent development was made in generalizing the DFS picture when E. Knill, [[Raymond Laflamme|R. Laflamme]], and L. Viola introduced the concept of a "noiseless subsystem".<ref name="Lidar and Whaley"/> Knill extended to higher-dimensional [[irreducible representation]]s of the [[algebra]] generating the dynamical symmetry in the system-environment interaction. Earlier work on DFSs described DF states as [[Singlet state|singlets]], which are one-dimensional irreducible representations. This work proved to be successful, as a result of this analysis was the lowering of the number of qubits required to build a DFS under collective decoherence from four to three.<ref name="Lidar and Whaley"/> The generalization from subspaces to subsystems formed a foundation for combining most known decoherence prevention and nulling strategies.
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| ==Conditions for the existence of decoherence-free subspaces==
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| ===Hamiltonian formulation===
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| Consider an N-dimensional quantum system S coupled to a bath B and described by the combined system-bath Hamiltonian as follows:
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| :<math>\hat{H} = \hat{H}_{S}\otimes\hat{I}_{B} + \hat{I}_{S}\otimes\hat{H}_{B} + \hat{H}_{I}</math> ,
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| where the interaction Hamiltonian <span style="vertical-align:25%;"><math>\hat{H}_{I}</math></span> is given in the usual way as
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| :<math>\hat{H}_{I} = \sum_{i}\hat{S}_{i}\otimes\hat{B}_{i},</math>
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| and where <math>\hat{S}_{i}\big(\hat{B}_{i}\big)</math> act upon the system(bath) only, and <math>\hat{H}_{S} \big(\hat{H}_{B}\big)</math> is the system(bath) Hamiltonian, and <math>\hat{I}_{S}\big(\hat{I}_{B}\big)</math> is the identity operator acting on the system (bath). | |
| Under these conditions, the dynamical evolution within <span style="vertical-align:25%;"><math>\tilde{\mathcal{H}}_{S}\subset\mathcal{H}_{S}</math></span>, where <math>\mathcal{H}_{S}</math> is the system Hilbert space, is completely unitary <math>\forall|\phi\rangle</math> (all possible bath states) if and only if:
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| (i) <math>\hat{S}_{i}|\phi\rangle = s_{i}|\phi\rangle, s_{i}\in\mathbb{C}</math>
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| <math>\forall|\phi\rangle</math> that [[Linear span|span]] <math>\mathcal{\tilde{H}}_{S}</math> and <span style="vertical-align:10%;"><math>\forall\hat{S}_{i}\in\mathcal{O}_{SB}(\mathcal{H}_{SB})</math></span>, the space of bounded system-bath operators on <span style="vertical-align:10%;"><math>\mathcal{H}_{SB}</math></span>, | |
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| (ii) the system and bath are not coupled at first (i.e. they can be represented as a product state),
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| (iii) there is no "leakage" of states out of <span style="vertical-align:10%;"><math>\mathcal{\tilde{H}}_{S}</math></span>; that is, the system Hamiltonian <span style="vertical-align:10%;"><math>\hat{H}_{S}</math></span> does not map the states <math>|\phi\rangle</math> out of <span style="vertical-align:10%;"><math>\mathcal{\tilde{H}}_{S}</math></span>.
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| In other words, if the system begins in <span style="vertical-align:10%;"><math>\mathcal{\tilde{H}}_{S}</math></span>(i.e. the system and bath are initially decoupled) and the system Hamiltonian <span style="vertical-align:10%;"><math>\hat{H}_{S}</math></span> leaves <span style="vertical-align:10%;"><math>\mathcal{\tilde{H}}_{S} = span\big[\big\{|\phi_{k}\rangle\big\}_{k=1}^{N}\big]</math></span> invariant, then <span style="vertical-align:10%;"><math>\mathcal{\tilde{H}}_{S}</math></span> is a DFS if and only if it satisfies (i).
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| These states are [[Degenerate energy levels|degenerate]] [[Eigenvector|eigenket]]s of <span style="vertical-align:10%;"><math>\hat{S}_{i}\in\mathcal{O}_{SB}(\mathcal{H}_{SB})</math></span> and thus are distinguishable, hence preserving information in certain decohering processes. Any subspace of the system Hilbert space that satisfies the above conditions is a decoherence-free subspace. However, information can still "leak" out of this subspace if condition (iii) is not satisfied. Therefore, even if a DFS exists under the Hamiltonian conditions, there are still non-unitary actions that can act upon these subspaces and take states out of them into another subspace, which may or may not be a DFS, of the system Hilbert space.
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| ====Operator-sum representation formulation====
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| Let <span style="vertical-align:20%;"><math>\mathcal{\tilde{H}}_{S}\subset\mathcal{H}_{S}</math></span> be an N-dimensional DFS, where <span style="vertical-align:10%;"><math>\mathcal{H}_{S}</math></span> is the system's (the quantum system alone) Hilbert space. The [[Quantum decoherence#Operator-sum representation|Kraus operators]] when written in terms of the N basis states that [[Linear algebra|span]] <span style="vertical-align:5%;"><math>\mathcal{H}_{S}</math></span> are given as:
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| :<math>\mathbf{A}_{l} =
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| \begin{pmatrix}
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| g_{l}\mathbf{\tilde{U}} & \mathbf{0} \\
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| \mathbf{0} & \mathbf{\bar{A}}_{l}
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| \end{pmatrix},\quad g_{l} = \sqrt{a_{j}}\langle k|\mathbf{U}_{C}|j\rangle</math>
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| where <span style="vertical-align:20%;"><math>\mathbf{U}_{C} = \mathit{exp}\big(\frac{-i\mathbf{H}_{C}t}{\hbar}\big)</math></span> (<span style="vertical-align:5%;"><math>\mathbf{H}_{C}</math></span> is the combined system-bath Hamiltonian), <span style="vertical-align:10%;"><math>\mathbf{\tilde{U}}</math></span> acts on <span style="vertical-align:10%;"><math>\mathcal{\tilde{H}}_{S}\subset\mathcal{H}_{S}</math></span>, and <span style="vertical-align:10%;"><math>\mathbf{\bar{A}}_{l}</math></span> is an arbitrary matrix that acts on <span style="vertical-align:10%;"><math>\mathcal{\tilde{H}^{\bot}}_{S}</math></span> (the [[orthogonal complement]] to <span style="vertical-align:10%;"><math>\mathcal{\tilde{H}}_{S}</math></span>). Since <span style="vertical-align:-5%;"><math>\mathbf{\bar{A}}_{l}</math></span> operates on <span style="vertical-align:10%;"><math>\mathcal{\tilde{H}^{\bot}}_{S}</math></span>, then it will not create decoherence in <span style="vertical-align:10%;"><math>\mathcal{\tilde{H}}_{S}</math></span>; however, it can (possibly) create decohering effects in <span style="vertical-align:10%;"><math>\mathcal{\tilde{H}^{\bot}}_{S}</math></span>. Consider the basis kets <span style="vertical-align:10%;"><math>\big\{|j\rangle\big\}_{j=1}^{N}</math></span> which span <span style="vertical-align:10%;"><math>\mathcal{\tilde{H}}_{S}</math> and, furthermore, they fulfill:
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| :<math>\mathbf{\bar{A}}_{l}|j\rangle = g_{l}\mathbf{\tilde{U}}|j\rangle,\quad \forall{l}.</math>
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| <span style="vertical-align:10%;"><math>\mathbf{\tilde{U}}</math></span> is an arbitrary [[unitary operator]] and may or may not be time-dependent, but it is independent of the indexing variable <math>\mathbf{\mathit{l}}</math>. The <math>\mathbf{\mathit{g}}_{l}</math>'s are [[Complex number|complex]] constants. Since <span style="vertical-align:10%;"><math>\big\{|j\rangle\big\}_{j=1}^{N}</math></span> spans <span style="vertical-align:10%;"><math>\mathcal{\tilde{H}}_{S}</math>, then any [[Quantum states#Pure states as rays in a Hilbert space|pure state]] <span style="vertical-align:10%;"><math>|\psi\rangle\in\mathcal{\tilde{H}}_{S}</math> can be written as a [[linear combination]] of these basis kets:
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| :<math>|\psi\rangle = \sum_{j=1}^{N}b_{j}|j\rangle,\quad b_{j}\in\mathbb{C}.</math>
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| This state will be decoherence-free; this can be seen by considering the action of <span style="vertical-align:10%;"><math>\mathbf{\bar{A}}_{l}</math></span> on <span style="vertical-align:10%;"><math>|\psi\rangle</math></span>:
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| :<math>
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| \begin{align}
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| \mathbf{\bar{A}}_{l}|\psi\rangle &= \sum_{j=1}^{N}b_{j}(\mathbf{\bar{A}}_{l}|j\rangle)\\
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| &= \sum_{j=1}^{N}b_{j}(g_{l}\mathbf{\tilde{U}}|j\rangle)\\
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| \mathbf{\bar{A}}_{l}|\psi\rangle &= g_{l}\mathbf{\tilde{U}}|\psi\rangle.
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| \end{align}</math>
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| Therefore, in terms of the [[density operator]] representation of <math>|\psi\rangle</math>, <math>\rho_{initial} = |\psi\rangle\langle\psi|</math>, the evolution of this state is:
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| :<math>
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| \begin{align}\rho_{final} &= \sum_{l}\mathbf{A}_{l}\rho_{initial}\mathbf{A}^{\dagger}_{l}\\
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| &= \sum_{l}g_{l}\mathbf{\tilde{U}}|\psi\rangle\langle\psi|h_{l}\mathbf{\tilde{U}}^{\dagger} \\
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| &= \mathbf{\tilde{U}}|\psi\rangle\langle\psi|\mathbf{\tilde{U}}^{\dagger}.
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| \end{align}</math>
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| The above expression says that <math>\mathbf{\mathit{\rho}}_{final}</math> is a pure state and that its evolution is unitary, since <span style="vertical-align:10%;"><math>\mathbf{\tilde{U}}</math></span> is unitary. Therefore, ''any'' state in <span style="vertical-align:10%;"><math>\mathcal{\tilde{H}}_{S}</math></span> will not decohere since its evolution is governed by a unitary operator and so its dynamical evolution will be completely unitary. Thus <span style="vertical-align:10%;"><math>\mathcal{\tilde{H}}_{S}</math></span> is a decoherence-free subspace.
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| The above argument can be generalized to an initial arbitrary [[Quantum states#Mixed states|mixed state]] as well.<ref name="Lidar and Whaley"/>
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| ====Semigroup formulation====
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| This formulation makes use of the [[Quantum decoherence#Semigroup approach|semigroup approach]]. The [[Quantum decoherence#Semigroup approach|Lindblad decohering term]] determines when the dynamics of a quantum system will be unitary; in particular, when <span style="vertical-align:10%;"><math>\mathbf{\mathit{L}}_{D}[\rho] = 0</math></span>, where <math>\mathbf{\mathit{\rho}}</math> is the density operator representation of the state of the system, the dynamics will be decoherence-free.
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| Let <span style="vertical-align:10%;"><math>\big\{|j\rangle\big\}_{j=1}^{N}</math></span> span <span style="vertical-align:10%;"><math>\mathcal{\tilde{H}}_{S}\subset\mathcal{H}_{S}</math></span>, where <span style="vertical-align:10%;"><math>\mathcal{H}_{S}</math></span> is the system's Hilbert space. Under the assumptions that:
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| *(i) the [[Quantum decoherence#Semigroup approach|noise parameters]] of the coefficient matrix of the Lindblad decohering term are not fine-tuned (i.e. no special assumptions are made about them)
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| *(ii) there is no dependence on the initial conditions of the initial state of the system
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| a necessary and sufficient condition of for <span style="vertical-align:10%;"><math>\mathcal{\tilde{H}}_{S}</math></span> to be a DFS is <math>\forall{|j\rangle}</math>:
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| :<math>\mathbf{F}_{\alpha}|j\rangle = \lambda_{\alpha}|j\rangle,\quad\forall\alpha.</math>
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| The above expression states that ''all'' basis states <span style="vertical-align:10%;"><math>|j\rangle</math></span> are degenerate eigenstates of the [[Quantum decoherence#Semigroup approach|error generators]] <span style="vertical-align:10;"><math>\big\{\mathbf{F}_{\alpha}\big\}_{\alpha=1}^{M=N\times{N}}.</math></span> As such, their respective [[Quantum decoherence#Collective dephasing|coherence terms]] do not decohere. Thus states within <span style="vertical-align:10%;"><math>\mathcal{\tilde{H}}_{S}</math></span> will remain mutually distinguishable after a decohering process since their respective [[eigenvalues]] are degenerate and hence identifiable after action under the error generators.
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| ==DFSs as a special class of information-preserving structures (IPS) and quantum error-correcting codes (QECCs)==
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| ===Information-preserving structures (IPS)===
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| DFSs can be thought of as "encoding" information through its set of states. To see this, consider a d-dimensional open quantum system that is prepared in the state <math>\mathbf{\rho}</math>-a non-negative (i.e. its eigenvalues are positive), trace-preserving <math>\big(\mathbf{\mathit{Tr}}[\rho]=1\big)</math>, <math>d\times d</math> density operator that belongs to the system's [[Hilbert-Schmidt operator|Hilbert-Schmidt]] space, the space of [[bounded operator]]s on <math>\mathcal{H}</math> <math>\big(\mathcal{B(\mathcal{H})}\big)</math>. Suppose that this density operator(state) is selected from a set of states <span style="vertical-align:10%;"><math>S = \big\{\rho_{i}\big\}_{i=1}^{n}\in\mathcal{\tilde{H}}_{S}</math></span>, a DFS of <math>\mathcal{H}_{S}</math> (the system's Hilbert space) and where <span style="vertical-align:10%;"><math>\mathbf{\mathit{n}}<\mathbf{\mathit{d}}</math></span>.
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| This set of states is called a ''code'', because the states within this set ''encode'' particular kind of information;<ref name="Blume-Kohout, Khoon Ng, Poulin, and Viola">[http://arxiv.org/abs/0705.4282 The structure of preserved information in quantum processes] from [[arXiv]]</ref> that is, the set ''S'' encodes information through its states. This information that is contained within <span style="vertical-align:10%;"><math>\mathbf{\mathit{S}}</math></span> must be able to be accessed; since the information is encoded in the states in <span style="vertical-align:10%;"><math>\mathbf{\mathit{S}}</math></span>, these states must be distinguishable to some process, <math>\mathbf{\zeta}</math> say, that attempts to acquire the information. Therefore, for two states <math>\mathbf{\rho}_{i},\mathbf{\rho}_{j}\in\mathit{S} \big(i\ne j\big)</math>, the process <math>\mathbf{\zeta}</math> is ''information preserving'' for these states if the states <math>\mathbf{\rho}_{i},\mathbf{\rho}_{j}</math> remain ''as'' distinguishable after the process as they were before it. Stated in a more general manner, a code <span style="vertical-align:10%;"><math>\mathbf{\mathit{S}}</math></span> (or DFS) is preserved by a process <math>\mathbf{\zeta}</math> iff each pair of states <math>\mathbf{\rho}_{i},\mathbf{\rho}_{j}\in\mathit{S}</math> is as distinguishable after <math>\mathbf{\zeta}</math> is applied as they were before it was applied.<ref name="Blume-Kohout, Khoon Ng, Poulin, and Viola"/> A more practical description would be: <span style="vertical-align:10%;"><math>\mathbf{\mathit{S}}</math></span> is preserved by a process <math>\mathbf{\zeta}</math> if and only if <span style="vertical-align:10%;"><math>\forall\mathbf{\rho,\rho'}\in\mathit{S}</math></span> and <span style="vertical-align:25%;"><math>\mathit{x}\in\mathbb{R}^{+}</math></span>
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| :<math>\big\|\mathbf{\zeta}\big(\mathbf{\rho}-\mathit{x}\mathbf{\rho'}\big)\big\|_{1} = \big\|\mathbf{\rho}-\mathit{x}\mathbf{\rho'}\big\|_{1}.</math>
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| This just says that <math>\mathbf{\zeta}</math> is a 1:1 trace-distance-preserving map on <span style="vertical-align:10%;"><math>\mathbf{\mathit{S}}</math></span>.<ref name="Blume-Kohout, Khoon Ng, Poulin, and Viola"/> In this picture DFSs are sets of states (codes rather) whose ''mutual distinguishability'' is unaffected by a process <math>\mathbf{\zeta}</math>.
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| ===Quantum error-correcting codes(QECCs)===
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| Since DFSs can encode information through their sets of states, then they are secure against errors (decohering processes). In this way DFSs can be looked at as a special class of QECCs, where information is encoded into states which can be disturbed by an interaction with the environment but retrieved by some reversal process.<ref name="Lidar and Whaley"/>
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| Consider a code <span style="vertical-align:10%;"><math>C = span\big[\big\{|j_{k}\rangle\big\}\big]</math></span>, which is a subspace of the system Hilbert space, with encoded information given by <span style="vertical-align:10%;"><math>\big\{|j_{k}\rangle\big\}</math></span> (i.e. the "codewords"). This code can be implemented to protect against decoherence and thus prevent loss of information in a small section of the system's Hilbert space. The errors are caused by interaction of the system with the environment (bath) and are represented by the Kraus operators.<ref name="Lidar and Whaley"/> After the system has interacted with the bath, the information contained within <math>\mathbf{\mathit{C}}</math> must be able to be "decoded"; therefore, to retrieve this information a '''recovery operator''' <span style="vertical-align:10%;"><math>\mathbf{R}</math></span> is introduced. So a QECC is a subspace <math>\mathbf{\mathit{C}}</math> along with a set of recovery operators <span style="vertical-align:10%;"><math>\big\{\mathbf{R}_{r}\big\}.</math></span>
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| Let <span style="vertical-align:10%;"><math>\mathbf{\mathit{C}}</math></span> be a QECC for the error operators represented by the Kraus operators <span style="vertical-align:10%;"><math>\big\{\mathbf{A}_{l}\big\}</math></span>, with recovery operators <span style="vertical-align:10%;"><math>\big\{\mathbf{R}_{r}\big\}.</math></span> Then <span style="vertical-align:10%;"><math>\mathbf{\mathit{C}}</math></span> is a DFS if and only if upon restriction to <span style="vertical-align:10%;"><math>\mathbf{\mathit{C}}</math></span>, then <span style="vertical-align:10%;"><math>\mathbf{R}_{r}\propto\mathbf{\tilde{U}}_{S}^{\dagger}, \forall{r}</math></span>,<ref name="Lidar and Whaley"/> where <span style="vertical-align:10%;"><math>\mathbf{\tilde{U}}_{S}^{\dagger}</math></span> is the inverse of the system evolution operator.
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| In this picture of reversal of quantum operations, DFSs are a special instance of the more general QECCs whereupon restriction to a given a code, the recovery operators become proportional to the inverse of the system evolution operator, hence allowing for unitary evolution of the system.
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| Notice that the subtle difference between these two formulations exists in the two words ''preserving'' and ''correcting''; in the former case, error-''prevention'' is the method used whereas in the latter case it is error-''correction''. Thus the two formulations differ in that one is a ''passive'' method and the other is an ''active'' method.
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| ==Example of a decoherence-free subspace==
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| ===Collective dephasing===
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| Consider a two-qubit Hilbert space, spanned by the basis qubits <span style="vertical-align:10%;"><math>\big\{|0\rangle_{1}\otimes|0\rangle_{2}, |0\rangle_{1}\otimes|1\rangle_{2}, |1\rangle_{1}\otimes|0\rangle_{2}, |1\rangle_{1}\otimes|1\rangle_{2}\big\}</math></span> which undergo [[Quantum decoherence#Collective dephasing|collective dephasing]]. A random phase <math>\mathbf{\mathit{\phi}}</math> will be created between these basis qubits; therefore, the qubits will transform in the following way:
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| :<math>
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| \begin{align}|0\rangle_{1}\otimes|0\rangle_{2} & \longrightarrow |0\rangle_{1}\otimes|0\rangle_{2} \\
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| |0\rangle_{1}\otimes|1\rangle_{2} & \longrightarrow e^{i\phi}|0\rangle_{1}\otimes|1\rangle_{2} \\
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| |1\rangle_{1}\otimes|0\rangle_{2} & \longrightarrow e^{i\phi}|1\rangle_{1}\otimes|0\rangle_{2} \\
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| |1\rangle_{1}\otimes|1\rangle_{2} & \longrightarrow e^{2i\phi}|1\rangle_{1}\otimes|1\rangle_{2}
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| \end{align}</math>.
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| Under this transformation the basis states <span style="vertical-align:10%;"><math>|0\rangle_{1}\otimes|1\rangle_{2}, |1\rangle_{1}\otimes|0\rangle_{2}</math></span> obtain the same phase factor <span style="vertical-align:20%;"><math>\mathbf{\mathit{e}}^{i\phi}</math></span>. Thus in consideration of this, a state <math>|\psi\rangle</math> can be encoded with this information (i.e. the phase factor) and thus evolve unitarily under this dephasing process, by defining the following encoded qubits:
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| :<math>
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| \begin{align} |0_{E}\rangle &= |0\rangle_{1}\otimes|1\rangle_{2} \\
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| |1_{E}\rangle &= |1\rangle_{1}\otimes|0\rangle_{2}
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| \end{align}</math>.
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| Since these are basis qubits, then any state can be written as a linear combination of these states; therefore,
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| :<math>|\psi_{E}\rangle = l|0_{E}\rangle + m|1_{E}\rangle,\quad l,m\in\mathbb{C}.</math>
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| This state will evolve under the dephasing process as:
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| :<math>|\psi_{E}\rangle\longrightarrow l|0\rangle_{1}\otimes e^{i\phi}|1\rangle_{2} + e^{i\phi}m|1\rangle_{1}\otimes|0\rangle_{2} = e^{i\phi}|\psi_{E}\rangle.</math>
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| However, the ''overall'' phase for a quantum state is unobservable and, as such, is irrelevant in the description of the state. Therefore, <math>|\psi_{E}\rangle</math> remains invariant under this dephasing process and hence the basis set <span style="vertical-align:10%;"><math>\big\{|0\rangle_{1}\otimes|1\rangle_{2}, |1\rangle_{1}\otimes|0\rangle_{2}\big\}</math></span> is a ''decoherence-free subspace'' of the 4-dimensional Hilbert space. Similarly, the subspaces <span style="vertical-align:10%;"><math>\big\{|0\rangle_{1}\otimes|0\rangle_{2}\big\}, \big\{|1\rangle_{1}\otimes|1\rangle_{2}\big\}</math></span> are also DFSs.
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| ==Alternative: decoherence-free subsystems==
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| Consider a quantum system with an N-dimensional system Hilbert space <span style="vertical-align:10%;"><math>\mathcal{H}_{C}</math></span> that has a general subsystem decomposition <span style="vertical-align:10%:"><math>\mathcal{H}_{C} = \oplus_{j=1}^{N}(\otimes_{i=1}^{l_{N}}\mathcal{H}_{ji}).</math></span> The subsystem <span style="vertical-align:10%;"><math>\mathcal{H}_{ji}</math></span> is a '''decoherence-free subsystem''' with respect to a system-environment coupling if every pure state in <span style="vertical-align:10%;"><math>\mathcal{H}_{ji}</math></span> remains unchanged with respect to this subsystem under the OSR evolution. This is true for any possible initial condition of the environment.<ref name="Bacon">[http://arxiv.org/abs/quant-ph/0305025 Decoherence, Control, and Symmetry in Quantum Computers] from [[arXiv]]</ref> To understand the difference between a decoherence-free ''subspace'' and a decoherence-free ''subsystem'', consider encoding a single qubit of information into a two-qubit system. This two-qubit system has a 4-dimensional Hilbert space; one method of encoding a single qubit into this space is by encoding information into a subspace that is spanned by two [[orthogonal]] qubits of the 4-dimensional Hilbert space. Suppose information is encoded in the orthogonal state <span style="vertical-align:10%;"><math>\alpha|0\rangle + \beta|1\rangle</math></span> in the following way:
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| :<math>\alpha|0\rangle_{1} + \beta|1\rangle_{2}\longrightarrow \alpha|0\rangle_{1}\otimes|1\rangle_{2} + \beta|1\rangle_{1}\otimes|0\rangle_{2}.</math>
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| This shows that information has been encoded into a ''subspace'' of the two-qubit Hilbert space. Another way of encoding the same information is to encode ''only'' one of the qubits of the two qubits. Suppose the first qubit is encoded, then the state of the second qubit is completely arbitrary since:
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| :<math>\alpha|0\rangle_{1} + \beta|1\rangle_{2}\longrightarrow \big(\alpha|0\rangle_{1} + \beta|1\rangle_{2}\big)\otimes|\psi\rangle.</math>
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| This mapping is a ''one-to-many'' mapping from the one qubit encoding information to a two-qubit Hilbert space.<ref name="Bacon"/> Instead, if the mapping is to <math>|\psi\rangle</math>, then it is identical to a mapping from a qubit to a subspace of the two-qubit Hilbert space.
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| ==See also==
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| [[Quantum decoherence]]
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| [[Quantum measurement]]
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| ==References==
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| {{reflist}}
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| {{DEFAULTSORT:Decoherence-Free Subspaces}}
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| [[Category:Quantum measurement]]
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| [[Category:Quantum information science]]
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