View source for The operator sum representation of open quantum system dynamics

Modern atomic physics rests on engineering the physics of quantum
systems, and one of the greatest parts of this challenge is control of
decoherence.  Beginning with the ideas born from manipulating
spontaneous emission rates with optical cavities, the study of open
quantum system dynamics has grown to encompass laser physics, cooling
of atoms, and nascent thoughts on quantum thermodynamics.  And one of
the most important of the new reaches of this study is the development
of quantum error correction, by which open quantum system dynamics can
be reversed, under certain circumstances.

The master equation and quantum trajectories approaches to this study
have in common their use of continuous time in describing open
quantum system dynamics.  However, it is often not important to be
able to know the state of a system at all times.  In fact, in
increasingly many experiments, what is needed is knowledge of the
state only at discrete times, such as before and after some
operation.  For example, in quantum information science, such
operations can be individual logic gates, or full communication
protocols such as teleportation or entanglement distillation.

Very generally, a discrete time picture is exceedingly useful in
modeling and analyzing noisy channels, such as are used in
communication.  In the world of classical physics, Shannon's theory of
how to communicate reliably over noisy channels led to the development
of modern information theory, and error correction codes.
Analogously, quantum information theory has developed to describe how
to communicate reliably over noisy quantum channels, with a slew of
novel quantum error correction codes.  Among other things, such codes
allow reliable information processing systems to be realized from
unreliable components, provided the component failure probabilities
are beneath a fixed threshold set by code properties.

In this section, we present a discrete time model of open quantum
system dynamics, known as the operator sum representation
("OSR").  One nice feature of this representation is that the
Lindblad form of the master equation can be derived from it, as we
show.  The OSR also provides a comprehensive way to grasp the
different unravelings possible for open system dynamics, as we
describe with examples demonstrating quantum error correction.

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=== OSR: definition and properties ===

Suppose the system under study is described by state <math>|\psi{\rangle}</math>.  It
interacts with an environment, initially in some state <math>|e{\rangle}</math>, through
a unitary interaction <math>U</math>, producing
:<math> 
	|e{\rangle}|\psi \rangle  \longrightarrow U|e{\rangle}|\psi{\rangle}
\,.
</math>
Suppose the output state of the environment is measured in the basis
<math>\{|e_k{\rangle}\}</math> (all mutually orthogonal).  Since the environment is
doing the measuring, the measurement results are left unknown, but
this does not change the fact that the resulting state of the system
is given by averaging over the environment's measurement results.
Using the notation
:<math> 
	\rho = \sum_k p_k |\phi_k{\rangle}{\langle}\phi_k|   \longleftrightarrow \rho =
	\sqrt{p_0} |\phi_0 \rangle  \oplus \sqrt{p_1} |\phi_1 \rangle  \oplus \cdots
\,,
</math>
we can write the resulting system as the statistical mixture of states
:<math> 
	 \langle e_0|U|e{\rangle}\, |\psi{\rangle}
	\oplus  \langle e_1|U|e{\rangle}\, |\psi{\rangle}
	\oplus  \langle e_2|U|e{\rangle}\, |\psi \rangle  \cdots
\,.
</math>
Let <math>E_k =  \langle e_k|U|e{\rangle}</math> be the operator (in the Hilbert space of the
system) given by the partial matrix element over the environment with
<math>|e{\rangle}</math> and <math>|e_k{\rangle}</math>.  The output state of the system is thus
{{Eq
|math=<math> 
	\rho_{out} = { \mathcal  E}(\rho_{in}) = \sum_k E_k \rho_{in} E_k^\dagger
\,,
</math>
|num=6.3.1
}}
where <math>\rho_{in} = |\psi{\rangle}{\langle}\psi|</math> is the input system state.  Note
that by construction,
:<math> 
	\sum_k E_k^\dagger E_k = \sum_k  \langle e|U^\dagger|e_k{\rangle}
	 \langle e_k|U|e \rangle  = I
\,,
</math>
where <math>I</math> is the identity operator acting in the Hilbert space of the
system.

Eq.(6.3.1) is known as the operator sum representation,
and the <math>E_k</math> are known as operation elements, or ''Kraus operators.''  When the operation elements satisfy <math>\sum_k E_k^\dagger
E_k = I</math>, a constraint known as the completeness relation, the
corresponding <math>{ \mathcal  E}(\cdot)</math> satisfies three important properties,
that
<blockquote>
* <math>{\rm Tr}[ { \mathcal  E}(\rho) ] = 1</math> for all <math>\rho</math> with <math>{\rm Tr}(\rho)=1</math>.
* <math>{ \mathcal  E}</math> is manifestly convex and linear, that is
:<math> 
	{ \mathcal  E}\left( \sum_k p_k \rho_k \right) = \sum_k p_k { \mathcal  E}(\rho_k)
\,.
</math>
* <math>{ \mathcal  E}</math> is completely positive, meaning that if <math>\rho\geq 0</math> then <math>{ \mathcal  E}(\rho)\geq 0</math> (it is a positive operator), and <math>(I_R\otimes { \mathcal  E}_Q)(\rho_{RQ})\geq 0</math> for any <math>\rho_{RQ}\geq 0</math> (acting on just a part of system still produces a legitimate state).
</blockquote>
Proving that <math>{ \mathcal  E}</math> leaves the trace unchanged is easy, but
providing complete positivity is outside of the scope of this brief
introduction (see, for example, Chapter 8 of Nielsen and Chuang, for a
proof).  Moreover, it turns out that if a quantum operation is ''defined'' as being a map satisfying these three properties, then it
must have an operator sum representation, so the OSR is indeed very
fundamental.

The practical importance of the operator sum representation lies in
the fact that any set of operators satisfying <math>\sum_k
E_k^\dagger E_k = I</math> defines a valid OSR, and that all interactions
with an environment can be described by some set of operation elements
satisfying <math>\sum_k E_k^\dagger E_k = I</math>.  It is also useful to know
that for a system of <math>N</math> dimensions, the maximum number of operation
elements needed to describe an arbitrary interaction with an
environment is <math>N^2</math>.

=== OSR: examples ===

Let us consider some examples of some simple operator sum
representations of interactions.

1. When there is no interaction with an environment, the operation element is a unitary transform <math>U</math>, such that <math>{ \mathcal 
E}(\rho) = U\rho U^\dagger</math>.

2. When the interaction is measurement with projectors <math>\{P_k\}</math>, then <math>{ \mathcal  E}(\rho) = \sum_k P_k \rho P_k</math>.

3. Suppose the interaction is

::[[Image:The_operator_sum_representaton_of_open_quantum_system_dynamics-pdamp-circ2.png|thumb|306px|none|]]
where the box with line represents a controlled rotation about the
<math>\hat{y}</math> axis by angle <math>\theta</math>, and the "environment" is modeled
as a single qubit.  The resulting operation elements are
:<math>
\begin{align}
	E_0 &=  \left[	\begin{array}{cc}{1}&{0}\\{0}&{\sqrt{1-\lambda}}\end{array}\right] 
\\
	E_1 &=	 \left[	\begin{array}{cc}{0}&{0}\\{0}&{\sqrt{\lambda}}\end{array}\right] 
\,,
\end{align}</math>
where <math>\sqrt{\lambda} =	\sin\theta/2</math>.  Direct calculations shows that
:<math> 
	{ \mathcal  E}\left(  \left[	\begin{array}{cc}{a}&{b}\\{c}&{d}\end{array}\right]  \right)
	= \sum_k E_k  \left[	\begin{array}{cc}{a}&{b}\\{c}&{d}\end{array}\right]  E_k^\dagger
	=  \left[	\begin{array}{cc}{a}&{b\sqrt{\lambda}}\\{c\sqrt{\lambda}}&{d}\end{array}\right] 
\,,
</math>
which is the transformation expected for pure phase damping (<math>T_2</math>).

4. For amplitude damping of a single qubit, the operation
elements are similar to those for phase damping:
:<math>
\begin{align}
	E_0 &=  \left[	\begin{array}{cc}{1}&{0}\\{0}&{\sqrt{1-\lambda}}\end{array}\right] 
\\	E_1 &=  \left[	\begin{array}{cc}{0}&{\sqrt{\lambda}}\\{0}&{0}\end{array}\right] 
\,.
\end{align}</math>

=== Unitary degree of freedom in the operator sum representation ===

Multiple choices of operation elements lead to the same density matrix
mapping <math>\rho \rightarrow { \mathcal  E}(\rho)</math>.  This is a consequence of
the infinite number of unravelings of density matrices into pure state
ensembles, manifested in this representation by the fact that the map
cannot depend on the environment's choice of measurement basis:
::[[Image:The_operator_sum_representaton_of_open_quantum_system_dynamics-quantum-operation-freedom.png|thumb|408px|none|]]
In this diagram, the unitary transform <math>U'</math> inserted before the
environment's measurement parameterizes this unitary degree of freedom
in the OSR.  It leads to the fact that if <math>\{E_k\}</math> are a valid set of
operation elements for <math>{ \mathcal  E}</math>, then
:<math> 
	F_j = \sum_k U_{jk} E_k
</math>
also give a valid set of operation elements, for any unitary matrix
with elements <math>U_{jk}</math>.

An example of this is the fact that
:<math>
\begin{align}
	F_0 &= \sqrt{\alpha}  \left[	\begin{array}{cc}{1}&{0}\\{0}&{1}\end{array}\right] 
\\	F_1 &= \sqrt{1-\alpha}  \left[	\begin{array}{cc}{0}&{1}\\{1}&{0}\end{array}\right] 
\end{align}
</math>
defines an OSR equivalent to that given earlier, for phase damping,
for appropriate choice of <math>\alpha</math>.

=== From OSR to master equation in Lindblad form ===

We have previously seen that open quantum systems can be described by
a master equation in Lindblad form, namely
:<math> 
    \dot{\rho} = -i [H,\rho]
		+\sum_k \left[ { L_k \rho L^\dagger _k
			- \frac{1}{2}( L^\dagger _k L_k \rho
					+ \rho L^\dagger _k L_k ) } \right] \,,
</math>
where <math>L_k</math> are jump operators.  It has been stated that this form is
necessary to guarantee physical validity (trace preservation and
complete positivity), but no derivation origin has been given.

A master equation in Lindblad form can be derived from an operator sum
representation <math>{ \mathcal  E}</math> for an interaction with an environment, in
the following way.  Let <math>\tau</math>, parameterize the strength of the
interaction, such that we may approximate
:<math> 
	\frac{d\rho}{dt} \approx \frac{{ \mathcal  E}(\rho)-\rho}{\tau}
\,,
 
</math>
in the same spirit of coarse-grained evolution in which master
equations are derived.  Without loss of generality, one of the
operation elements for <math>{ \mathcal  E}</math> should be of the order of unity,
such that to leading order,
:<math> 
	E_0 = I - iK\tau + O(\tau^2)
\,,
</math>
where <math>K</math> is independent of <math>\tau</math>.

Define <math>H = (K+ K^\dagger )/2</math> and <math>J=i(K- K^\dagger )/2</math> as being the Hermitian
and anti-Hermitian parts of <math>K</math>, such that <math>H</math> describes the
first-order Hamiltonian evolution under <math>{ \mathcal  E}</math>.  Specifically, to
leading order in <math>\tau</math>,
:<math> 
	E_0 \rho E_0^\dagger = \rho - i\tau [H,\rho] - \tau(J\rho+\rho J)
\,.
</math>
Because of the completeness relation <math>\sum_k E_k^\dagger E_k = I</math>,
:<math> 
	\sum_{k\neq 0} E_k^\dagger E_k
	= I - E_0^\dagger E_0
	\approx I-(I+iK^\dagger \tau)(I-iK \tau)
	\approx 2J\tau
</math>
The terms in the OSR with <math>k\neq 0</math> are of order <math>\tau</math>, so that we
may define
:<math> 
	L_k = \frac{E_k}{\sqrt{\tau}}
\,.
</math>
Thus, to leading order in <math>\tau</math>, we may express <math>J</math> in terms of <math>L_k</math> as
:<math> 
	J = \frac{1}{2} \sum_{k\neq 0} L_k^\dagger L_k
\,,
</math>
giving
:<math> 
	{ \mathcal  E}(\rho) = \sum_k E_k \rho E_k^\dagger
	\approx \rho - i\tau[H,\rho] + \tau \sum_{k\neq 0}
		  \left[    {
			L_k \rho  L^\dagger _k
			-\frac{1}{2}  L^\dagger _k L_k \rho -\frac{1}{2} \rho  L^\dagger _k L_k
		} \right] 
\,.
</math>
Finally, this produces the coarse-grained equation of motion
:<math> 
	\dot{\rho} = - i[H,\rho] + \sum_{k\neq 0} L_k \rho  L^\dagger _k
			-\frac{1}{2}  L^\dagger _k L_k \rho -\frac{1}{2} \rho  L^\dagger _k L_k
\,,
</math>
which is the desired master equation in Lindblad form.

The key steps to keep in mind, in this identification between the
operator sum representation <math>{ \mathcal  E}</math> and the master equation, are
<blockquote>
* The Hamiltonian changes from the original system Hamiltonian <math>H_0</math> to becoming <math>H_0+H</math>, where <math>H</math> is derived from the unity-order operation element in <math>{ \mathcal  E}</math>.  Often, <math>H</math> may be small, causing a small renormalization of energies, such as in the Lamb shift associated with spontaneous emission decay.  However, it can also be substantial and not commute with <math>H_0</math>, as demonstrated by other unravelings of spontaneous emission.
* The remaining operation elements in <math>{ \mathcal  E}</math> define the jump operators, when expanded to leading order in <math>\sqrt{\tau}</math>, for small interaction parameter <math>\tau</math>.
</blockquote>

=== Quantum error correction ===

Perhaps one of the most surprising and fundamentally interesting
results from quantum information is quantum error correction.
Classical error correction is a process used to reverse random errors
which happen, say in a communication channel, which undesirably
distorts a message being transmitted.  This can be accomplished, for
example, using redundancy, say by repeating the transmission of a
given message three or more times, and using majority voting to
recover the original at the receiver.

However, before 1995, it was believed that such an error correction
concept could not apply to quantum systems, because
<blockquote>
* Quantum states collapse when measured
* Errors are continuous
* Quantum states cannot be cloned
</blockquote>
In particular, for example, one cannot encode a quantum state
<math>|\psi{\rangle}</math> against errors simply by replicating it three or more times,
as <math>|\psi{\rangle}|\psi{\rangle}|\psi{\rangle}\cdots</math>.  Such perfect "cloning" is known
to be fundamentally impossible within the framework of quantum
mechanics, and furthermore, even with such a state in hand, errors to
<math>|\psi{\rangle}</math> cannot in general be corrected simply by using redundant
copies.

Nevertheless, it turns out there are ways around all of these
objections.  The part of the Hilbert space containing the information
about the state you wish to preserve need not be measured; only the
effect of the environment need be determined by a measurement.  And
using entangled states allow errors to be made orthogonal and
distinguishable; entanglement also replaces the role played by
redundant copies in classical error correction.

Here, we illustrate the basic ideas behind quantum error correction
using a set of specific examples.

==== 3-qubit bit-flip code ====

Suppose errors occur to a qubit as described by the quantum operation
<math>{ \mathcal  E}(\rho) = (1-p)\rho + pX\rho X</math>, where <math>p</math> is a probability.
This process is known as a "bit-flip" channel, because the qubit is
left alone with probability <math>1-p</math>, and flipped by <math>X</math> with probability
<math>p</math>.  We provide in the following an encoding and recovery method,
allowing bit-flip errors on an unknown qubit to be corrected.

The first key idea is to encode using entangled states, rather than
using redundancy.  Define "logical zero" as <math>|0_L \rangle  = |000{\rangle}</math> and
"logical one" as <math>|1_L \rangle  = |111{\rangle}</math>, and suppose instead of sending
<math>a|0{\rangle}+b|1{\rangle}</math> through the channel, instead we send <math>a|0_L{\rangle}+b|1_L{\rangle}</math>.
Much like for a classical three-bit majority voting code, we find that
the effect of the quantum noise process (acting independently and
identically on the three qubits in the code) performs the
transformation
:<math> 
	a|0_L \rangle  + b|1_L \rangle  \stackrel{{ \mathcal  E}}{\rightarrow} \left\{
	\begin{array}{lc}
	a|000{\rangle}+b|111{\rangle}~~~~ & (1-p)^3
\\	a|001{\rangle}+b|110{\rangle}~~~~ & p(1-p)^2
\\	a|010{\rangle}+b|101{\rangle}~~~~ & p(1-p)^2
\\	a|100{\rangle}+b|011{\rangle}~~~~ & p(1-p)^2
\\	a|011{\rangle}+b|100{\rangle}~~~~ & p^2(1-p)
\\	a|101{\rangle}+b|010{\rangle}~~~~ & p^2(1-p)
\\	a|110{\rangle}+b|001{\rangle}~~~~ & p^2(1-p)
\\	a|111{\rangle}+b|000{\rangle}~~~~ & p^3
	\end{array}
	\right.
\,,
 
</math>
where the probabilities of each pathway are given in the column on the
right.  The output state of the channel is the resulting statistical
mixture of states.

Classical error correction would proceed by measuring the noisy
output, processing the measurement, then reconstructing the original
with high fidelity.  Quantum error correction, however, must be more
subtle.  Here is the second key idea: instead of projectively
measuring the output state, we perform partial measurements, to
determine only the effect of the environment, and leave the encoded
qubit untouched, with high probability.

Specifically, suppose that pairs of qubits can be measured, to
determine if they are identical (either <math>00</math> or <math>11</math>, or opposite,
<math>01</math> or <math>10</math>), without revealing the specific values of the qubits.
Define two such parity measurements,
:<math> 
	U_1 = Z_1 Z_2 = ZZI
~~~~~~{\rm and}~~~~~~
	U_2 = Z_2 Z_3 = IZZ
\,,
</math>
corresponding to measuring the parities of the first and second
qubits, and the second and third qubits, respectively.  Let the output
of these measurements be <math>0</math> if the two qubits are identical, and <math>1</math>
otherwise.

Considering just the four high probability output states, we find the
following:
:<math> 
	\begin{array}{cccc}
	{\rm State} & U_1~{\rm result} & U_2~{\rm result} & {\rm
							Recovery~operation}
\\
	a|000 \rangle  + b|111 \rangle  & 0 & 0 & I
\\	a|001 \rangle  + b|110 \rangle  & 0 & 1 & X_3
\\	a|010 \rangle  + b|101 \rangle  & 1 & 1 & X_2
\\	a|100 \rangle  + b|011 \rangle  & 1 & 0 & X_1
	\end{array}
\,,
</math>
where the second and third columns give the measurement results for
<math>U_1</math> and <math>U_2</math>, known as the syndrome.  Note that none of the
measurement results distinguish between <math>a</math> and <math>b</math>, and thus the
encoded qubit is left intact.  After doing this measurement, we can
recover the initial state <math>a|0_L{\rangle}+b|1_L{\rangle}</math> by applying the unitary
transformation given in the fourth column.

The final output state <math>\rho</math> after performing this correction
procedure is not perfect, because of the probability of two or more
errors occurring.  Quantitatively, the fidelity of the reconstructed
state is
:<math> 
	F(\rho,|\psi \rangle ) = \sqrt{{\langle}\psi|\rho|\psi{\rangle}} = \sqrt{1-3p^2+2p^3}
\,.
</math>

Interestingly, this scheme also corrects a small rotational error,
even though the error is continuous!  Let the error be described by
the quantum operation
:<math> 
	{ \mathcal  E}(\rho) = e^{-i\epsilon X_1} \rho e^{i\epsilon X_1}
\,.
</math>
We have that <math>|\psi' \rangle  = R_{x_1}(2\epsilon) |\psi \rangle  = |\psi \rangle  -
i\epsilon X_1 |\psi{\rangle}</math>, so the syndrome measurement result collapses
this state into either <math>|\psi{\rangle}</math> or <math>-i\epsilon X_1 |\psi{\rangle}</math>.  The
fidelity of the resulting output, after recovery <math>{ \mathcal  R}</math>, is
:<math> 
	F({ \mathcal  R}({ \mathcal  E}(|\psi \rangle )),|\psi \rangle ) \approx \sqrt{1-\epsilon^4}
	\approx 1-\epsilon^2
\,.
</math>
In comparison, if no encoding were done then the output fidelity would
have been <math>\approx 1-\epsilon</math>.  This reduction of error from
<math>O(\epsilon)</math> to <math>O(\epsilon^2)</math> is a signature of the success of the
error correction, even for continuous errors.

==== 3-qubit phase-flip code ====

A similar procedure can be used to correct phase flip errors.  Let
<math>{ \mathcal  E}(\rho) = (1-p)\rho + pZ\rho Z</math>.  Recall that <math>HZH=X</math> and
<math>HXH=Z</math>.  Thus, <math>H{ \mathcal  E}(H\rho H) H = { \mathcal  E}_{\rm bit~flip}</math>.
Thus, we can correct phase flip errors by the encoding <math>|0_L \rangle  =
|{\rm {{+++}}}{\rangle}</math>, <math>|1_L \rangle  = |{\rm {{{-}}}{-}{-}}{\rangle}</math>, where <math>|{{{\pm}}}{\rangle}
\equiv (|0{\rangle}\pm|1 \rangle )/\sqrt{2}</math>, and we perform the syndrome
measurements <math>XXI</math> and <math>IXX</math>.

==== Arbitrary errors are superpositions of bit-flip and phase-flip errors ====

The reason phase flip errors are significant and interesting is
because of the following.

bit-flip (<math>X</math>), phase-flip (<math>Z</math>), and bit-phase flip (<math>XZ</math>) errors.

can be written as <math>{ \mathcal  E}(\rho) = \sum_k E_k \rho E_k^\dagger</math>.  Let
<math>\rho</math> be a single qubit, and define <math>E_k = \sum_j c_{jk} \sigma_j</math>,
where <math>\sigma_j \in \{I,X,Y,Z\}</math>.  Then
:<math>\begin{array}{rcl}  
	{ \mathcal  E}(\rho) &=&
		 \sum_{kjj'} c_{kj} x^*_{kj'} \sigma_j \rho \sigma_{j'}
\\
	&=& \sum_{kl} \chi_{jl} \sigma_j \rho \sigma_{l}
\,,
\end{array}</math>
where <math>\chi_{jl} = \sum_k c_{kj} c^*{kj'}</math> is a matrix of
c-number coefficients.

For example, for <math>R_x(2\epsilon)|\psi \rangle  \approx |\psi \rangle  - i\epsilon
X|\psi{\rangle}</math>, we have <math>{ \mathcal  E}(\rho) = \rho - i\epsilon (X\rho - \rho X)
+ \epsilon^2 X\rho X</math>.  The syndrome measurement then projects the
environment into definite error states, removing the off-diagonal
terms (proportional to <math>\epsilon</math> instead of to <math>\epsilon^2</math>) in this
expression.

=== Quantum error correction: Shor's 9-qubit code ===

A combination of the bit-flip and phase-flip codes allows correction
of any single qubit error.  This code encodes a single qubit as
:<math>\begin{align}
	|0_L \rangle  &= \frac{(|000 \rangle  + |111 \rangle )^{\otimes 3}}{\sqrt{8}}
\\	|1_L \rangle  &= \frac{(|000 \rangle  - |111 \rangle )^{\otimes 3}}{\sqrt{8}}
\,.
\end{align}</math>
The syndrome measurement operators are
:<math> 
	Z_1Z_2,
~~	Z_2,Z_3,
~~	Z_4,Z_5,
~~	Z_5,Z_6,
~~	Z_7,Z_8,~~{\rm and}
~~	Z_8,Z_9
</math>
for bit flip errors, and
:<math> 
	X_1X_2X_3X_4X_5X_6
~~~~~~  X_4X_5X_6X_7X_8X_9
</math>
for phase flip errors.  It is simple to see that any single qubit bit
flip or phase flip or bit-phase flip error gives a unique syndrome
measurement result, by direct examination.  It corrects to leading
order errors resulting from small rotations, as well.  And perhaps,
not so obviously, the code also perfectly corrects for errors
resulting from removal of any single qubit!

=== References ===

[[Category:Optical Bloch Equations]]