A decoherence-free subspace (DFS) is a subspace of a quantum system's Hilbert space that is invariant to non-unitary dynamics. Alternatively stated, they are a small section of the system Hilbert space where the system is decoupled from the environment and thus its evolution is completely unitary. DFSs can also be characterized as a special class of 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 (coherent) control of quantum systems is the desired goal. 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 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.
The study of DFSs began with a search for structured methods to avoid decoherence in the subject of quantum information processing (QIP). The methods involved attempts 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 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.[1] Originally the term "sub-decoherence" was used by Palma to describe this situation. Noteworthy is also independent work by 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.[2]
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 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 Hamiltonians. 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 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 perturbations 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 Kraus operator-sum representation (OSR). Later, A. Shabani and Lidar generalized the DFS framework relaxing the requirement that the initial state needs to be a DF-state and modified some known conditions for DFS.[3]
A subsequent development was made in generalizing the DFS picture when E. Knill, R. Laflamme, and L. Viola introduced the concept of a "noiseless subsystem".[1] Knill extended to higher-dimensional irreducible representations of the algebra generating the dynamical symmetry in the system-environment interaction. Earlier work on DFSs described DF states as 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.[1] The generalization from subspaces to subsystems formed a foundation for combining most known decoherence prevention and nulling strategies.
Consider an N-dimensional quantum system S coupled to a bath B and described by the combined system-bath Hamiltonian as follows:where the interaction Hamiltonian
\hat{H}I
\hat{S}i(\hat{B}i)
\hat{H}S(\hat{H}B)
\hat{I}S(\hat{I}B)
\tilde{l{H}}S\subsetl{H}S
l{H}S
\forall|\phi\rangle
In other words, if the system begins in
l{\tilde{H}}S
\hat{H}S
l{\tilde{H}}S
These states are degenerate eigenkets of
\hat{S}i\inl{O}SB(l{H}SB)
Let
l{\tilde{H}}S\subsetl{H}S
l{H}S
l{H}S
HC
\tilde{U
l{\tilde{H}}S\subsetl{H}S
\bar{A
l{\tilde{H}\bot
l{\tilde{H}}S
\bar{A
l{\tilde{H}\bot
l{\tilde{H}}S
l{\tilde{H}\bot
N | |
\left\{|j\rangle\right\} | |
j=1 |
l{\tilde{H}}S
\tilde{U
l
gl
N | |
\left\{|j\rangle\right\} | |
j=1 |
l{\tilde{H}}S
|\psi\rangle\inl{\tilde{H}}S
This state will be decoherence-free; this can be seen by considering the action of
\bar{A
|\psi\rangle
Therefore, in terms of the density operator representation of
|\psi\rangle
\rhoinitial=|\psi\rangle\langle\psi|
The above expression says that
\rhofinal
\tilde{U
l{\tilde{H}}S
l{\tilde{H}}S
This formulation makes use of the semigroup approach. The Lindblad decohering term determines when the dynamics of a quantum system will be unitary; in particular, when
LD[\rho]=0
\rho
N | |
\left\{|j\rangle\right\} | |
j=1 |
l{\tilde{H}}S\subsetl{H}S
l{H}S
a necessary and sufficient condition for
l{\tilde{H}}S
\forall{|j\rangle}
The above expression states that all basis states
|j\rangle
\left\{F\alpha
M=N x {N | |
\right\} | |
\alpha=1 |
l{\tilde{H}}S
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
\boldsymbol{\rho}
\operatorname{Tr}[\rho]=1
d x d
l{H}
l{B(l{H})}
S=\left\{\rhoi
n | |
\right\} | |
i=1 |
\inl{\tilde{H}}S
l{H}S
n<d
S
S
\boldsymbol{\zeta}
\boldsymbol{\rho}i,\boldsymbol{\rho}j\inS, (i\nej)
\boldsymbol{\zeta}
\boldsymbol{\rho}i,\boldsymbol{\rho}j
S
\boldsymbol{\zeta}
\boldsymbol{\rho}i,\boldsymbol{\rho}j\inS
\boldsymbol{\zeta}
S
\boldsymbol{\zeta}
\forall\boldsymbol\rho,\boldsymbol\rho'\inS
x\in\R+
This just says that
\boldsymbol{\zeta}
S
\boldsymbol{\zeta}
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.[1]
Consider a code
C=\operatorname{span}\left[\left\{|jk\rangle\right\}\right]
\left\{|jk\rangle\right\}
C
R
C
\left\{Rr\right\}.
Let
C
\left\{Al\right\}
\left\{Rr\right\}.
C
C
Rr\propto\tilde{U
\tilde{U
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.
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.
Consider a two-qubit Hilbert space, spanned by the basis qubits
\left\{|0\rangle1 ⊗ |0\rangle2,|0\rangle1 ⊗ |1\rangle2,|1\rangle1 ⊗ |0\rangle2,|1\rangle1 ⊗ |1\rangle2\right\}
\phi
Under this transformation the basis states
|0\rangle1 ⊗ |1\rangle2,|1\rangle1 ⊗ |0\rangle2
ei\phi
|\psi\rangle
Since these are basis qubits, then any state can be written as a linear combination of these states; therefore,
This state will evolve under the dephasing process as:
However, the overall phase for a quantum state is unobservable and, as such, is irrelevant in the description of the state. Therefore,
|\psiE\rangle
\{|0\rangle1 ⊗ |1\rangle2,|1\rangle1 ⊗ |0\rangle2\}
\{|0\rangle1 ⊗ |0\rangle2\},\{|1\rangle1 ⊗ |1\rangle2\}
Consider a quantum system with an N-dimensional system Hilbert space
l{H}C
l{H}ji
l{H}ji
\alpha|0\rangle+\beta|1\rangle
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:
This mapping is a one-to-many mapping from the one qubit encoding information to a two-qubit Hilbert space.[5] Instead, if the mapping is to
|\psi\rangle