Three Quantum Spin-$1/2$ Particles

Quantum physics provides a rich source of systems with many opportunities for representing and protecting quantum information. Sometimes it is possible to encode information in such a way that it is protected from the errors indefinitely, without intervention. An example is the trivial two-qubit system discussed before. Whenever error protection without intervention is possible, there is an information-carrying subsystem such that errors act only on the associated syndrome subsystem regardless of the current state. An information-carrying subsystem with this property is called ``noiseless''. A physically motivated example of a one-qubit noiseless subsystem can be found in three spin-$1\over 2$ particles with errors due to random fluctuations in an external field.

A spin-$1\over 2$ particle's state space is spanned by two states $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{{\color{brown}\uparrow}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$ and $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{{\color{brown}\downarrow}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$. Intuitively, these states correspond to the spin pointing ``up'' ( $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{{\color{brown}\uparrow}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$) or ``down'' ( $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{{\color{brown}\downarrow}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$) in some chosen reference frame. The state space is therefore the same as that of a qubit and we can make the identifications $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{{\color{brown}\uparrow}}\m... ...t$}{\mathfrak{0}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$ and $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{{\color{brown}\downarrow}}... ...t$}{\mathfrak{1}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$. An external field causes the spin to ``rotate'' according to an evolution of the form

$$\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi_t}... ...mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}.$$ (16)

The vector $\vec u = (u_x,u_y,u_z)$ characterizes the direction of the field and the strength of the spin's interaction with the field. This situation arises, for example, in nuclear magnetic resonance with spin-$1\over 2$ nuclei, where the fields are magnetic fields (see [17]).

Now consider the physical system composed of three spin-$1\over 2$ particles with errors acting as identical rotations of the three particles. Such errors occur if they are due to a uniform external field that fluctuates randomly in direction and strength. The evolution caused by a uniform field is given by

$$\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi_t}\mbox... ...ngle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {123}}}} = e^{-i(u_x{\sigma_x}^{({\mathsf {1}})}+u_y{\sigma_y}^{({\mathsf {1... ...ngle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {123}}}}$$

$$= e^{-i(u_x({\sigma_x}^{({\mathsf {1}})}+{\sigma_x}^{({\mathsf {2}}... ...ngle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {123}}}}$$

$$= e^{-i(u_x J_x+u_y J_y + u_z J_z)t}\mbox{$\vert\hspace*{-3pt}\vert... ...gle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {123}}}},$$ (17)

with $J_u=\left({\sigma_u}^{({\mathsf {1}})}+{\sigma_u}^{({\mathsf {2}})}+{\sigma_u}^{({\mathsf {3}})}\right)/2$ for $u=x,y$ and $z$. We can exhibit the error operators arising from a uniform field in a compact form by defining $\vec J=(J_x,J_y,J_z)$ and $\vec v = (u_x,u_y,u_z)t$. Then the error operators are given by ${\color{red}E}(\vec v)= e^{-i\vec v\cdot\vec J}$, where the dot product in the exponent is calculated like the standard vector dot product.

For a one-qubit noiseless subsystem, the key property of the error model is that the errors are symmetric under any permutation of the three particles. A permutation of the particles acts on the particles' state space by permuting the labels in the logical states. For example, the permutation $\pi$ that swaps the first two particles acts on logical states as

$$\pi\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{a}\m... ...3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {3}}}}.$$ (18)

To say that the errors are symmetric under particle permutations means that each error ${\color{red}E}$ satisfies $\pi^{-1}{\color{red}E}\pi = {\color{red}E}$, or equivalently ${\color{red}E}\pi=\pi{\color{red}E}$ ( ${\color{red}E}$ ``commutes'' with $\pi$). To see that this condition is satisfied, write

$$\pi^{-1}{\color{red}E}(\vec v)\pi = \pi^{-1}e^{-i\vec v\cdot\vec J}\pi$$

$$= e^{-i\pi^{-1}(\vec v\cdot\vec J)\pi}$$

$$= e^{-i\vec v\cdot (\pi^{-1}\vec J\pi)}.$$ (19)

If $\pi$ permutes particle $\mathsf {a}$ to $\mathsf {b}$, then $\pi^{-1}{\sigma_u}^{({\mathsf {a}})}\pi = {\sigma_u}^{({\mathsf {b}})}$. It follows that $\pi^{-1}\vec J\pi=\vec J$. This expression shows that the errors commute with the particle permutations and therefore cannot distinguish between the particles. An error model satisfying this property is called a ``collective'' error model.

If a noiseless subsystem exists, then it suffices to learn the symmetries of the error model to construct the subsystem. This procedure is explained in Sect. 6.2. For the three spin-${1\over 2}$ system, the procedure results in a one-qubit noiseless subsystem protected from all collective errors. We first exhibit the subsystem identification and then discuss its properties to explain why it is noiseless. As in the case of the seven-state cyclic system, the identification involves a proper subspace of the physical system's state space. The subsystem identification involves a four-dimensional subspace and is defined by the following correspondence:

$$\begin{array}{rcrcrcl} {1\over \sqrt{3}}\Big(\mbox{$\vert\hs... ...gle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$} \end{array}$$ (20)

The state labels for the syndrome subsystem (before the dot in the expressions on the right side) identify it as a spin-${1\over 2}$ subsystem. In particular, it responds to the errors caused by uniform fields in the same way as the physical spin-${1\over 2}$ particles. This behavior is caused by $2J_u$ acting as the $u$-Pauli operator on the syndrome subsystem. To confirm this property, we apply $2J_u$ to the logical states of Eq. 20 for $u=z,x$. The property for $u=y$ then follows because $i\sigma_y=\sigma_z\sigma_x$. Consider $2J_z$. Each of the four states shown in Eq. 20 is an eigenstate of $2J_z$. For example, the physical state for $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{{\color{brown}\uparrow}}\m... ...t$}{\mathfrak{0}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$ is a superposition of states with two spins up ( ${\color{brown}\uparrow}$) and one spin down ( ${\color{brown}\downarrow}$). The eigenvalue of such a state with respect to $2J_z$ is the difference $\Delta$ between the number of spins that are up and down. Thus, $2J_z\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{{\color{brown}\uparrow... ...t$}{\mathfrak{0}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$. The difference is also $\Delta=1$ for $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{{\color{brown}\uparrow}}\m... ...t$}{\mathfrak{1}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$ and $\Delta=-1$ for $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{{\color{brown}\downarrow}}... ...t$}{\mathfrak{0}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$ and $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{{\color{brown}\downarrow}}... ...t$}{\mathfrak{1}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$. Therefore, $2J_z$ acts as the $z$-Pauli operator on the syndrome subsystem. To confirm this behavior for $2J_x$, we compute $2J_x\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{{\color{brown}\uparrow... ...t$}{\mathfrak{0}}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$.

$$\begin{array}[b]{rcl} 2J_x\mbox{$\vert\hspace*{-3pt}\vert\hs... ...le\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}. \end{array}$$ (21)

Similarly, one can check that, for the other logical states, the effect of $2J_x$ is to flip the orientation of the syndrome spin. The fact that the subsystem identified in Eq. 20 is noiseless now follows from the fact that the errors ${\color{red}E}(\vec v)$ are exponentials of sums of the syndrome spin operators $J_u$. The errors therefore act as the identity on the information-carrying subsystem.

The noiseless qubit supported by three spin-$1\over 2$ particles with collective errors is another example in which the subsystem identification does not involve the whole state space of the system. In this case, the errors of the error model cannot remove amplitude from the subspace. As a result, if we detect an error, that is, if we find that the system's state is in the orthogonal complement of the subspace of the subsystem identification, we can deduce that either the error model is inadequate, or we introduced errors in the manipulations required for transferring information to the noiseless qubit.

The noiseless subsystem of three spin-${1\over 2}$ particles can be physically motivated by an analysis of quantum spin numbers. The physical motivation is outlined in Fig. 8.

FIG. 8: One noiseless qubit from three spin-$1\over 2$ particles. The left side shows the three particles, with errors caused by fluctuations in a uniform magnetic field depicted by a noisy coil. The spin along direction $u$ ($u=x,y,z$) can be measured and its expectation is given by $\mbox{$\langle\hspace*{-4.3pt}\langle\hspace*{-4.3pt}\langle$}{\psi}\mbox{$\ver... ...3pt}\vert$}{\psi}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$, where $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$ is the quantum state of the particles and $J_u$ is the total spin observable along the $u$-axis given by the half-sum of the $u$-Pauli matrices of the particles as defined in the text. The squared magnitude of the total spin is given by the expectation of the observable $J^2=\vec J \cdot \vec J = J_x^2+ J_y^2+J_z^2$. The observable $J^2$ commutes with the $J_u$ and therefore also with the errors ${\color{red}E}(\vec v)=e^{-i\vec v \cdot \vec J}$ caused by uniform field fluctuations. This can be verified directly, or one can note that ${\color{red}E}(\vec v)$ acts on $\vec J$ as a rotation in three dimensions, and as one would expect, such rotations preserve the ``squared length'' $J^2$ of $\vec J$. It now follows that the eigenspaces of $J^2$ are invariant under the errors, and therefore that the eigenspaces are good places to look for noiseless subsystems. The eigenvalues of $J^2$ are of the form $j(j+1)$, where $j$ is the spin quantum number of the corresponding eigenspace. There are two eigenspaces, one with spin $j={1\over 2}$ and the other with spin $j={3\over 2}$. The figure shows a thought experiment that involves passing the three-particle system through a type of beam splitter (BS) or Stern-Gerlach apparatus sensitive to $J^2$. Using such a beam splitter, the system of particles can be made to go in one of two directions depending on $j$. In the figure, if the system's state is in the the spin-$3\over 2$ subspace, it passes through the beam splitter; if it is in the spin-$1\over 2$ subspace, the system is reflected up. It can be shown that the subspace with $j={3\over 2}$ is four-dimensional and spanned by the states that are symmetric under particle permutations. Unfortunately, there is no noiseless subsystem in this subspace (see Sect. 6.2). The spin-$1\over 2$ subspace is also four dimensional and spanned by the states in Eq. 20. The spin-$1\over 2$ property of the subspace implies that the spin operators $J_u$ act in a way that is algebraically identical to the way $\sigma_u/2$ acts on a single spin-$1\over 2$ particle. This property implies the existence of the syndrome subsystem introduced in the text. Conventionally, the spin-$1\over 2$ subspace is thought of as consisting of two orthogonal two-dimensional subspaces each behaving like a spin-$1\over 2$ with respect to the $J_u$. This choice of subspaces is not unique, but by associating them with two logical states of a noiseless qubit, one can obtain the subsystem identification of Eq.20. Some care needs to be taken to ensure that the noiseless qubit operators commute with the $J_u$, as they should (see Sect. 6.2). In the thought experiment, one can imagine unitarily rotating the system emerging in the upper path to make explicit the syndrome spin-$1\over 2$ subsystem and the noiseless qubit with which it must be paired. The result of this rotation is shown.