# Conditional Action and imperfect Erasure of Qubits

Heinz-Jürgen Schmidt  Universität Osnabrück, Fachbereich Physik, D - 49069
Osnabrück, Germany

###### Abstract

We consider state changes in quantum theory due to “conditional action" and
relate these to the discussion of entropy decrease due to interventions of
“intelligent beings" and the principles of Szilard and Landauer/Bennett. The
mathematical theory of conditional actions is a special case of the theory of
“instruments” which describes changes of state due to general measurements and
will therefore be briefly outlined in the present paper. As a detailed example
we consider the imperfect erasure of a qubit that can also be viewed as a
conditional action and will be realized by the coupling of a spin to another
small spin system in its ground state.

##  I Introduction

According to a widespread opinion, there are two types of state change in
quantum mechanics: Time evolution in closed systems, and state changes due to
measurements. The mathematical description of these two processes is known in
principle:

  1. (i)

Time evolution in closed systems can be described by means of unitary
operators $U(t)$ according to

$\rho\mapsto U(t)\,\rho\,U(t)^{\ast}\;,$ (1)

where $U(t)$ is obtained by solutions of the time-dependent Schrödinger
equation and $\rho$ denotes any statistical operator.

  2. (ii)

Conditional state changes according to the outcome of a measurement of an
observable $A$ will be described, in the simplest case, by maps of the form

$\rho\mapsto P_{n}\,\rho\,P_{n}\;,$ (2)

where $\left(P_{n}\right)_{n\in{\mathcal{N}}}$ is the family of
eigenprojections of a self-adjoint operator $A=\sum_{n}a_{n}P_{n}$. Without
selection according to the outcomes of the measurement the total state change
will be

$\rho\mapsto\sum_{n\in{\mathcal{N}}}P_{n}\,\rho\,P_{n}\;,$ (3)

In a recent article S20  we have suggested a third type of state change,
called “conditional action", that combines the two afore-mentioned ones
insofar as it describes a state change depending on the result of a preceding
measurement.

  1. (iii)

In the simplest case a conditional action is mathematically described by maps
of the form

$\rho\mapsto U_{n}\,P_{n}\,\rho\,P_{n}\,U_{n}^{\ast}\;,$ (4)

with the same notation as in (2) and a family of unitary operators
$\left(U_{n}\right)_{n\in{\mathcal{N}}}$. Without selection according to the
outcomes of the measurement the total state change will be

$\rho\mapsto\sum_{n\in{\mathcal{N}}}U_{n}\,P_{n}\,\rho\,P_{n}\,U_{n}^{\ast}\;,$
(5)

Before explaining the details of this suggestion and suitable generalizations
we will fix some general notation used in the present paper. A measurement
leading to the state change (2) is called a “Lüders measurement" in accordance
with BLM96 , sometimes also called “projective measurements" in the
literature. In order not to have to go into technical intricacies the quantum
system $\Sigma$ will be described by a finite-dimensional Hilbert space
${\mathcal{H}}$. In this case the index set ${\mathcal{N}}$ will also be
finite. Let $B({\mathcal{H}})$ denote the real linear space of Hermitean
operators $A:{\mathcal{H}}\rightarrow{\mathcal{H}}$, and
$B_{1}^{+}({\mathcal{H}})$ the convex subset of statistical operators, i. e.,
Hermitean operators $\rho$ with non-negative eigenvalues and
$\mbox{Tr}\rho=1$. The state changes considered above in (2) and (3) can be
viewed as a map $\mathfrak{L}:{\mathcal{N}}\times B({\mathcal{H}})\rightarrow
B({\mathcal{H}})$ defined by

$\mathfrak{L}(n)(\rho):=P_{n}\,\rho\,P_{n}\;,$ (6)

that will be called a “Lüders instrument" and the corresponding map

$\mathfrak{L}({\mathcal{N}})(\rho):=\sum_{n\in{\mathcal{N}}}P_{n}\,\rho\,P_{n}\;,$
(7)

the “total Lüders operation". A difference of the two state changes according
to (i) and (ii) arises when we consider the change of the von Neumann entropy

$S(\rho):=-\mbox{Tr}\left(\rho\,\log\rho\right),\quad\mbox{for }\rho\in
B_{1}^{+}({\mathcal{H}})\;.$ (8)

Under unitary time evolutions (i) the entropy remains constant,

$S(\rho)=S\left(U_{t}\,\rho\,U_{t}^{\ast}\right)\;,$ (9)

whereas for Lüders measurements (ii) the entropy may increase and we can only
state that

$S(\rho)\leq S(\mathfrak{L}({\mathcal{N}})(\rho))\;,$ (10)

see vN32  – SG20 , in accordance with the second law of thermodynamics.

In contrast to closed systems, time evolution in open systems can take a more
general form. An obvious model to account for the time evolution in open
systems is to consider the extension of the system $\Sigma$ with Hilbert space
${\mathcal{H}}$ by another, auxiliary system $E$ (environment, heat bath,
measurement apparatus, …) with Hilbert space ${\mathcal{K}}$ and the unitary
time evolution $V$ of the total system $\Sigma+E$. If the total system is
initially in the state $\rho\otimes\sigma$ it will generally evolve into an
entangled state $V\left(\rho\otimes\sigma\right)V^{\ast}$. In the end, we
again consider the system $\Sigma$ and find its reduced state $\rho_{1}$ given
by the partial trace

$\rho_{1}=\mbox{Tr}_{\mathcal{K}}\left(V\left(\rho\otimes\sigma\right)V^{\ast}\right)\;.$
(11)

The corresponding state change $\rho\mapsto\rho_{1}$ will, in general, not be
of unitary type (1), but represents a natural extension (ie) of the state
changes according to (i). In general, the entropy balance for these state
changes is ambivalent: $S(\rho_{1})$ can be smaller or larger than $S(\rho)$.
In fact, the initial entropy of the total system is $S(\rho)+S(\sigma)$ and
the unitary time evolution $V$ leaves this invariant. But the separation of
the total system into its parts $\rho_{1}$ according to (11) and

$\rho_{2}=\mbox{Tr}_{\mathcal{H}}\left(V\left(\rho\otimes\sigma\right)V^{\ast}\right)\;,$
(12)

increases the entropy (or leaves it constant) according to “subadditivity" of
$S$, see NC00 , 11.3.4., and hence

$S(\rho)+S(\sigma)\leq S(\rho_{1})+S(\rho_{2})\;.$ (13)

But $S(\rho_{1})-S(\rho)$ may assume positive or negative values. This can be
physically understood as the phenomenon that, apart from a possible increase
of the total entropy according to (13), there may be an entropy flow from the
system $\Sigma$ into the environment $E$ or vice versa.

An analogous extension of the system $\Sigma$ to $\Sigma+E$ can also be
considered for Lüders measurements. We again start with an initial total state
$\rho\otimes\sigma$, where $\sigma\in B_{1}^{+}({\mathcal{K}})$, and a unitary
time evolution $V$ of the total system. Then a Lüders measurement
corresponding to a complete family $\left(Q_{n}\right)_{n\in{\mathcal{N}}}$ of
mutually orthogonal projections of the auxiliary system is performed and the
post-measurement total state is reduced to the system $\Sigma$ leading to the
final state

$\mathfrak{I}(n)(\rho):=\mbox{Tr}_{\mathcal{K}}\left(\left(\mathbbm{1}\otimes
Q_{n}\right)V\left(\rho\otimes\sigma\right)V^{\ast}\left(\mathbbm{1}\otimes
Q_{n}\right)\right)\;,$ (14)

or, without selection, to

$\mathfrak{I}({\mathcal{N}})(\rho):=\mbox{Tr}_{\mathcal{K}}\left(\sum_{n}\left(\mathbbm{1}\otimes
Q_{n}\right)V\left(\rho\otimes\sigma\right)V^{\ast}\left(\mathbbm{1}\otimes
Q_{n}\right)\right)\;.$ (15)

Thus we obtain extensions (iie) of the state changes (ii) due to Lüders
measurements by maps $\mathfrak{I}:{\mathcal{N}}\times
B({\mathcal{H}})\rightarrow B({\mathcal{H}})$ of the form (14), that are
called “instruments" in the literature, see BLPY16  and Section II for more
precise mathematical definitions. Lüders instruments are idealized special
cases of general instruments that, in some sense, minimize the perturbation of
the $\Sigma$ system by the measurement, but “real measurements" are better
described by general instruments. Analogous remarks as in the case of open
systems apply for the entropy balance: It is well-known, see L73  or NC00 ,
Exercise 11.15, that general measurements may decrease the system’s entropy.

The latter observation has lead us to the suggestion S20  that the entropy
decrease of systems due to the “intervention of intelligent beings" as, e. g.,
Maxwell’s demon, can be explained by the same mechanism. Originally, the
notion of “conditional action" was developed to describe the intervention of
Maxwell’s demon in the energy distribution of a gas with two chambers:
Depending on the result of an energy measurement on a gas molecule approaching
the partition between the two chambers, a door is opened or shut. Thus, the
further time evolution of the gas depends on the result of the measurement.
Similarly, the result of measuring whether a single molecule is in the left or
right chamber can be used to trigger an isothermal expansion to the right or
left (Szilard’s engine). Szilard argues S29  that the entropy decrease of the
system is compensated by the entropy costs of acquiring information about the
position of the gas particle (“Szilard’s principle"). His arguments are
formulated within classical physics and not easy to understand, see also the
analysis and reconstruction of Szilard’s reasoning in LR94 , EN98  and EN99 .
Nevertheless, it seems possible that the entropy decrease due to such external
interventions is a special case of the well-understood entropy decrease due to
state changes described by general instruments.

In fact, it can be easily confirmed, that the maps of the form (4) describing
“conditional action" are special cases of instruments and hence are called
“Maxwell instruments" in S20 . The mathematical notion of state changes
described by instruments is sufficiently general to cover not only changes due
to inevitable measurement disturbances but also “deliberate" state changes
depending on the result of a measurement.

This notion of “conditional action" will be slightly generalized in the
present paper, and then comprises not only interventions of Maxwell’s demon or
cycles of Szilard’s engine S29 , Z84 , S20 , but also quantum teleportation
NC00  Ch. 1.3.7, quantum error correction NC00  Ch. 10 or erasure of qubits
S20 .

Relative to the choice of a suitable basis a qubit has two values, “0" or “1".
Consider a “yes-no"-measurement corresponding to said basis. If the result is
“1" the two states are swapped, hence $``1"\mapsto``0"$. If the result is “0"
then nothing is done, hence $``0"\mapsto``0"$. This constitutes the
conditional action which sets the qubit state to its default value “0" at any
case and hence can be legitimately considered as an “erasure of a qubit".

The latter example contains an ironic punch line in that the erasure of memory
contents with measurement results and the corresponding entropy costs are
usually considered to resolve the apparent contradiction between the actions
of Maxwell’s demon and the second law (Landauer’s principle). If memory
erasure itself were taken as a conditional action, we would seem to enter an
infinite circle of creating and erasing new memory contents. The obvious
resolution to this problem is the observation that the entropy decrease in the
system $\Sigma$ is due to some flow of entropy from $\Sigma$ to the auxiliary
system $E$ as described above. If $E$ can be viewed as a “memory device" then,
at the end of the conditional action, it already contains the missing entropy.
It is not necessary to erase the content of the memory. The latter would not
create the missing entropy, but only make it visible.

As in S20  it seems sensible to distinguish between the principle that erasure
of memory produces entropy L61  (“Landauer’s principle" in the narrow sense)
and the position that this effect constitutes the solution of the apparent
paradox of Maxwell’s demon B82  (henceforward called “Landauer/Bennett
principle"). Moreover, it will be a matter of substantiating our critique of
the Landauer/Bennett principle (not of the Landauer principle) outlined above
with a more realistic model of qubit erasure than that given in S20 .

To this end we realize the qubit (the system $\Sigma$) by a single spin with
spin quantum number $s=1/2$ described by a Hilbert space
${\mathcal{H}}\cong{\mathbbm{C}}^{2s+1}={\mathbbm{C}}^{2}$ and model the
erasure of the qubit by the coupling of the single spin with a “heat bath" $E$
consisting of $N=6$ spins such that the time evolution of the total system can
be analytically calculated. The quotation marks refer to the fact that the
“heat bath" is pretty small and not macroscopic, as usually required, and
that, moreover, it is rather a “cold bath". This is due to the choice of the
default value "0" of the qubit as the ground state $\downarrow$ of the single
spin. Thus, erasing the qubit is physically equivalent to cooling the system
$\Sigma$ to the temperature $T=0$. Although this is, strictly speaking,
impossible due to the third law of thermodynamics, it can be approximately
accomplished by coupling the single spin to a system of $N=6$ spins in its
ground state. Here we ignore the physical impossibility to prepare a system in
its ground state and consider the ground state of the “heat bath" as a
suitable approximation to a state of very low temperature. This approximation
has the advantage of providing fairly simple expressions for the relevant
quantities considered in this paper. The corresponding calculations are
presented in Section V.

As a side effect of this account results the necessity to define the concept
of “conditional action" somewhat more generally than in S20 . This is done in
Section II where we also recapitulate the basic notions of quantum measurement
theory required for the present work. A critical account of the Szilard
principle in the realm of quantum theory is given in Section III, where we
also formulate an upper bound for the entropy decrease due to conditional
action that is compatible with Szilard’s reasoning but only valid under
certain restrictions. A similar bound is derived in Section IV where the
connections of the present theory with the OLR approach A17  are considered.
The proofs are moved to the appendix, as is the explicit construction of a
“standard" measurement dilation for a general instrument. This measurement
dilation is well-known but nevertheless reproduced here since some arguments
given in this paper depend on its details. We close with a Summary and Outlook
in Section VI.

##  II General definitions and results

In the following we will heavily rely upon the mathematical notions of
operations and instruments. Although these notions are well-known, see, e. g.,
K83  – P13b  and BLPY16 , it will be in order to recall the pertinent
definitions adapted to the present purposes and their interpretations in the
context of measurement theory. For readability, we sometimes will repeat
definitions already presented in the introduction I.

Let ${\mathcal{H}}$ be a $d$-dimensional Hilbert space, $B({\mathcal{H}})$
denote the space of Hermitean operators
$A:{\mathcal{H}}\longrightarrow{\mathcal{H}}$ and $B^{+}({\mathcal{H}})$ the
cone of positively semi-definite operators, i. e., having only non-negatives
eigenvalues. The convex subset $B_{1}^{+}({\mathcal{H}})\subset
B^{+}({\mathcal{H}})$ consists of statistical operators $\rho$ with
$\mbox{Tr}\rho=1$. Such operators physically describe (mixed) states. Pure
states are represented by one-dimensional projectors $P_{\psi}$, where
$\psi\in{\mathcal{H}}$ with $\|\psi\|=1$.

According to NC00 , 8.2.1, there are three equivalent ways to define
operations:

  * •

By considering the system $\Sigma$ coupled to environment $E$,

  * •

by an operator-sum representation, or

  * •

via physically motivated axioms.

Here we follow the second approach and define an “operation" to be a map
$A:B({\mathcal{H}})\longrightarrow B({\mathcal{H}})$ of the form

$A(\rho)=\sum_{i\in{\mathcal{I}}}A_{i}\,\rho\,A_{i}^{\ast}\;,$ (16)

with the Kraus operators $A_{i}:{\mathcal{H}}\rightarrow{\mathcal{H}}$ and a
finite index set ${\mathcal{I}}$, see K83 . It follows that an operation is
linear and maps $B^{+}({\mathcal{H}})$ into itself. It may be trace-preserving
or not.

Operations are intended to describe state changes due to measurements. For
example, the total Lüders operation (7) is a trace-preserving operation in the
above sense with ${\mathcal{I}}={\mathcal{N}}$ and $A_{n}=P_{n}$ for all
$n\in{\mathcal{N}}$. An operation $A:B({\mathcal{H}})\rightarrow
B({\mathcal{H}})$ will be called pure iff the representation (16) of $A$ can
be reduced to a single Kraus operator, i. e.,

$A(\rho)=A_{1}\,\rho\,A_{1}^{\ast}\;.$ (17)

Physically, this means that a pure operation maps pure states onto pure
states, up to a positive factor.

There exists a so-called statistical duality between states and observables,
see BLPY16 , chapter 23.1. In the finite-dimensional case $B({\mathcal{H}})$
can be identified with its dual space $B({\mathcal{H}})^{\ast}$ by means of
the Euclidean scalar product $\mbox{Tr}\,(A\,B)$. Physically, we may
distinguish between the two spaces in the sense that $B({\mathcal{H}})$ is
spanned by the subset of statistical operators representing states and
$B({\mathcal{H}})^{\ast}$ is spanned by the subset of operators with
eigenvalues in the interval $[0,1]$ representing effects. Effects describe
yes-no-measurements including the subset of projectors, which are the extremal
points of the convex set of effects, see BLPY16 .

Every operation $A:B({\mathcal{H}})\rightarrow B({\mathcal{H}})$, viewed as a
transformation of states (Schrödinger picture) gives rise to the dual
operation $A^{\ast}:B({\mathcal{H}})^{\ast}\longrightarrow
B({\mathcal{H}})^{\ast}$ viewed as a transformation of effects (Heisenberg
picture). Reconsider the representation (16) of the operation $A$ by means of
the Kraus operators $A_{i}$. Then the dual operation $A^{\ast}$ has the
corresponding representation

$A^{\ast}(X)=\sum_{i\in{\mathcal{I}}}A_{i}^{\ast}\,X\,A_{i}\;,$ (18)

for all $X\in B({\mathcal{H}})^{\ast}$.

Let ${\mathcal{N}}$ be a finite set of outcomes. Then the map
${\mathfrak{I}}:{\mathcal{N}}\times B({\mathcal{H}})\longrightarrow
B({\mathcal{H}})$ will be called an instrument iff

  * •

${\mathfrak{I}}(n)$ is an operation for all $n\in{\mathcal{N}}$, and

  * •

$\mbox{Tr}\left(\sum_{n\in{\mathcal{N}}}{\mathfrak{I}}(n)(\rho)\right)=\mbox{Tr}\rho$
for all $\rho\in B({\mathcal{H}})$.

The first condition can be re-written as

${\mathfrak{I}}(n)(\rho)=\sum_{i\in{\mathcal{I}}_{n}}A_{ni}\,\rho\,A_{ni}^{\ast}\quad\mbox{for
all }n\in{\mathcal{N}},$ (19)

with suitable Kraus operators $A_{ni}:{\mathcal{H}}\rightarrow{\mathcal{H}}$.
The second condition can be rephrased by saying that the total operation
${\mathfrak{I}}({\mathcal{N}})$ defined by

${\mathfrak{I}}({\mathcal{N}})(\rho)\equiv\sum_{n\in{\mathcal{N}}}{\mathfrak{I}}(n)(\rho)$
(20)

will be trace-preserving. An instrument ${\mathfrak{I}}$ will be called “pure"
iff each operation ${\mathfrak{I}}(n),\;n\in{\mathcal{N}},$ is pure.

Examples of pure instruments are given by Lüders instruments (6) and “Maxwell
instruments" (4).

Similarly as for operations, every instrument ${\mathfrak{I}}$ gives rise to a
dual instrument ${\mathfrak{I}}^{\ast}:{\mathcal{N}}\times
B({\mathcal{H}})^{\ast}\longrightarrow B({\mathcal{H}})^{\ast}$ defined by

${\mathfrak{I}}^{\ast}(n)(X):={\mathfrak{I}}(n)^{\ast}(X)$ (21)

for all $n\in{\mathcal{N}}$ and $X\in B({\mathcal{H}})^{\ast}$. The condition
that the total operation (20) will be trace-preserving translates into

${\mathfrak{I}}^{\ast}({\mathcal{N}})({\mathbbm{1}})=\sum_{n\in{\mathcal{N}}}{\mathfrak{I}}^{\ast}(n)({\mathbbm{1}})=\sum_{n\in{\mathcal{N}}}\sum_{i\in{\mathcal{I}}_{n}}A_{ni}^{\ast}\,A_{ni}={\mathbbm{1}}\;.$
(22)

Thus every dual instrument yields a resolution of the identity by means of
effects

$F_{n}:={\mathfrak{I}}^{\ast}(n)({\mathbbm{1}})=\sum_{i\in{\mathcal{I}}_{n}}A_{ni}^{\ast}\,A_{ni}\;,$
(23)

and hence to a generalized observable in the sense of a positive operator-
valued measure $F=\left(F_{n}\right)_{n\in{\mathcal{N}}}$, see BLPY16 .Note,
however, that compared to the general definition in BLPY16  we will have to
consider generalized observables only in the discrete, finite-dimensional
case. The traditional notion of “sharp" observables represented by self-
adjoint operators corresponds to the special case of a projection-valued
measure $\left(P_{n}\right)_{n\in{\mathcal{N}}}$ satisfying
$\sum_{n\in{\mathcal{N}}}P_{n}={\mathbbm{1}}$. From now on, by "observables"
we always want to understand the generalized case.

It can be shown S20  that “Maxwell instruments" are just pure instruments
corresponding to sharp observables. The example of imperfect erasure of qubits
considered in Section V suggest that the class of “Maxwell instruments" is too
narrow to describe realistic conditional actions. First, imperfect erasure
cannot be described by a pure instrument, since the initial state of the “heat
bath" is not pure. Moreover, the measurement of a sharp “heat bath" observable
does not give rise to a sharp qubit observable. Hence it seems sensible to use
general instruments to describe conditional action. Fortunately, the main
results on the entropy balance of conditional action in S20  can be easily
generalized to general instruments.

To this end we reconsider the map $\mathfrak{I}$ defined in (14) by means of
coupling the system $\Sigma$ to some environment $E$. Recall that the
environment $E$ is described by some Hilbert space ${\mathcal{K}}$ and an
initial state $\sigma\in B_{1}^{+}\left({\mathcal{K}}\right)$. Moreover, $V$
denotes the unitary time evolution of the total system and
$\left(Q_{n}\right)_{n\in{\mathcal{N}}}$ a sharp environment observable. It
can be shown that (i) (14) defines an instrument in the above sense and (ii)
every instrument can be obtained in this way, see Theorem 7. 14 of BLPY16 ,
Exercise 8. 9 of NC00 , or Appendix A. The special instrument defined in (14)
will be referred to as a “measurement dilation"
$\mathfrak{D}_{{\mathcal{K},\sigma,V,Q}}$ of a given instrument
$\mathfrak{I}$. If the initial state $\sigma$ of the environment is pure,
$\sigma=P_{\phi}$, the measurement dilation will also be denoted by
$\mathfrak{D}_{{\mathcal{K},\phi,V,Q}}$. A measurement dilation of a given
instrument $\mathfrak{I}$ is hence a physical realization of $\mathfrak{I}$ by
a Lüders instrument of the extended system $\Sigma+E$ and a subsequent
reduction to $\Sigma$.

Let a conditional action be described by the instrument $\mathfrak{I}$ with
measurement dilation $\mathfrak{I}=\mathfrak{D}_{{\mathcal{K},\sigma,V,Q}}$.
W. r. t. this measurement dilation we define the two reduced states

$\rho_{1}:=\mbox{Tr}_{\mathcal{K}}\left(\sum_{n\in{\mathcal{N}}}\left({\mathbbm{1}}\otimes
Q_{n}\right)V\left(\rho\otimes\sigma\right)V^{\ast}\left({\mathbbm{1}}\otimes
Q_{n}\right)\right)\;,$ (24)

and

$\rho_{2}:=\mbox{Tr}_{\mathcal{H}}\left(\sum_{n\in{\mathcal{N}}}\left({\mathbbm{1}}\otimes
Q_{n}\right)V\left(\rho\otimes\sigma\right)V^{\ast}\left({\mathbbm{1}}\otimes
Q_{n}\right)\right)\;.$ (25)

Then the analogous arguments leading to the entropy balance (13) also prove:

###### Proposition 1

Under the preceding conditions the following holds:

$\Delta S:=S(\rho)-S(\rho_{1})\leq S(\rho_{2})-S(\sigma)\;.$ (26)

In connection with the Szilard principle discussed in the next Section the
following proposition will be of some interest:

###### Proposition 2

The total operation $\mathfrak{I}({\mathcal{N}})$ of an instrument with
measurement dilation $\mathfrak{I}=\mathfrak{D}_{{\mathcal{K},\sigma,V,Q}}$ is
independent of the environment observable $Q$.

This means in particular that $\mathfrak{I}({\mathcal{N}})$ could even be
realized by a coupling of $\Sigma$ to some environment $E$, unitary time
evolution and final state reduction, without any measurement at all. The proof
of Proposition 2 can be found in Appendix B.1.

##  III The Szilard principle revisited

As mentioned in the Introduction, the ideas of L. Szilard S29  to resolve the
apparent contradiction between the results of the “intervention of intelligent
beings" and the second law are published more than nine decades ago and are
confined to classical physics. Therefore, the reconstruction of “Szilard’s
principle" for quantum mechanics could appear as somewhat daring. In this
section, nevertheless, we will reconsider what we understand by “Szilard’s
principle” from the point of view developed in the present article.

According to this principle the entropy decrease of the system is compensated
by the entropy costs of acquiring information about the system’s state. Recall
that we have considered a so-called measurement dilation of the instrument
$\mathfrak{I}$ describing state changes due to conditional action which
extends the system $\Sigma$ by an auxiliary system $E$ (environment). For the
“standard dilation" given in Appendix A the dimension of the Hilbert space
${\mathcal{K}}$ corresponding to the auxiliary system $E$ equals the number of
outcomes $\left|{\mathcal{N}}\right|$ of the Lüders measurement of $Q$ if the
instrument $\mathfrak{I}$ is pure. It is therefore tempting to consider the
auxiliary system $E$ as a “memory” that holds the information about the result
of the measurement and to interpret the “entropy cost of information
acquisition” as the entropy $S(\rho_{2})$ of the final state $\rho_{2}$ of $E$
after the measurement. The probabilities $p_{n},\,n\in{\mathcal{N}},$ of the
various outcomes are given by $p_{n}=\mbox{Tr}\left(\rho\,F_{n}\right)$, where
$F=\left(F_{n}\right)_{n\in{\mathcal{N}}}$ is the observable (23)
corresponding to the conditional action and $\rho$ is the initial state of the
system $\Sigma$. The Shannon entropy $H(p)$ of the probability distribution
$\left(p_{n}\right)_{n\in{\mathcal{N}}}$ is independent of any measurement
dilation and will be called the “Shannon entropy of the experiment"

$H(\rho,F):=H(p):=-\sum_{n\in{\mathcal{N}}}p_{n}\,\log p_{n}\;.$ (27)

Then we can prove the following inequality, which confirms Szilard’s principle
in the above given version:

###### Theorem 1

(Szilard’s principle - quantum case)
The entropy decrease $\Delta S=S(\rho)-S(\rho_{1})$ of a conditional action
corresponding to a pure instrument $\mathfrak{I}$ is bounded by the Shannon
entropy of the experiment, i. e.,

$\Delta S\leq H(\rho,F)\;.$ (28)

For the proof see Appendix B.2. It is worth noting that the bound in (28) is
independent of the pure instrument describing the conditional action and
depends only on the probabilities $p_{n}=\mbox{Tr}\left(\rho F_{n}\right)$.
The theorem is trivially satisfied if the conditional action leads to an
increase of entropy, i. e., $\Delta S\leq 0$, as in the case of a Lüders
measurement without any conditional action. Another trivial case is given if
the observable $F$ is sharp and the projections $F_{n}$ are one-dimensional.
In this case $S(\rho)\leq S\left(\sum_{n}F_{n}\rho F_{n}\right)=H(\rho,F)$ and
the theorem holds since $S(\rho_{1})\geq 0$. In other words: Entropy cannot
fall below the value of zero.

Otherwise the bound (28) is non-trivial. Consider the example of
${\mathcal{H}}={\mathbb{C}}^{3}$ with three mutually orthogonal one-
dimensional projections $P_{1},P_{2},P_{3}$,

$\rho={\textstyle\frac{1}{2}}P_{1}+{\textstyle\frac{3}{10}}P_{2}+{\textstyle\frac{1}{5}}P_{3}\;,$
(29)

and $F=(P_{1},P_{2}+P_{3})$. It follows that

$S(\rho)=-{\textstyle\frac{1}{2}}\log\left({\textstyle\frac{1}{2}}\right)-{\textstyle\frac{3}{10}}\log\left({\textstyle\frac{3}{10}}\right)-{\textstyle\frac{1}{5}}\log\left({\textstyle\frac{1}{5}}\right)\approx
1.02965>H(\rho,F)=\log 2\approx 0.693147\;,$ (30)

and hence, in this example, Theorem 1 says more than just that the entropy of
$\rho$ cannot drop to negative values. It is straightforward to construct a
Maxwell instrument corresponding to the observable $F$ such that
$\rho_{1}={\textstyle\frac{4}{5}}\,P_{1}+{\textstyle\frac{1}{5}}\,P_{2}$ and
hence $\Delta S\approx 0.529251<\log 2\approx 0.693147$ in accordance with
Theorem 1.

A slight generalization of Theorem 1 is the following:

###### Corollary 1

The upper bound (28) also holds if the instrument $\mathfrak{I}$ can be
written as a convex linear combination of pure instruments with the same set
of outcomes ${\mathcal{N}}$.

For the proof see Appendix B.3. A pure instrument has a standard dilation with
one-dimensional projections $Q_{n},\;n\in{\mathcal{N}},$ and a pure initial
state $\sigma=P_{\phi}$ of $E$. If we extend this standard dilation by
considering a real mixed initial state $\sigma$ of $E$ we obtain a convex
combination of pure instruments for which Corollary 1 holds. But not every
instrument is a convex linear combination of pure ones. Actually, there exist
instruments where the bound of entropy decrease given in (28) is violated.

To provide an example we consider the (perfect) erasure of two qubits. Thus
${\mathcal{H}}={\mathbbm{C}}^{2}\otimes{\mathbbm{C}}^{2}\cong{\mathbbm{C}}^{4}$
and we consider an orthonormal basis of ${\mathcal{H}}$ denoted by
$\left(\psi_{1}=\uparrow\uparrow,\,\psi_{2}=\uparrow\downarrow,\,\psi_{3}=\downarrow\uparrow,\,\psi_{4}=\downarrow\downarrow\right)$.
The conditional action maps all these four basis states onto the default state
$\psi_{4}=\downarrow\downarrow$. It has the following measurement dilation:
${\mathcal{K}}={\mathcal{H}}$, initial auxiliary state
$\phi=\downarrow\downarrow$, unitary time evolution $V$ of the total system
defined by $V(\Phi\otimes\Psi)=\Psi\otimes\Phi$ and Lüders measurement of the
auxiliary observable
$Q=\left(Q_{\nu}\right)_{\nu=1,\ldots,4}=\left(|\psi_{\nu}\rangle\langle\psi_{\nu}|\right)_{\nu=1,\ldots,4}$.
The corresponding instrument
$\mathfrak{I}=\mathfrak{D}_{{\mathcal{K}},\phi,V,Q}$ is pure and hence
satisfies (28).

Then we consider another instrument $\widetilde{\mathfrak{I}}$ by changing the
auxiliary observable to
$\widetilde{Q}=\left(Q_{1}+Q_{2},Q_{3}+Q_{4}\right)=\left(|\psi_{1}\rangle\langle\psi_{1}|\otimes{\mathbbm{1}},|\psi_{2}\rangle\langle\psi_{2}|\otimes{\mathbbm{1}}\right)$.
The corresponding system observable
$\widetilde{F}=\left(|\uparrow\rangle\langle\uparrow|\otimes{\mathbbm{1}},|\downarrow\rangle\langle\downarrow|\otimes{\mathbbm{1}}\right)$
is also two-valued and corresponds to a measurement of the first qubit w. r.
t. the considered basis. All other components of
$\mathfrak{D}_{{\mathcal{K}},\phi,V,Q}$ are left unchanged. Consider the
initial state $\rho={\textstyle\frac{1}{4}}{\mathbbm{1}}$ of the system with
$S(\rho)=\log 4$ and $H(\rho,\widetilde{F})=\log 2$. It follows that
$V(\rho\otimes P_{\phi})V^{\ast}=P_{\phi}\otimes\rho$ and hence
$\rho_{1}=P_{\phi}$ and $S(\rho_{1})=0$. Consequently, $\Delta
S=S(\rho)-S(\rho_{1})=\log 4>\log 2=H(\rho,\widetilde{F})$ in contrast to
(28).

Similar examples abound: Whenever the entropy decrease $\Delta S$ due to a
conditional action is larger than $\log 2$ then a corresponding measurement
dilation can be modified to yield $S(\rho,\widetilde{F})=\log 2$ without
changing $\Delta S$ due to Proposition 2. The modified instrument
$\widetilde{\mathfrak{I}}$ cannot be written as a convex linear combination of
pure instruments according to Corollary 1.

As a conclusion for the evaluation of Szilard’s principle we can state that
there are examples of conditional actions where the entropy decrease in the
system can be explained by an entropy increase at least as large in a memory,
as well as counter examples. In the counterexamples, however, we have no
violation of the second law, but only an impossibility to reduce the auxiliary
system to its function as a memory. This is especially true for the limiting
case of an entropy reduction without measurement.

The example of Section V, see Figure 3, shows that the upper bound (28) of the
entropy decrease holds for a larger class of conditional actions than given by
Theorem 1 or Corollary 1. It remains an open task to determine this class more
precisely.

##  IV Connections to the OLR approach

There exists a vast amount of literature on Maxwell’s demon and related
questions footnote . Among them is an article that comes rather close to the
results of the present work, namely A13 , that deals with Szilard’s engine and
where we read in the abstract:

> In this paper, Maxwell’s Demon is analyzed within a “referential" approach
> to physical information that defines and quantifies the Demon’s information
> via correlations between the joint physical state of the confined molecule
> and that of the Demon’s memory. On this view […] information is erased not
> during the memory reset step of the Demon’s cycle, but rather during the
> expansion step, when these correlations are destroyed.

The mentioned notion of “observer-local referential (OLR) information" is
further outlined in A17 . A detailed comparison of the “conditional action
approach" and the “OLR approach" is beyond the scope of this paper. Arguably,
a key difference is that we could not model the formation of a correlation by
a measurement and the subsequent destruction of that correlation by a
conditional action as a sequence of state changes and instead had to use
measurement dilation as a surrogate, see Section VI.

To illustrate the nevertheless existing connections between the two
approaches, we will derive another upper bound for the entropy decrease
analogous to Theorem 1 using the notion of OLR information. We will restrict
ourselves to the case of conditional actions described by Maxwell instruments
$\mathfrak{M}$, i. e., instruments of the form (4). Let $\mathfrak{L}$ denote
the corresponding Lüders instrument of the form (2) such that $\mathfrak{M}$
and $\mathfrak{L}$ share the same Hilbert space ${\mathcal{H}}$ and the same
sharp observable $P=\left(P_{n}\right)_{n\in{\mathcal{N}}}$. Further consider
the standard measurement dilations

$\mathfrak{L}=\mathfrak{D}_{{\mathcal{K}},\phi^{\prime},V^{\prime},Q^{\prime}},\quad\mbox{and}\quad\mathfrak{M}=\mathfrak{D}_{{\mathcal{K}},\phi,V,Q}\;,$
(31)

as explicitly constructed in Appendix A but specialized to the case of pure
instruments, see also S20 . W. r. t. these measurement dilations we further
define

$\displaystyle\rho_{12}$ $\displaystyle:=$
$\displaystyle\sum_{n\in{\mathcal{N}}}\left(\mathbbm{1}-Q_{n}\right)\,V\,\left(\rho\otimes
P_{\phi}\right)\,V^{\ast}\,\left(\mathbbm{1}-Q_{n}\right)\,,$ (32)
$\displaystyle\rho_{1}$ $\displaystyle:=$
$\displaystyle\mbox{Tr}_{\mathcal{K}}\,\rho_{12},\quad\mbox{and}\quad\rho_{2}:=\mbox{Tr}_{\mathcal{H}}\,\rho_{12}\;,$
(33)

and analogously for the primed quantities:

$\displaystyle\rho_{12}^{\prime}$ $\displaystyle:=$
$\displaystyle\sum_{n\in{\mathcal{N}}}\left(\mathbbm{1}-Q^{\prime}_{n}\right)\,V^{\prime}\,\left(\rho\otimes
P_{\phi^{\prime}}\right)\,V^{\prime\ast}\,\left(\mathbbm{1}-Q^{\prime}_{n}\right)\,,$
(34) $\displaystyle\rho_{1}^{\prime}$ $\displaystyle:=$
$\displaystyle\mbox{Tr}_{\mathcal{K}}\,\rho_{12}^{\prime},\quad\mbox{and}\quad\rho_{2}^{\prime}:=\mbox{Tr}_{\mathcal{H}}\,\rho_{12}^{\prime}\;.$
(35)

In accordance with A17  we define the OLR information

$\displaystyle{\mathcal{I}}$ $\displaystyle:=$ $\displaystyle
S(\rho_{1})+S(\rho_{2})-S(\rho_{12})=:S_{1}+S_{2}-S_{12}\;,$ (36)
$\displaystyle{\mathcal{I}}^{\prime}$ $\displaystyle:=$ $\displaystyle
S(\rho_{1}^{\prime})+S(\rho_{2}^{\prime})-S(\rho_{12}^{\prime})=:S_{1}^{\prime}+S_{2}^{\prime}-S_{12}^{\prime}\;,$
(37) $\displaystyle\Delta{\mathcal{I}}$ $\displaystyle:=$
$\displaystyle{\mathcal{I}}^{\prime}-{\mathcal{I}}\;.$ (38)

Further, let

$S_{0}:=S(\rho)=S(\rho\otimes P_{\phi})=S(\rho\otimes P_{\phi^{\prime}})\;,$
(39)

and

$\Delta S:=S_{0}-S_{1}\;.$ (40)

Then we can prove the following

###### Proposition 3

Under the preceding conditions the entropy decrease $\Delta S$ is bounded from
above by

$\Delta S\leq\Delta{\mathcal{I}}\;.$ (41)

The proof of Proposition 3 can be found in Appendix B.4. Of course, this is
only a first step to analyze the mentioned relations, since the conditions of
Proposition 3 are rather limited, e. g., by the fact that only the standard
dilation is considered and not an arbitrary measurement dilation.

##  V Imperfect erasure of a qubit

The role of the detailed example considered in this Section is twofold: First,
we can explain and illustrate the definitions of the previous sections using a
non-trivial but still computable example. Second, this reasonably realistic
case also demonstrates the viability of the general theory.

###  V.1 Definition of the model

We consider a system of $N$ spins with spin quantum number $s=1/2$ equipped
with a uniform anti-ferromagnetic Heisenberg coupling and a Zeeman term. This
leads to a Hamiltonian

$H_{N}=J\,\sum_{1\leq\mu<\nu\leq
N}\\!\hbox{}\mathop{\vphantom{\mathbf{s}}\smash{\mathbf{s}}}\limits_{\sim}\\!{}_{\mu}\cdot\\!\hbox{}\mathop{\vphantom{\mathbf{s}}\smash{\mathbf{s}}}\limits_{\sim}\\!{}_{\nu}\,+B\sum_{\mu=1}^{N}\\!\hbox{}\mathop{\vphantom{s}\smash{s}}\limits_{\sim}\\!{}_{\mu}^{z}\;,$
(42)

where $J>0$ and $B>0$ are dimensionless physical parameter characterizing the
spin system.
$\\!\hbox{}\mathop{\vphantom{\mathbf{s}}\smash{\mathbf{s}}}\limits_{\sim}\\!{}_{\mu}=\left(\\!\hbox{}\mathop{\vphantom{s}\smash{s}}\limits_{\sim}\\!{}_{\mu}^{x},\\!\hbox{}\mathop{\vphantom{s}\smash{s}}\limits_{\sim}\\!{}_{\mu}^{y},\\!\hbox{}\mathop{\vphantom{s}\smash{s}}\limits_{\sim}\\!{}_{\mu}^{z}\right)$
represents the vector of spin operators at the site $\mu$. It is well-known
that the corresponding time evolution can be analytically calculated since we
may write the Hamiltonian in the form

$H_{N}=\frac{J}{2}\left(\\!\hbox{}\mathop{\vphantom{S}\smash{S}}\limits_{\sim}\\!{}^{2}-\frac{3N}{4}\mathbbm{1}\right)+B\,\\!\hbox{}\mathop{\vphantom{S}\smash{S}}\limits_{\sim}\\!{}^{z}\;,$
(43)

where
$\\!\hbox{}\mathop{\vphantom{\mathbf{S}}\smash{\mathbf{S}}}\limits_{\sim}\hbox{}\\!:=\sum_{\mu=1}^{N}\\!\hbox{}\mathop{\vphantom{\mathbf{s}}\smash{\mathbf{s}}}\limits_{\sim}\\!{}_{\mu}$
denotes the total vector of spin operators and
$\\!\hbox{}\mathop{\vphantom{S}\smash{S}}\limits_{\sim}\\!{}^{z}$ its
$z$-component. Since
$\\!\hbox{}\mathop{\vphantom{S}\smash{S}}\limits_{\sim}\\!{}^{2}$ and
$\\!\hbox{}\mathop{\vphantom{S}\smash{S}}\limits_{\sim}\\!{}^{z}$ commute they
possess a system of common eigenvectors $\left|\alpha;S,M\right\rangle$
satisfying the eigenvalue equations

$\displaystyle\\!\hbox{}\mathop{\vphantom{S}\smash{S}}\limits_{\sim}\\!{}^{2}\,\left|\alpha;S,M\right\rangle$
$\displaystyle=$ $\displaystyle S(S+1)\,\left|\alpha;S,M\right\rangle\;,$ (44)
$\displaystyle\\!\hbox{}\mathop{\vphantom{S}\smash{S}}\limits_{\sim}\\!{}^{z}\,\left|\alpha;S,M\right\rangle$
$\displaystyle=$ $\displaystyle M\,\left|\alpha;S,M\right\rangle\;,$ (45)

and hence

$H_{N}\,\left|\alpha;S,M\right\rangle=\left({\textstyle\frac{J}{2}}\left(S(S+1)-{\textstyle\frac{3N}{4}}\right)+B\,M\right)\left|\alpha;S,M\right\rangle=:E_{N}(S,M)\,\left|\alpha;S,M\right\rangle\;.$
(46)

The theory of coupling angular momenta treated in many textbooks yields that
the quantum number $S$ assumes the values
$\frac{1}{2},1,\frac{3}{2},\ldots,\frac{N}{2}$ for odd $N$ and
$0,1,2,\ldots,\frac{N}{2}$ for even $N$ and $M=-S,-S+1,\ldots,S-1,S$. The
symbol “$\alpha$" stands for further quantum numbers that allow for the
degeneracy $D_{N}(S)$ of the eigenspaces with common eigenvalues $S(S+1)$ and
$M$ of $\\!\hbox{}\mathop{\vphantom{S}\smash{S}}\limits_{\sim}\\!{}^{2}$ and
$\\!\hbox{}\mathop{\vphantom{S}\smash{S}}\limits_{\sim}\\!{}^{z}$, resp., such
that the normalized eigenvectors $\left|\alpha;S,M\right\rangle$ will be
unique up to a phase. For a selection of such degeneracies see Figure 1.

Figure 1: “Half Galton Board": The degeneracies $D_{N}(S)$ of states
$\left|S,M\right\rangle$ generated by coupling $N$ spins with $s=1/2$ can be
obtained by the (red) number of coupling paths that start at $(N=0,S=0)$ and
terminate at $(N,S)$.

We will consider a single spin with $s=1/2$ with Hilbert space
${\mathcal{H}}\cong{\mathbbm{C}}^{2}$ representing one qubit and try to
realize the erasure of the qubit by coupling the single spin to a “heat bath"
consisting of $N=6$ uniformly coupled spins such that the total Hamiltonian
$H$ will be of the form $H_{7}$. Moreover, we choose $J=B=1$ thereby fixing a
natural energy unit and corresponding physical units of time and temperature
by setting $\hbar=k_{B}=1$. The choice of the “heat bath" with $N=6$ spins has
the pleasant consequence that all relevant quantities can be directly
calculated by the means of computer-algebraic means without resorting to the
theory of coupling angular momenta.

The “heat bath" with Hilbert space ${\mathcal{K}}\cong{\mathbbm{C}}^{64}$ has
a ground state with energy $E_{6}(0,0)=E_{6}(1,-1)=-\frac{9}{4}$ that is
$14$-fold degenerate. This follows from the degeneracies $D_{6}(S=0)=5$ and
$D_{6}(S=1)=9$, see Figure 1. Let $Q_{0}$ denote the projector onto the
corresponding eigenspace and $Q_{1}$ the complementary projector such that
$Q_{0}+Q_{1}={\mathbbm{1}}_{\mathcal{K}}$.

We will assume that initially the “heat bath" is in its ground state
$\sigma:={\textstyle\frac{1}{14}}Q_{0}$ corresponding to the temperature $T=0$
whereas the single spin is in an arbitrary mixed state $\rho$. Then a unitary
time evolution $U_{t}:=\exp\left(-{\sf i}\,t\,H\right)$ takes place followed
by a Lüders measurement of the sharp heat bath observable $(Q_{0},Q_{1})$.
After this measurement we consider the two reduced states

$\rho_{1}=\mbox{Tr}_{\mathcal{K}}\left(\sum_{n=0,1}\left({\mathbbm{1}}\otimes
Q_{n}\right)U_{t}\left(\rho\otimes\sigma\right)U_{t}^{\ast}\left({\mathbbm{1}}\otimes
Q_{n}\right)\right)\;,$ (47)

and

$\rho_{2}=\mbox{Tr}_{\mathcal{H}}\left(\sum_{n=0,1}\left({\mathbbm{1}}\otimes
Q_{n}\right)U_{t}\left(\rho\otimes\sigma\right)U_{t}^{\ast}\left({\mathbbm{1}}\otimes
Q_{n}\right)\right)\;.$ (48)

Obviously, $\rho_{1}$ is the result of the total operation
$\rho_{1}=\mathfrak{I}({\mathcal{N}})(\rho)$ corresponding to the instrument

$\mathfrak{I}(n)(\rho):=\mbox{Tr}_{\mathcal{K}}\left(\left({\mathbbm{1}}\otimes
Q_{n}\right)U_{t}\left(\rho\otimes\sigma\right)U_{t}^{\ast}\left({\mathbbm{1}}\otimes
Q_{n}\right)\right)\;,$ (49)

where $n\in{\mathcal{N}}=\\{0,1\\}$.

It turns out that for the special model we have considered the matrix elements
of $\rho_{n}$ are $4\pi$-periodic functions of $t$. Instead of dwelling into a
debate how to cope with these oscillating terms we simply make the choice
$t=2\pi$, i.e., we consider the time evolution of a half period before
performing the final measurement. This choice gives reasonable results which
suffices to constructing an example of the general theory outlined in this
paper. Now all parameters of our model for imperfect erasure are fixed and we
proceed by presenting the relevant results without explicating the further
details of the computer-algebraic calculation.

###  V.2 Results on the instrument $\mathfrak{I}$

The first results concern the calculation and visualization of the total
trace-preserving operation
$\rho\mapsto\rho_{1}=\mathfrak{I}({\mathcal{N}})(\rho)$. Recall that
$\rho\in{\mathcal{B}}({\mathcal{H}})$ where the latter is a $4$-dimensional
space spanned by the four Pauli matrices

$\sigma_{0}=\left(\begin{array}[]{cc}1&0\\\
0&1\end{array}\right),\;\sigma_{1}=\left(\begin{array}[]{cc}0&1\\\
1&0\end{array}\right),\;\sigma_{2}=\left(\begin{array}[]{cc}0&-{\sf i}\\\ {\sf
i}&0\end{array}\right),\;\sigma_{3}=\left(\begin{array}[]{cc}1&0\\\
0&-1\end{array}\right)\;.$ (50)

They are mutually orthogonal w. r. to the Euclidean scalar product
$(A,B)\mapsto\mbox{Tr}\left(AB\right)$ of Hermitean $2\times 2$-matrices and
have the length $\sqrt{2}$. W. r. t. this basis the total operation
$\mathfrak{I}({\mathcal{N}})$ can be represented by the $4\times 4$ matrix

${\mathbf{I}}={\textstyle\frac{1}{7}}\left(\begin{array}[]{cccc}7&0&0&0\\\
0&1&0&0\\\ 0&0&1&0\\\ -4&0&0&3\\\ \end{array}\right)\;.$ (51)

Note that $\mathfrak{I}({\mathcal{N}})$ being trace preserving is equivalent
to the property $\mathfrak{I}^{\ast}({\mathcal{N}})(\sigma_{0})=\sigma_{0}$ of
the dual instrument $\mathfrak{I}^{\ast}$. The latter is represented by the
transposed matrix ${\mathbf{I}}^{\top}$ and hence the first row of
${\mathbf{I}}$, that corresponds to the first column of ${\mathbf{I}}^{\top}$,
must be necessarily of the form $(1,0,0,0)$.

The density matrices $\rho$ in the Hilbert space
${\mathcal{H}}\cong{\mathbbm{C}}^{2}$ can be represented by the points
$(x_{1},x_{2},x_{3})^{\top}$ of a unit ball in ${\mathbbm{R}}^{3}$ such that
the pure states corresponding to one-dimensional projectors form its surface
${\mathcal{S}}^{2}$, the so-called “Bloch sphere". This representation is
given by

$\rho={\textstyle\frac{1}{2}}\left(\sigma_{0}+\sum_{i=1}^{3}x_{i}\,\sigma_{i}\right)\;.$
(52)

Figure 2: Visualization of the total operation $\mathfrak{I}({\mathcal{N}})$
of imperfect erasure of a qubit as the affine mapping of the Bloch sphere
${\mathcal{S}}^{2}$ onto an ellipsoid ${\mathcal{E}}$ that touches
${\mathcal{S}}^{2}$ at the south pole.

Under the total operation $\mathfrak{I}({\mathcal{N}})$, which is an affine
map on states, the Bloch sphere is mapped onto an ellipsoid ${\mathcal{E}}$
lying inside ${\mathcal{S}}^{2}$, see Figure 2. Due to
$\det({\mathbf{I}})={\textstyle\frac{1}{7}}\times{\textstyle\frac{1}{7}}\times{\textstyle\frac{3}{7}}={\textstyle\frac{3}{343}}$
the volume of ${\mathcal{S}}^{2}$ is compressed to less than one percent. This
volume compression is typical for conditional action. The invariance of
${\mathcal{E}}$ under rotations about the $3$-axis is due to the azimuthal
symmetry of the Hamiltonian (42) and of the initial state of the “heat bath".

Some properties of the mapping $\mathfrak{I}({\mathcal{N}})$ can be read off
the matrix (51): The south pole of ${\mathcal{S}}^{2}$ is mapped onto itself
and this is the only point where ${\mathcal{E}}$ touches the Bloch sphere.
Since this south pole corresponds to the state
$\rho={\textstyle\frac{1}{2}}\left(\sigma_{0}-\sigma_{3}\right)$ its
invariance under $\mathfrak{I}({\mathcal{N}})$ is reflected by the equation
$\mathbf{I}(1,0,0,-1)^{\top}=(1,0,0,-1)^{\top}$. Physically, the south pole
represents the ground state of the qubit corresponding to the Gibbs state with
temperature $T=0$. Its invariance under $\mathfrak{I}({\mathcal{N}})$ hence
means that the quit remains in its ground state if it is coupled to a “heat
bath" of temperature $T=0$, which is very plausible.

The orientation relative to the coordinate system and the semi-axes
$({\textstyle\frac{1}{7}},{\textstyle\frac{1}{7}},{\textstyle\frac{3}{7}})$ of
the ellipsoid ${\mathcal{E}}$ can be read off the lower right $3\times
3$-submatrix of ${\mathbf{I}}$, see (51). The center of ${\mathcal{E}}$ lies
at $x_{3}=-{\textstyle\frac{4}{7}}$ corresponding to the state

$\rho_{1}^{\prime}={\textstyle\frac{1}{2}}\left(\sigma_{0}-{\textstyle\frac{4}{7}}\sigma_{3}\right)={\textstyle\frac{1}{14}}\left(\begin{array}[]{cc}3&0\\\
0&11\\\ \end{array}\right)\;,$ (53)

and the north pole of ${\mathcal{S}}^{2}$ is mapped onto

$\rho_{1}^{\prime\prime}={\textstyle\frac{1}{2}}\left(\sigma_{0}-{\textstyle\frac{1}{7}}\sigma_{3}\right)={\textstyle\frac{1}{7}}\left(\begin{array}[]{cc}3&0\\\
0&4\\\ \end{array}\right)\;.$ (54)

By a perfect erasure of a qubit the Bloch sphere would be completely mapped
onto the south pole; a more realistic scenario corresponds to a mapping onto a
small ellipsoid close to the south pole. The present example may not yield the
best possible result; however its virtue lies in the fact that ${\mathcal{E}}$
can be analytically calculated and is rather simple in form.

After having analyzed the total operation $\mathfrak{I}({\mathcal{N}})$ we
proceed by considering the two components $\mathfrak{I}(n),\;n=0,1,$ of the
instrument $\mathfrak{I}$. We will determine the corresponding Kraus operators
$A_{nmj}$ such that

$\mathfrak{I}(n)(\rho)=\sum_{mj}A_{nmj}\,\rho\,A_{nmj}^{\ast},\quad
n\in{\mathcal{N}}\;.$ (55)

The index $i$ occurring in (19) has been replaced here by a multi-index
$i=(m,j)$. According to the general theory the Kraus operators $A_{nmj}$ can
be derived from the measurement dilation
$\mathfrak{I}={\mathfrak{D}}_{{\mathcal{K}},\sigma,V,Q}$ by means of the
equation

$\left\langle a\right|A_{nmj}\left|b\right\rangle=\sqrt{q_{j}}\left\langle
a\otimes\phi_{m}\right|V\left|b\otimes\psi_{j}\right\rangle,\quad
a,b\in{\mathcal{H}},\;m\in{\mathcal{M}}_{n}\;,$ (56)

see NC00 , 8.35. Here we have used the spectral decomposition of the initial
state $\sigma$ of the auxiliary system

$\sigma=\sum_{j}q_{j}\,\left|\psi_{j}\right\rangle\left\langle\psi_{j}\right|\;,$
(57)

and that of the projector $Q_{n}$

$Q_{n}=\sum_{m\in{\mathcal{M}}_{n}}\left|\phi_{m}\right\rangle\left\langle\phi_{m}\right|\;.$
(58)

The latter is defined w. r. t. a suitable partition
${\mathcal{M}}=\biguplus_{n}{\mathcal{M}}_{n}$ of the index set
${\mathcal{M}}$ corresponding to an orthonormal basis
$\left(\phi_{m}\right)_{m\in{\mathcal{M}}}$ of ${\mathcal{K}}$ adapted to the
sharp observable $\left(Q_{n}\right)_{n\in{\mathcal{N}}}$.

Figure 3: Plot of the final entropy $S_{1}$ versus the initial one $S_{0}$ for
an imperfect erasure of a qubit. The blue dots correspond to $10,000$ randomly
chosen initial states $\rho$ of the single spin. The enveloping green curve is
analogously calculated for the one-dimensional family $\rho(p)$ according to
(62,63) and reaches its maximum of $S_{0}=\log 2$ for
$p={\textstyle\frac{1}{2}}$ (dashed green line). The corresponding value
$S_{1/2}$ of $S_{1}$ is given by (64). Only for the points below the (red)
line $S_{0}=S_{1}$ there occurs a decrease of entropy due to the conditional
action. For the green curve this will happen if $0<p<p_{1}$, where $p_{1}$ is
given by (65). The dashed blue curve represents $S_{0}-H(\rho(p),F)$ according
to (27). Therefore, the bound (28) holds although the conditions of Theorem 1
or Corollary 1 are not satisfied.

For the present example it appears at first sight that we would need $14\times
64=896$ Kraus operators. Fortunately, only $94$ Kraus operators do not vanish.
Also, they can be combined and simplified so that only the following three
operators remain:

$A_{1}=\left(\begin{array}[]{cc}\sqrt{\textstyle\frac{5}{14}}&0\\\
0&\sqrt{\textstyle\frac{5}{14}}\\\ \end{array}\right),\quad
A_{2}=\left(\begin{array}[]{cc}-{\textstyle\frac{1}{\sqrt{14}}}&0\\\
0&{\textstyle\frac{3}{\sqrt{14}}}\\\ \end{array}\right),\quad
A_{3}=\left(\begin{array}[]{cc}0&0\\\ {\textstyle\frac{2}{\sqrt{7}}}&0\\\
\end{array}\right)\;.$ (59)

Here the first two operators $A_{1}$ and $A_{2}$ belong to $\mathfrak{I}(0)$
and $A_{3}$ to $\mathfrak{I}(1)$. Hence the observable $F=(F_{0},F_{1})$ given
by the instrument $\mathfrak{I}$ is obtained as

$F_{0}=A_{1}^{\ast}\,A_{1}+A_{2}^{\ast}\,A_{2}=\left(\begin{array}[]{cc}{\textstyle\frac{3}{7}}&0\\\
0&1\\\ \end{array}\right)\quad\mbox{and}\quad
F_{1}=A_{3}^{\ast}\,A_{3}=\left(\begin{array}[]{cc}{\textstyle\frac{4}{7}}&0\\\
0&0\\\ \end{array}\right)\;,$ (60)

satisfying $F_{0}+F_{1}={\mathbbm{1}}$, as it is required for $F$ being an
observable. The fact that $F$ is not a sharp observable means that, despite
energy conservation, there is no perfect correlation between the energy of the
individual spin and that of the “heat bath”. The latter would be expected only
on the basis of time-dependent perturbation theory (Fermi’s “Golden Rule”) and
does not hold for a finite interaction between the spin and the “heat bath”.

###  V.3 Results on the entropy balance

Figure 4: Plot of the initial total entropy $S_{0}+S_{i}$ (red curve) and the
final entropy $S_{1}+S_{2}$ (blue curve) calculated for the one-parameter
family $\rho(p)$ according to (61) such that $S_{0}=S(\rho(p))$. Obviously,
$S_{0}+S_{i}<S_{1}+S_{2}$ for $0<p\leq 1$. $S_{i}=\log 14$ denotes the initial
entropy of the “heat bath" and hence $S_{0}+S_{i}$ assumes its maximum $\log
2+\log 14=\log 28\approx 3.3322$ at $p=1/2$. At $p=1$ the final entropy
$S_{1}+S_{2}$ assumes the value $S_{f}\approx 3.54621$ according to (66).

We now turn to the entropy balance. First we plot $S_{1}:=S(\rho_{1})$ versus
$S_{0}:=S(\rho)$, see Figure 3. For the one-parameter family of states

$\rho(p):=\left(\begin{array}[]{cc}p&0\\\ 0&1-p\end{array}\right)\;,$ (61)

where $0\leq p\leq 1$, we obtain a curve with parametric representation

$\displaystyle S_{0}(\rho(p))$ $\displaystyle=$
$\displaystyle-p\log(p)-(1-p)\log(1-p),$ (62) $\displaystyle S_{1}(\rho(p))$
$\displaystyle=$
$\displaystyle{\textstyle\frac{1}{7}}\left((3p-7)\log\left(1-{\textstyle\frac{3p}{7}}\right)-3p\log\left({\textstyle\frac{3p}{7}}\right)\right)\;,$
(63)

see the green curve in Figure 3. The value $p=1/2$ corresponds to the maximum
$\log 2$ of $S_{0}$ and the value

$S_{1/2}={\textstyle\frac{1}{7}}\left({\textstyle\frac{11}{2}}\log\left({\textstyle\frac{14}{11}}\right)+{\textstyle\frac{3}{2}}\log\left({\textstyle\frac{14}{3}}\right)\right)$
(64)

of $S_{1}$ corresponding to the entropy of the center of the ellipsoid
${\mathcal{E}}$. This curve is the envelope of the set of all points
$\left(S_{0}(\rho),S_{1}(\rho)\right)$ as can be seen as follows.

The surfaces with constant entropy (“adiabatic surfaces") are the concentric
spheres ${\mathcal{S}}$ inside the Bloch sphere (together with the center
considered as a degenerate sphere). The set of states $\rho$ corresponding to
such a concentric sphere ${\mathcal{S}}$ is mapped under
$\mathfrak{I}({\mathcal{N}})$ onto an ellipsoid
${\mathcal{E}}^{\prime}\subset{\mathcal{E}}$ that is also invariant under
rotations about the $3$-axis. The north pole $N$ of ${\mathcal{S}}$
corresponding to the state $\rho(p)$ is mapped onto the north pole
$N^{\prime}$ of ${\mathcal{E}}^{\prime}$. Similarly, the south pole $S$ of
${\mathcal{S}}$ corresponding to the state $\rho(1-p)$ is mapped onto the
south pole $S^{\prime}$ of ${\mathcal{E}}^{\prime}$. The total ellipsoid
${\mathcal{E}}^{\prime}$ is bounded by the two concentric spheres through
$N^{\prime}$ and $S^{\prime}$ and hence the entropy of all states
corresponding to ${\mathcal{E}}^{\prime}$ is bounded by $S_{1}(\rho(p))$ and
$S_{1}(\rho(1-p))$.

Decrease of entropy, i. e., $S(\rho_{1})<S(\rho_{0})$, will not always occur.
For example, the north pole of ${\mathcal{S}}^{2}$ corresponds to a pure state
of vanishing entropy and is mapped onto a mixed state with positive entropy.
For the states $\rho(p)$ decrease of entropy is equivalent to

$0<p<p_{1}:={\textstyle\frac{1}{7}}\left({\textstyle\frac{11}{2}}\log\left({\textstyle\frac{14}{11}}\right)+{\textstyle\frac{3}{2}}\log\left({\textstyle\frac{14}{3}}\right)\right)\approx
0.51958\;,$ (65)

see Figure 3. For the value $p=p_{1}$ the initial state $\rho(p_{1})$ is
mapped under $\mathfrak{I}({\mathcal{N}})$ onto $\rho(1-p_{1})$ which has the
same entropy. Recall that within the family $\rho(p),\;0\leq p\leq p_{1}$ only
states with $0<p<1/2$ have a positive temperature and hence for these entropy
decrease is guaranteed.

Although the instrument $\mathfrak{I}$ does not satisfy the conditions of
Theorem 1 or Corollary 1 its entropy decrease satisfies the same bound given
in (28), see Figure 3.

According to the considerations of Section II and Proposition 1 it is clear
that a possible decrease of entropy will be compensated by an equal or larger
increase of entropy of the auxiliary system, i. e., of the “heat bath".
Nevertheless, it will be instructive to check this result for the considered
example, see Figure 4. We have plotted the initial total entropy $S_{0}+S_{i}$
(red curve) and the final entropy $S_{1}+S_{2}$ (blue curve) calculated for
the one-parameter family $\rho(p)$ according to (61) such that
$S_{0}=S(\rho(p))$. Obviously, $S_{0}+S_{i}<S_{1}+S_{2}$ for $0<p\leq 1$ in
accordance with the second law. $S_{i}=\log 14$ denotes the initial entropy of
the “heat bath" due to $14$-fold degeneracy of the ground state of $H_{6}$. At
$p=1$ the final entropy $S_{1}+S_{2}$ assumes the value

$S_{f}={\textstyle\frac{4}{7}}\log\left({\textstyle\frac{7}{4}}\right)+{\textstyle\frac{3}{7}}\log\left({\textstyle\frac{7}{3}}\right)+{\textstyle\frac{5\log(14)}{14}}+{\textstyle\frac{4}{7}}\
\log\left({\textstyle\frac{63}{4}}\right)+{\textstyle\frac{\log(126)}{14}}\approx
3.54621\;,$ (66)

see Figure 4.

##  VI Summary and Outlook

In this paper we have elaborated a recent proposal S20  to describe the
“intervention of intelligent beings" in quantum systems in terms of
“conditional action". This is a genuinely physical concept. Mathematically,
the notion of general “instruments", originally intended to explain state
changes due to measurements, is already broad enough to include conditional
action.

A fundamental assumption here is that it is not necessary to describe the
inner life of “intelligent beings" in more detail; it is sufficient to analyze
the workings of apparatuses built to realize measurements and conditional
actions. Ideally, such an analysis includes the original measurement on the
system $\Sigma$, the storage of the measurement result in a classical memory,
and the subsequent unitary time evolution of $\Sigma$ conditioned by the
memory contents. But the construction of such a complete model of a
conditional action would, in my opinion, require a solution of the quantum
measurement problem and hence is impossible at present.

We must therefore confine ourselves to considering physical realizations of
conditional actions restricted to so-called “measurement dilations”. These are
well-known mathematical constructions BLPY16  that reduce general instruments
acting on $\Sigma$ to special Lüders instruments acting on a larger system
$\Sigma+E$. These tools also open the way to understanding the (possible)
entropy decrease in $\Sigma$ due to the conditional action as an entropy flow
from $\Sigma$ to the auxiliary system $E$, in the same sense as the (possible)
entropy decrease due to a general measurement can be explained.

The latter explanation can also be related to existing approaches to resolving
the apparent contradiction of said entropy decrease with a tentative second
law of quantum thermodynamics. Among such approaches are the Szilard
principle, the Landauer-Bennett principle, and the recent OLR approach. Due to
the Szilard principle the entropy decrease in $\Sigma$ is, at least,
compensated by the entropy production associated with the measurement of the
system’s state. This principle has been confirmed by the present conditional
action approach in the special case where the auxiliary system $E$ can be
conceived as a memory device, see Theorem 1, but not in general. Also a
partial compatibility to the OLR approach has been shown, in so far as, in
special cases, the entropy decrease in $\Sigma$ is bounded by the loss of
mutual information due to conditional action, see Proposition 3. Similarly,
the approach based on the entropy costs of memory erasure (Landauer-Bennett
principle) is compatible with our approach, but cannot be viewed as the
ultimate solution of the apparent paradox.

We have analyzed the imperfect erasure of a qubit by means of a physical
model. This model descries the cooling of a single spin by coupling it to a
“cold bath" consisting of six other spins such that the total time evolution
can be analytically calculated. This model thus represents a more or less
realistic measurement dilatation of imperfect erasure conceived as a
conditional action, and as such motivates the slight generalization of this
concept compared to S20 . At the same time this example reveals some problems
of the mentioned principles based on acquisition or deletion of information,
since imperfect erasure of a qubit can be achieved without any measurement at
all. This is even more plausible if one considers the physical interpretation
of the erasure as a cooling of a single spin. The OLR approach appears to
avoid this problem because it relies on an information concept that is
independent of possible measurements, see A08  for a corresponding treatment
of imperfect memory erasure.

For future investigations it seems to be a desirable goal to extend the
conditional action approach to the field of classical physics. First steps
toward this goal restricted to discrete state spaces have been made in S20 .
The role of measurement is different in classical theories because, unlike in
quantum theory, there are always idealized measurements that do not change the
state of the system. However, there exist non-trivial instruments describing
conditional action even in classical theories, and it should be possible to
realize them by extending the system analogously to the quantum case.

##  Appendix A Construction of the standard measurement dilation for a general
instrument

Let an instrument of the form (19) be given, i. e.,

$\mathfrak{J}(n)(\rho)=\sum_{i\in{\mathcal{I}}_{n}}A_{ni}\,\rho
A_{ni}^{\ast},\quad n\in{\mathcal{N}}\;,$ (67)

such that the corresponding observable
$F=\left(F_{n}\right)_{n\in{\mathcal{N}}}$ is given by

$F_{n}=\sum_{i\in{\mathcal{I}}_{n}}A_{ni}^{\ast}A_{ni},\quad\mbox{for all
}n\in{\mathcal{N}}\;,$ (68)

satisfying $\sum_{n\in{\mathcal{N}}}F_{n}={\mathbbm{1}}_{\mathcal{H}}$.

Following NC00  we want to explicitly construct a measurement dilation of
$\mathfrak{J}$ of the form (15), see also the analogous construction for a
Maxwell instrument in S20 .

To this end we define
${\mathcal{N}}^{\prime}:=\\{(n,i)\left|\right.n\in{\mathcal{N}}\mbox{ and
}i\in{\mathcal{I}}_{n}\\}$ and choose
${\mathcal{K}}={\mathbbm{C}}^{{\mathcal{N}}^{\prime}}$ and an orthonormal
basis $\left(|ni\rangle\right)_{n\in{\mathcal{N},\,i\in{\mathcal{I}}_{n}}}$ in
${\mathcal{K}}$. Let $\phi\in{\mathcal{K}}$ be one of these basis vectors,
say, $\phi=|11\rangle$. Further, let $\left(Q_{n}\right)_{n\in{\mathcal{N}}}$
denote the complete family of projectors in the Hilbert space ${\mathcal{K}}$
defined by

$Q_{n}=\sum_{i\in{\mathcal{I}}_{n}}|ni\rangle\langle ni|\;,\quad\mbox{for all
}n\in{\mathcal{N}}\;.$ (69)

Moreover, let $\check{Q}_{ni}$ be the subspace of
${\mathcal{H}}\otimes{\mathcal{K}}$ formed by vectors of the form
$\psi\otimes|ni\rangle$ for all $\psi\in{\mathcal{H}}$ and fixed
$n\in{\mathcal{N}},\,i\in{\mathcal{I}}_{n}$ and define a linear map
$V_{11}:\check{Q}_{11}\rightarrow{\mathcal{H}}\otimes{\mathcal{K}}$ by

$V_{11}\left|\psi
11\right\rangle:=V_{11}\left(\psi\otimes|11\rangle\right):=\sum_{n\in{\mathcal{N}},i\in{\mathcal{I}}_{n}}A_{ni}\psi\otimes|ni\rangle\;,$
(70)

for all $\psi\in{\mathcal{H}}$.

###### Lemma 1

The map $V_{11}:\check{Q}_{11}\rightarrow{\mathcal{H}}\otimes{\mathcal{K}}$ is
a partial isometry, i. e., satisfies
$V_{11}^{\ast}\,V_{11}={\mathbbm{1}}_{\check{Q}_{11}}$.

Proof: Let $\varphi,\,\psi$ be two arbitrary vectors of ${\mathcal{H}}$ and
consider the scalar products

$\displaystyle\left\langle\varphi 11\right|V_{11}^{\ast}V_{11}\left|\psi
11\right\rangle$ $\displaystyle\stackrel{{\scriptstyle(\ref{E2})}}{{=}}$
$\displaystyle\sum_{nimj}\left\langle\varphi\right|A_{ni}^{\ast}\,A_{mj}\left|\psi\right\rangle\underbrace{\left\langle
ni\right|\left.mj\right\rangle}_{\delta_{nm}\delta_{ij}}$ (71)
$\displaystyle=$
$\displaystyle\sum_{ni}\left\langle\varphi\right|A_{ni}^{\ast}\,A_{ni}\left|\psi\right\rangle$
(72) $\displaystyle\stackrel{{\scriptstyle(\ref{Eobs})}}{{=}}$
$\displaystyle\left\langle\varphi\right|\underbrace{\sum_{n}F_{n}}_{{\mathbbm{1}}_{\mathcal{H}}}\left|\psi\right\rangle=\left\langle\varphi
11\right|\left.\psi 11\right\rangle\;,$ (73)

which completes the proof of Lemma 1. $\Box$

Next we extend the partial isometry $V_{11}$ to a unitary operator
$V:{\mathcal{H}}\otimes{\mathcal{K}}\rightarrow{\mathcal{H}}\otimes{\mathcal{K}}$.
This completes the definition of the quantities ${\mathcal{K}},\phi,V,Q$
required for the measurement dilation. It remains to show that
$\mathfrak{J}={\mathfrak{D}}_{{\mathcal{K}},\phi,V,Q}$. To this end we
introduce an orthonormal basis
$\left(\left|\ell\right\rangle\right)_{\ell=1,\ldots,d}$ in ${\mathcal{H}}$
and write

$\rho=\sum_{k\ell}|k\rangle\langle k|\rho|\ell\rangle\langle\ell|\;.$ (74)

Hence

$\rho\otimes P_{\phi}=\sum_{k\ell}|k11\rangle\langle
k|\rho|\ell\rangle\langle\ell 11|\;,$ (75)

and, further,

$\displaystyle V\left(\rho\otimes P_{\phi}\right)V^{\ast}$
$\displaystyle\stackrel{{\scriptstyle(\ref{E5})}}{{=}}$
$\displaystyle\sum_{k\ell}V\,|k11\rangle\langle k|\rho|\ell\rangle\langle\ell
11|\,V^{\ast}$ $\displaystyle\stackrel{{\scriptstyle(\ref{E2})}}{{=}}$
$\displaystyle\sum_{k\ell nimj}A_{ni}|k\rangle\langle
k|\rho|\ell\rangle\langle\ell|A_{mj}^{\ast}\otimes|ni\rangle\langle mj|$
$\displaystyle\stackrel{{\scriptstyle(\ref{E4})}}{{=}}$
$\displaystyle\sum_{nimj}A_{ni}\,\rho\,A_{mj}^{\ast}\otimes|ni\rangle\langle
mj|\;.$ (77)

Using

$Q_{r}|ni\rangle\langle mj|Q_{r}=\delta_{rn}\,\delta_{rm}\,|ri\rangle\langle
rj|\;,$ (78)

for all $r\in{\mathcal{N}}$, (77) implies

$\displaystyle\left({\mathbbm{1}}\otimes Q_{r}\right)V\left(\rho\otimes
P_{\phi}\right)V^{\ast}\left({\mathbbm{1}}\otimes Q_{r}\right)$
$\displaystyle=$
$\displaystyle\sum_{ij}\left(A_{ri}\,\rho\,A_{rj}^{\ast}\right)\otimes|ri\rangle\langle
rj|\;,$ (79)

and

$\displaystyle{\mathcal{D}}_{{\mathcal{K}},\phi,V,Q}(r)(\rho)$
$\displaystyle=$
$\displaystyle\mbox{Tr}_{\mathcal{K}}\left(\left({\mathbbm{1}}\otimes
Q_{r}\right)V\left(\rho\otimes
P_{\phi}\right)V^{\ast}\left({\mathbbm{1}}\otimes Q_{r}\right)\right)$ (80)
$\displaystyle\stackrel{{\scriptstyle(\ref{E8})}}{{=}}$
$\displaystyle\sum_{ij}\left(A_{ri}\,\rho\,A_{rj}^{\ast}\right)\mbox{Tr}_{\mathcal{K}}\left(|ri\rangle\langle
rj|\right)$ (81) $\displaystyle=$
$\displaystyle\sum_{i}A_{ri}\,\rho\,A_{ri}^{\ast}\;,$ (82)

since $\mbox{Tr}_{\mathcal{K}}\left(|ri\rangle\langle rj|\right)=\delta_{ij}$
for all $r\in{\mathcal{N}}$. The latter expression equals

$\mathfrak{J}(r)(\rho)\stackrel{{\scriptstyle(\ref{E1})}}{{=}}\sum_{i\in{\mathcal{I}}_{r}}A_{ri}\,\rho\,A_{ri}^{\ast}\;,$

thereby proving that the above construction is a correct measurement dilation
of ${\mathfrak{J}}$.

##  Appendix B Proofs

###  B.1 Proof of Proposition 2

We define

$\rho_{1}:=\mathfrak{I}({\mathcal{N}})(\rho)=\sum_{n\in{\mathcal{N}}}\mbox{Tr}_{\mathcal{K}}\left(\left({\mathbbm{1}}\otimes
Q_{n}\right)\rho^{\prime}\left({\mathbbm{1}}\otimes Q_{n}\right)\right)\;,$
(83)

where

$\rho^{\prime}:=V\left(\rho\otimes\sigma\right)V^{\ast}\;,$ (84)

and will prove Proposition 2 by showing that
$\rho_{1}=\mbox{Tr}_{\mathcal{K}}\left(\rho^{\prime}\right)$. Assume some
orthonormal basis
$\left(\ldots,|\alpha\rangle,\ldots,|\beta\rangle,\ldots\right)$ in
${\mathcal{H}}$ and another orthonormal basis
$\left(|m\rangle\right)_{m\in{\mathcal{M}}}$ in ${\mathcal{K}}$ that is
adapted to the environment observable $Q$ in the sense that

$Q_{n}=\sum_{m\in{\mathcal{M}}_{n}}|m\rangle\langle m|\;,\quad\mbox{for all
}n\in{\mathcal{N}},$ (85)

w. r. t. a partition
${\mathcal{M}}=\biguplus_{n\in{\mathcal{N}}}{\mathcal{M}}_{n}$ of the index
set ${\mathcal{M}}$. It follows that

$\left({\mathbbm{1}}\otimes Q_{n}\right)\left|\beta
m\right\rangle=|\beta\rangle\otimes\left\\{\begin{array}[]{r@{\quad: \quad}
l}|m\rangle&m\in{\mathcal{M}}_{n},\\\ 0&\mbox{else},\end{array}\right.,$ (86)

for all $m\in{\mathcal{M}}$ and any base vector $\left|\beta\right\rangle$ and
analogously for $\left\langle\alpha m\right|\left({\mathbbm{1}}\otimes
Q_{n}\right)$. Hence an arbitrary matrix element of $\rho_{1}$ assumes the
form

$\displaystyle\left\langle\alpha\right|\rho_{1}\left|\beta\right\rangle$
$\displaystyle\stackrel{{\scriptstyle(\ref{rhodef})}}{{=}}$
$\displaystyle\sum_{n\in{\mathcal{N}}}\sum_{m\in{\mathcal{M}}}\left\langle\alpha
m\right|\left({\mathbbm{1}}\otimes
Q_{n}\right)\rho^{\prime}\left({\mathbbm{1}}\otimes Q_{n}\right)\left|\beta
m\right\rangle$ (87)
$\displaystyle\stackrel{{\scriptstyle(\ref{betam})}}{{=}}$
$\displaystyle\sum_{n\in{\mathcal{N}}}\sum_{m\in{\mathcal{M}}_{n}}\left\langle\alpha
m\right|\rho^{\prime}\left|\beta m\right\rangle$ (88) $\displaystyle=$
$\displaystyle\sum_{m\in{\mathcal{M}}}\left\langle\alpha
m\right|\rho^{\prime}\left|\beta m\right\rangle$ (89) $\displaystyle=$
$\displaystyle\left\langle\alpha\right|\mbox{Tr}_{\mathcal{K}}\left(\rho^{\prime}\right)\left|\beta\right\rangle\;,$
(90)

thereby completing the proof of Proposition 2. $\Box$

###  B.2 Proof of Theorem 1

For a pure instrument, the construction of a standard dilation given in
Appendix A is simplified by omitting the indices $i,j\in{\mathcal{I}}_{n}$ and
the corresponding sums. Especially we obtain

$\rho_{12}:=\sum_{r}\left({\mathbbm{1}}\otimes Q_{r}\right)V\left(\rho\otimes
P_{\phi}\right)V^{\ast}\left({\mathbbm{1}}\otimes
Q_{r}\right)\stackrel{{\scriptstyle(\ref{E8})}}{{=}}\sum_{r}\left(A_{r}\,\rho\,A_{r}^{\ast}\right)\otimes|r\rangle\langle
r|\;.$ (91)

It follows that

$S(\rho)\leq S(\rho_{12})\;,$ (92)

since the von Neumann entropy vanishes for pure state like $P_{\phi}$, is
additive for tensor products and invariant under unitary transformations.
Moreover, it is non-decreasing under Lüders measurements. Further we consider
the two reduced states of $\rho_{12}$:

$\displaystyle\rho_{1}$ $\displaystyle=$
$\displaystyle\mbox{Tr}_{\mathcal{K}}\,\rho_{12}\stackrel{{\scriptstyle(\ref{rho12})}}{{=}}\sum_{r}A_{r}\,\rho\,A_{r}^{\ast},$
(93) $\displaystyle\rho_{2}$ $\displaystyle=$
$\displaystyle\mbox{Tr}_{\mathcal{H}}\,\rho_{12}\stackrel{{\scriptstyle(\ref{rho12})}}{{=}}\sum_{r}\mbox{Tr}\left(A_{r}\,\rho\,A_{r}^{\ast}\right)\,|r\rangle\langle
r|\stackrel{{\scriptstyle(\ref{eff1})}}{{=}}\sum_{r}\mbox{Tr}\left(\rho\,F_{r}\right)\,|r\rangle\langle
r|=\sum_{r}p_{r}\,|r\rangle\langle r|\;.$ (94)

Due to the subadditivity of the von Neumann entropy, see NC00 , 11.3.4, we
conclude

$S(\rho)\stackrel{{\scriptstyle(\ref{Srho12})}}{{\leq}}S(\rho_{12})\leq
S(\rho_{1})+S(\rho_{2})\;,$ (95)

see also Proposition 1, and hence

$\Delta S=S(\rho)-S(\rho_{1})\leq S(\rho_{2})=H(\rho,F)\;.$ (96)

The latter equation follows from the spectral composition of $\rho_{2}$ due to
(48) which implies

$S(\rho_{2})=-\sum_{r}p_{r}\,\log
p_{r}\stackrel{{\scriptstyle(\ref{ShannonExp})}}{{=}}H(\rho,F)\;,$ (97)

thereby completing the proof of Theorem 1. $\Box$

###  B.3 Proof of Corollary 1

Turning to the proof of Corollary 1 we assume that the instrument
$\mathfrak{I}$ can be written as a convex sum of pure instruments, i. e.,

$\mathfrak{I}(n)(\rho)=\sum_{i\in
I}\lambda_{i}\,\mathfrak{I}^{(i)}(n)(\rho)\;,$ (98)

for all $n\in{\mathcal{N}}$ and $\rho\in B_{1}^{+}({\mathcal{H}})$, such that

$\lambda_{i}>0\quad\mbox{for all }i\in I\quad\mbox{and}\quad\sum_{i\in
I}\lambda_{i}=1\;.$ (99)

We will apply Theorem 1 for each pure instrument $\mathfrak{I}^{(i)}$ and
obtain

$\Delta S^{(i)}=S(\rho)-S\left(\rho_{1}^{(i)}\right)\leq
H\left(p^{(i)}\right)\;,$ (100)

using some self-explaining notation. In particular, the Shannon entropy
$H\left(p^{(i)}\right)$ is calculated for the probability distribution

$p_{n}^{(i)}=\mbox{Tr}\left(\mathfrak{I}^{(i)}({n})(\rho)\right)\;,$ (101)

satisfying

$\sum_{n}p_{n}^{(i)}=1\;.$ (102)

Moreover,

$\sum_{i}\lambda_{i}\,p_{n}^{(i)}\stackrel{{\scriptstyle(\ref{PR1},\ref{PR6})}}{{=}}\mbox{Tr}\left(\mathfrak{I}(n)(\rho)\right)=p_{n}\;,$
(103)

for all $n\in{\mathcal{N}}$. Due to concavity of the Shannon entropy, see NC00
, Ex. 11.21, (103) implies

$H(\rho,F)=H(p)\geq\sum_{i}\lambda_{i}\,H\left(p^{(i)}\right)\;.$ (104)

Similarly, concavity of the von Neumann entropy, see NC00 , 11.3.5., yields

$S(\rho_{1})=S\left(\mathfrak{I}({\mathcal{N}})(\rho)\right)\stackrel{{\scriptstyle(\ref{PR1})}}{{=}}S\left(\sum_{i}\lambda_{i}\mathfrak{I}^{(i)}({\mathcal{N}})(\rho)\right)\geq\sum_{i}\lambda_{i}S\left(\mathfrak{I}^{(i)}({\mathcal{N}})(\rho)\right)=\sum_{i}\lambda_{i}S\left(\rho_{1}^{(i)}\right)\;.$
(105)

Finally,

$\Delta
S=S(\rho)-S\left(\rho_{1}\right)\stackrel{{\scriptstyle(\ref{PR10})}}{{\leq}}S(\rho)-\sum_{i}\lambda_{i}\,S\left(\rho_{1}^{(i)}\right){=}\sum_{i}\lambda_{i}\,\left(S(\rho)-S\left(\rho_{1}^{(i)}\right)\right)\stackrel{{\scriptstyle(\ref{PR5})}}{{\leq}}\sum_{i}\lambda_{i}\,H\left(p^{(i)}\right)\stackrel{{\scriptstyle(\ref{PR9})}}{{\leq}}H(p)\;,$
(106)

thereby completing the proof of Corollary 1. $\Box$

###  B.4 Proof of Proposition 3

By setting $A_{n}=U_{n}\,P_{n}$, resp. $A_{n}=P_{n}$, for all
$n\in{\mathcal{N}}$, we obtain from (91):

$\rho_{12}=\sum_{n}\left(U_{n}P_{n}\rho
P_{n}U_{n}^{\ast}\right)\otimes|n\rangle\langle
n|=:\sum_{n}p_{n}\,\rho_{n}\otimes|n\rangle\langle n|\;,$ (107)

where $p_{n}:=\mbox{Tr}\left(\rho\,P_{n}\right)$, and

$\rho_{12}^{\prime}=\sum_{n}\left(P_{n}\rho
P_{n}\right)\otimes|n\rangle\langle n|\;.$ (108)

Moreover, by means of (48),

$\rho_{2}=\rho_{2}^{\prime}=\sum_{n}p_{n}|n\rangle\langle n|\;,$ (109)

and hence

$S_{2}=S_{2}^{\prime}=H(p)=H(\rho,P)\;.$ (110)

Since the $\rho_{n}\otimes|n\rangle\langle n|$ in (107) as well as the
$\left(P_{n}\rho P_{n}\right)\otimes|n\rangle\langle n|$ in (108) have
orthogonal support, theorem 11.8 (4) of NC00  can be applied and yields:

$\displaystyle S_{12}$ $\displaystyle=$ $\displaystyle
S\left(\rho_{12}\right)=\sum_{n}p_{n}S\left(\rho_{n}\otimes|n\rangle\langle
n|\right)+H(p)$ (111) $\displaystyle=$
$\displaystyle\sum_{n}p_{n}S\left(\frac{1}{p_{n}}U_{n}P_{n}\rho
P_{n}U_{n}^{\ast}\right)+H(p)$ (112) $\displaystyle=$
$\displaystyle\sum_{n}p_{n}S\left(\frac{1}{p_{n}}P_{n}\rho P_{n}\right)+H(p)$
(113) $\displaystyle=$ $\displaystyle S_{12}^{\prime}\;,$ (114)

using the invariance of von Neumann entropy under unitary transformations in
(113). Finally,

$\displaystyle\Delta{\mathcal{I}}$
$\displaystyle\stackrel{{\scriptstyle(\ref{OLRa}-\ref{OLRc})}}{{=}}$
$\displaystyle\left(S_{1}^{\prime}+S_{2}^{\prime}-S_{12}^{\prime}\right)-\left(S_{1}+S_{2}-S_{12}\right)$
(115) $\displaystyle\stackrel{{\scriptstyle(\ref{S12d})}}{{=}}$ $\displaystyle
S_{1}^{\prime}+S_{2}^{\prime}-S_{1}-S_{2}\stackrel{{\scriptstyle(\ref{S2p})}}{{=}}S_{1}^{\prime}-S_{1}$
(116) $\displaystyle\geq$ $\displaystyle S_{0}-S_{1}=\Delta S\;,$ (117)

where we have used $S_{1}^{\prime}\geq S_{0}$ in (117) since a total Lüders
operation never decreases entropy. This completes the proof of Proposition 3.
$\Box$

###### Acknowledgements.

I thank all members of the DFG Research Unit FOR 2692 as well as Thomas
Bröcker for stimulating and insightful discussions.

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