Quantum Error Analysis

One of the most important consequences of the subsystems interpretation of encoding quantum information in a physical system is that the encoded quantum information can be error-free even though errors have severely changed the state of the physical system. Almost trivially, any error operator acting only on the syndrome subsystem has no effect on the quantum information. The goal of error correction is to actively intervene and maintain the syndrome subsystem in states where the dominant error operators continue to have little effect on the information of interest. An important issue in analyzing error correction methods is to estimate the residual error in the encoded information. A simple example of how that can be done was discussed for the quantum repetition code. The same ideas can be applied in general. Let $\mathsf {S}$ be the physical system in which the information is encoded and $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\psi}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {S}}}}$ an initial state containing such information with the syndrome subsystem appropriately prepared. Errors and error-correcting operations modify the state. The new state can be expressed using environment labeling as $\sum_e\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{e}\mbox{$\rangle\hsp... ...rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {S}}}}$. In view of the partitioning into information-carrying and syndrome subsystems, ``good'' states $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{e}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {E}}}}$ are those states for which ${A_e}^{({\mathsf {S}})}$ acts only on the syndrome subsystem given that the syndrome has been prepared. The remaining states $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{e}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {E}}}}$ form the set of ``bad'' states, ${\color{red}{\cal B}}$. The error probability $p_e$ can be bounded from above by

$$p_e \leq \left\vert\sum_{e\in{\color{red}{\cal B}}}\mbox{$\vert\hspace*{-3... ...*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {S}}}}\right\vert^2$$

$$\leq \left(\sum_{e\in{\color{red}{\cal B}}}\vert\mbox{$\vert\hspace*{-... ..._{{}_{\!\!{\mathsf {E}}}}\vert\; \vert{A_e}^{({\mathsf {S}})}\vert _1\right)^2,$$ (30)

where $\vert A\vert _1 = \max_{\phi}\mbox{$\langle\hspace*{-4.3pt}\langle\hspace*{-4.3... ...3pt}\vert$}{\phi}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$, the maximum being taken over normalized states. The second inequality usually leads to a gross overestimate but is independent of the encoded information and often suffices for obtaining good results. Because the environment-labeled sum is not unique, a goal of the representation of the errors acting on the system is to use ``good'' operators to the largest extent possible. The flexibility of these error-expansions makes them very useful for analyzing error models in conjunction with error-correction methods.

In principle, we can obtain better expressions for $p_e$ by calculating the density matrix $\rho$ of the state of the subsystem containing the desired quantum information. This calculation involves ``tracing over'' the syndrome subsystem. The matrix $\rho$ can then be compared to the intended state. If the intended state is pure, given by $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\phi}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$, the probability of error is given by $1-\mbox{$\langle\hspace*{-4.3pt}\langle\hspace*{-4.3pt}\langle$}{\phi}\mbox{$\v... ...3pt}\vert$}{\phi}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$, which is the probability that a measurement that distinguishes between $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\phi}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$ and its orthogonal complement fails to detect $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{\phi}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$. The quantity $\mbox{$\langle\hspace*{-4.3pt}\langle\hspace*{-4.3pt}\langle$}{\phi}\mbox{$\ver... ...3pt}\vert$}{\phi}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$ is called the ``fidelity'' of the state $\rho$.

For applications to communication, the goal is to be able to reliably transmit arbitrary states through a communication channel, which may be physical or realized via an encoding/decoding scheme. It is therefore important to characterize the reliability of the channel independent of the information transmitted. Eq. 30 can be used to obtain state-independent bounds on the error probability but does not readily provide a single measure of reliability. One way to quantify the reliability is to identify the error of the channel with the average error $\epsilon_a$ over all possible input states. The reliability is then given by the average fidelity $1-\epsilon_a$ Another elegant way appropriate for QIP is to use the ``entanglement fidelity'' [20]. Entanglement fidelity measures the error when the input is maximally entangled with an identical ``reference'' system. In this process, the reference system is imagined to be untouched, so that the state of the reference system together with the output state can be compared to the original entangled state. For a one-qubit channel labeled $\mathsf {S}$, the reference system is a qubit, which we label with $\mathsf {R}$. An initial, maximally entangled state is

$$\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{B}\mbox... ...ngle\hspace*{-4.3pt}\rangle$}_{{}_{\!\!{\mathsf {S}}}}\right).$$ (31)

The reference qubit is assumed to be perfectly isolated and not affected by any errors. The final state ${\rho}^{({\mathsf {R\,S}})}$ is compared to $\mbox{$\vert\hspace*{-3pt}\vert\hspace*{-3pt}\vert$}{B}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$, which gives the entanglement fidelity according to the formula $f_e=\mbox{$\langle\hspace*{-4.3pt}\langle\hspace*{-4.3pt}\langle$}{B}\mbox{$\ve... ...*{-3pt}\vert$}{B}\mbox{$\rangle\hspace*{-4.3pt}\rangle\hspace*{-4.3pt}\rangle$}$. The entanglement error is $\epsilon_e=1-f_e$. It turns out that this definition does not depend on the choice of maximally entangled state. Fortunately, the entanglement error and the average error $\epsilon_a$ are related by a linear expression:

$$\epsilon_a= {2\over 3}\epsilon_e.$$ (32)

For $k$-qubit channels, the constant ${2\over 3}$ is replaced by $2^k/(2^k+1)$. Experimental measurements of these fidelities do not require the reference system. There are simple averaging formulas to express them in terms of the fidelities for transmitting each of a sufficiently large set of pure states. An example of the experimental determination of the entanglement fidelity when the channel is realized by error-correction is provided in [21].