#  Subdifferential sets of operators

T. S. S. R. K. Rao Department of Mathematics
Shiv Nadar University
Delhi (NCR)
India, E-mail<EMAIL_ADDRESS>
###### Abstract.

Let $X,Y$ be complex Banach spaces. Let ${\mathcal{L}}(X,Y)$ be the space of
bounded operators . An important aspect of understanding differentiability is
to study the subdifferential of the norm at a point, say $x\in X$, this is the
set, $\\{f\in X^{\ast}:\|f\|=1~{},f(x)=\|x\|\\}$. See page 7 in [2]. Motivated
by recent results of Singla ([10]) in the context of Hilbert spaces, for
$T\in{\mathcal{L}}(X,Y)$, we determine the subdifferential of the operator
norm at $T$,
$\partial_{T}=\\{\Lambda\in{\mathcal{L}}(X,Y)^{\ast}:\Lambda(T)=\|T\|~{},~{}\|\Lambda\|=1\\}$.
Our approach is based on the ‘position’ of the space of compact operators and
the Calkin norm of $T$. Our ideas give a unified approach and extend several
results from [10] to the case of $\ell^{p}$-spaces for $1<p<\infty$. We also
investigate the converse, using the structure of the subdifferential set to
decide when the Calkin norm is a strict contraction. As an application of
these ideas, we partially solve the open problem of relating the
subdifferential of the operator norm at a compact operator $T$ to that of
$T(x_{0})$, where $x_{0}$ is a unit vector where $T$ attains its norm.

###### Key words and phrases:

Subdifferential of the norm, Spaces of operators, tensor product spaces

###### 2000 Mathematics Subject Classification:

Primary 47 L 05, 46 B28, 46B25

##  1\.  Introduction

Let $X,Y$ be complex Banach spaces. An interesting problem in operator theory
is to determine standard Banach space theoretic sets in ${\mathcal{L}}(X,Y)$,
based on geometric properties of $X,Y$ and their duals.

Let $X_{1}$ denote the closed unit ball and let $\partial_{e}X^{\ast}_{1}$ be
the set of extreme points of the dual unit ball. In the context of dual
spaces, closure operation is always with respect to the weak∗-topology. For
notational simplicity, we call the subdifferential of the norm at a vector
$x$, $\partial_{x}=\\{f\in X^{\ast}:\|f\|=1,~{}f(x)=\|x\|\\}$, as the
subdifferential set, with the vector under consideration appearing in the
subscript . This notation is consistent with [10]. Study of these sets in
various categories of Banach spaces has long history (an extensive
bibliography is available in [11]) and we only quote the results of Taylor and
Werner (see [12] and [13]) to illustrate.

For $y\in X$, the relation to the directional derivative of the norm at $x$ is
given by the formula (see [5] and [2], pages 7,8)

$lim_{t\rightarrow
0^{+}}\frac{\|x+ty\|-\|x\|}{t}=sup\\{re(f(y)):f\in\partial_{x}\\}.$

We recall that $\partial_{x}$ is an extreme, weak∗-closed convex subset of
$X^{\ast}_{1}$ ( called a face, see [1] for basic convexity theory) and
therefore the set of extreme points,
$\partial_{e}(\partial_{x})=\partial_{x}\cap\partial_{e}X^{\ast}_{1}$. Thus we
can replace the functionals in the formula by extreme points. The idea now is
to replace the functionals in the RHS set by those coming from an appropriate
subspace of $X$, where the extreme functionals have specific description. Note
that the limit on LHS is independent of this manipulation, which is the prime
motivation of this approach.

For $x^{\ast\ast}\in X^{\ast\ast}$, $y^{\ast}\in Y^{\ast}$,
$x^{\ast\ast}\otimes y^{\ast}$ denotes the functional $(x^{\ast\ast}\otimes
y^{\ast})(T)=x^{\ast\ast}(T^{\ast}(y^{\ast}))$. We recall from [4] (Theorem
VI.1.3) that for any subspace $J$ of compact operators ${\mathcal{K}}(X,Y)$ ,
$\partial_{e}J^{\ast}_{1}\subset\\{x^{\ast\ast}\otimes
y^{\ast}:x^{\ast\ast}\in\partial_{e}X^{\ast\ast}_{1},~{}y^{\ast}\in\partial_{e}Y^{\ast}_{1}\\}$
and equality holds if $J$ also contains all finite rank operators. For
$A\in{\mathcal{L}}(X,Y)$, such that $d(A,{\mathcal{K}}(X,Y))<1$, our main
result is to determine $\partial_{A}$ depending on the position of the space
of compact operators.

We recall from Chapter I of [4], that a closed subspace $J\subset X$ is said
to be a $M$-ideal, if there is a linear projection $P:X^{\ast}\rightarrow
X^{\ast}$ such that $ker(P)=J^{\bot}$ and
$\|x^{\ast}\|=\|P(x^{\ast})\|+\|x^{\ast}-P(x^{\ast})\|$ for all $x^{\ast}\in
X^{\ast}$. When this happens, functionals in $J^{\ast}$ have unique norm-
preserving extension in $X^{\ast}$ and $J^{\ast}$ is canonically identified
with the range of $P$ so that $X^{\ast}=J^{\bot}\bigoplus_{1}J^{\ast}$,
consequently
$\partial_{e}X^{\ast}_{1}=\partial_{e}J^{\ast}_{1}\cup\partial_{e}J^{\bot}_{1}$.

See [4] for examples of $M$-ideals from function spaces, Banach algebras and
spaces of operators. In particular Chapter VI deals with examples where
${\mathcal{K}}(X,Y)$ is a $M$-ideal in ${\mathcal{L}}(X,Y)$, which includes,
${\mathcal{K}}(\ell^{p})$ in ${\mathcal{L}}(\ell^{p})$ for $1<p<\infty$ and
${\mathcal{K}}(X,c_{0})$ is a $M$-ideal in ${\mathcal{L}}(X,c_{0})$ as well as
$\bigoplus_{\infty}X$, for any Banach space $X$. Note also $M$-ideals in
$C^{\ast}$-algebras are precisely closed two-sided ideals. This explains the
overlap with [10] and gives possible extensions of the results there beyond
$C^{\ast}$-algebras. See also [11].

We investigate conditions on $X,Y$ which will allow us to determine the Calkin
norm (i.e., the quotient norm of ${\mathcal{L}}(X,Y)/{\mathcal{K}}(X,Y)$) of
an operator, knowing the behaviour of the subdifferential set. We show that
under the general conditions assumed here, if $0$ is not a weak∗ accumulation
point of $\partial_{e}X^{\ast\ast}_{1}~{},~{}\partial_{e}Y^{\ast}_{1}$, then
the Calkin norm indeed gets determined.

Since the subdifferential set $\partial_{x}$ can be used to evaluate the
limit, for $y\in X$, $lim_{t\rightarrow 0^{+}}\frac{\|x+ty\|-\|x\|}{t}$, a
natural question is, when $X$ is canonically embedded in its bidual
$X^{\ast\ast}$, if one can replace $y$ in the above formula, by $\tau\in
X^{\ast\ast}$, while retaining $\partial_{x}$ in the RHS? We partially answer
this, using the notion of point of norm-weak upper semi-continuity of the
preduality map on $X^{\ast\ast}$ (see the preamble before Proposition 11).

We also show that for the special classes of compact operators
${\mathcal{K}}(X,Y)$, when they form a $M$-ideal in ${\mathcal{L}}(X,Y)$ and
${\mathcal{L}}(X,Y)$ is the bidual of ${\mathcal{K}}(X,Y)$ (for example, when
$X$ and $Y$ are $\ell^{p}$-spaces, $1<p<\infty$), the subdifferential
continues to behave well in all higher even order duals of
${\mathcal{K}}(X,Y)$.

Another interesting problem is to relate the subdifferential of the operator
norm at $T$ to that of $T(x_{0})$, where $x_{0}$ is a point where $T$ attains
its norm. Using some recent work from [9], we give conditions when this can be
solved for a $T\in{\mathcal{K}}(X,Y)$, for a reflexive space $X$, $J\subset Y$
is a $M$-ideal.

For a compact Hausdorff space $\Omega$, let $C(\Omega,Y)$ be the space of
$Y$-valued continuous functions, equipped with the supremum norm. For
$\omega\in\Omega$, $y^{\ast}\in Y^{\ast}$, $\delta(\omega)\otimes y^{\ast}$
denotes the functional $(\delta(\omega)\otimes y^{\ast})(f)=y^{\ast}(f(w))$.
For a $M$-ideal $J\subset Y$ and $f\in C(\Omega,Y)$, if
$\partial_{f}=\overline{CO}\\{\delta(\omega)\otimes
y^{\ast}:y^{\ast}(f(\omega))=\|f\|,~{}\omega\in\Omega~{},y^{\ast}\in\partial_{e}J^{\ast}_{1}\\}$,
(here and elsewhere, $CO$ stands for the convex hull of the set in $\\{.\\}$),
when $0$ is not a weak∗-closed accumulation point of
$\partial_{e}Y^{\ast}_{1}$, there is a $\omega_{0}\in\Omega$ such that
$\|f(\omega_{0})\|=\|f\|$ and
$\partial_{f(\omega_{0})}=\overline{CO}\\{y^{\ast}\in\partial_{e}J^{\ast}_{1}:y^{\ast}(f(\omega_{0}))=\|f\|\\}$.

##  2\. Geometry of the subdifferential set

We first give a general form of Theorem 1.4 from [10] (see also the Zbl review
by this author). Let $J\subset X$ be a closed subspace. We note that if $x\in
J$, then relative to $J$,
$\partial_{x}=\overline{CO}\\{f\in\partial_{e}J^{\ast}_{1}:f(x)=1\\}$, where
the closure is in the weak∗-topology of $J^{\ast}$.

######  Theorem 1.

Let $J\subset X$ be a $M$-ideal. Let $x\in X$ be a unit vector such that
$d(x,J)<1$. Then
$\partial_{x}=\overline{CO}\\{f\in\partial_{e}J^{\ast}_{1}:f(x)=1\\}$. Here
the closure is taken in the weak∗-topology of $X^{\ast}$. In particular
$\partial_{x}$ has the same description as when $x\in J$. Thus for $y\in X$,

$lim_{t\rightarrow
0^{+}}\frac{\|x+ty\|-\|x\|}{t}=sup\\{re(f(y)):f\in\partial_{e}J^{\ast}_{1},~{}f(x)=1\\}.$

###### Proof.

Since $J$ is a $M$-ideal, we have, $X^{\ast}=J^{\ast}\bigoplus_{1}J^{\bot}$.
Thus if the set $\\{f\in\partial_{e}J^{\ast}_{1}:f(x)=1\\}$ is non-empty, then
it is contained in $\partial_{x}$.

Since $\partial_{x}$ is a weak∗-closed extreme convex subset of
$X^{\ast}_{1}$, we will show that any extreme point $g\in\partial_{x}$ is in
$J^{\ast}$ . This completes the proof.

Since $g\in\partial_{e}X^{\ast}_{1}$, in view of decomposition of $X^{\ast}$,
suppose $g\in J^{\bot}$. Now $g(x)=1$ implies $d(x,J)=1$. Therefore,
$g\in\partial_{e}J^{\ast}_{1}$. ∎

######  Remark 2.

Since $y$ is weak∗-continuous on $J^{\ast}$, when $\partial_{e}J^{\ast}_{1}$
is weak∗-closed in the weak∗-topology of $J^{\ast}$, we have

$lim_{t\rightarrow 0^{+}}\frac{\|x+ty\|-\|x\|}{t}=re(f_{0}(y))$

for some $f_{0}\in\partial_{e}J^{\ast}_{1}$, $f_{0}(x)=1$. This in particular
always happens (under the above hypothesis) when $\partial_{e}X^{\ast}_{1}$ is
weak∗-closed.

The following corollary follows immediately from our specific knowledge of
extreme points of ${\mathcal{K}}(X,Y)^{\ast}_{1}$.

######  Corollary 3.

Let $X,Y$ be Banach spaces and let
$J\subset{\mathcal{K}}(X,Y)\subset{\mathcal{L}}(X,Y)$ be a $M$-ideal in
${\mathcal{L}}(X,Y)$. For any $A\in{\mathcal{L}}(X,Y)$ such that $d(A,J)<1$,
$\partial_{A}=\overline{CO}\\{x^{\ast\ast}\otimes
y^{\ast}\in\partial_{e}J^{\ast}_{1}:x^{\ast\ast}(A^{\ast}(y^{\ast}))=1,~{}x^{\ast\ast}\in\partial_{e}X^{\ast\ast}_{1}~{},y^{\ast}\in\partial_{e}Y^{\ast}_{1}\\}$.

######  Remark 4.

In particular we have $A^{\ast}$ attains its norm and hence so are all the
adjoints of higher order of $A$. More generally we note that if a
$B\in{\mathcal{L}}(X,Y)$, is such that $B^{\ast}$ attains its norm, then it
attains it at an extreme point $y^{\ast}\in Y^{\ast}_{1}$ (see [7]). Since
$\|B\|=\|B^{\ast}(y^{\ast})\|$, let
$x^{\ast\ast}\in\partial_{e}X^{\ast\ast}_{1}$ be such that
$x^{\ast\ast}(B^{\ast}(y^{\ast}))=(x^{\ast\ast}\otimes y^{\ast})(B)=\|B\|$.
Now if the $J$ above contains all finite rank operators, $x^{\ast\ast}\otimes
y^{\ast}\in\partial_{e}J^{\ast}_{1}$. It may be worth recalling here that the
set of operators whose adjoint attains its norm is dense in
${\mathcal{L}}(X,Y)$, see [14]. When $X=Y=\ell^{p}$ for $1<p<\infty$,
$J={\mathcal{K}}(\ell^{p})$, we have for $A\in{\mathcal{L}}(\ell^{p})$ with
$d(A,{\mathcal{K}}(\ell^{p}))<\|A\|$, $\partial_{A}=\overline{CO}\\{x\otimes
y:A(x)(y)=\|A\|,~{}x\in\ell^{p}_{1}~{},y\in\ell^{q}_{1}\\}$, thus extending
Theorem 1.4 from [10].

Let $Z\subset Y$ be a closed subspace. The following proposition identifies
subdifferential limits of vectors in $X/Z$. In what follows, by
$\pi:X\rightarrow X/Z$ we denote the quotient map.

######  Proposition 5.

Let $Z\subset Y\subset X$ be Banach spaces. Suppose $Y$ is a $M$-ideal in $X$
and $x\in X$ is such that $d(x,Y)<d(x,Z)$. Then for $x^{\prime}\in X$,

$lim_{t\rightarrow 0^{+}}\frac{\|\pi(x)+t\pi(x^{\prime})\|-\|\pi(x)\|}{t}$
$=sup\\{re(f(x^{\prime})):f\in\partial_{e}(Y^{\ast}\cap
Z^{\bot})_{1}~{},~{}f(x)=1\\}.$

###### Proof.

By Proposition I.1.7 in [4], we have that $Y/Z$ is a $M$-ideal in $X/Z$. Hence
$Z^{\bot}=Y^{\bot}\bigoplus_{1}(Y^{\ast}\cap Z^{\bot})$. For any $y\in Y$,
$\|\pi(x)-\pi(y)\|=\|\pi(x-y)\|\leq\|x-y\|$. Thus $d(\pi(x),Y/Z)\leq
d(x,Y)<d(x,Z)=\|\pi(x)\|$. Now identifying $(Y/Z)^{\ast}=Y^{\ast}\cap
Z^{\bot}\subset(X/Z)^{\ast}$, the conclusion follows from Theorem 1. ∎

We next consider the question, suppose $J$ is a $M$-ideal in $X$ and the
formula for $\partial_{x}$ is valid, when is $d(x,J)<1?$. The following
example illustrates how it can depend on the choice of the $M$-ideal.

######  Example 6.

Consider $c_{0}\subset\ell^{\infty}$, a well known example of a $M$-ideal
(algebraic ideal) of $\ell^{\infty}$. Let $x\in\ell^{\infty}$ be a unit
vector, with all but finitely many coordinates coming from the unit circle
$\Gamma$, the rest from the open disc. Using the identification of
$\ell^{\infty}=C(\beta({\bf N}))$, we see that
$\partial_{e}(\partial_{x})\subset\Gamma\\{\delta(\tau):\tau\in\beta({\bf
N})\\}$. Thus we get
$\partial_{x}=\overline{CO}\\{\overline{t}\delta(k):k\in{\bf
N}~{},t=x(k)~{},|x(k)|=1\\}$, where $\delta(\tau)$ is the Dirac measure.
Clearly $\delta(k)\in\partial_{e}(\ell^{1}=c_{0}^{\ast})_{1}$. Now for the
ideal
$J=\\{\alpha\in\ell^{\infty}:\alpha(k)=0~{}for~{}k~{}with~{}|x(k)|<1\\})$, $J$
determines $\partial_{x}$, as well as $c_{0}$. However $d(x,c_{0})=1$ and
$d(x,J)<1$.

We give two positive answers depending on the weak∗-closure of
$\partial_{e}J^{\ast}_{1}$ in $J^{\ast}$ and $X^{\ast}$.

######  Theorem 7.

Suppose for $X,Y,J$ as in Theorem 1, Suppose
$\partial_{x}=\overline{CO}\\{f\in\partial_{e}J^{\ast}_{1}:f(x)=1\\}$.

  1. (1)

Suppose $0\notin\overline{\partial_{e}J^{\ast}_{1}}$, closure in the
weak∗-topology of $J^{\ast}$.

  2. (2)

Suppose
$\overline{\partial_{e}J^{\ast}_{1}}\subset[0,1]\partial_{e}J^{\ast}_{1}$,
closure taken on the weak∗-topology of $X^{\ast}$.

then $d(x,J)<1$.

###### Proof.

To see this assuming 1), if $d(x,J)=1$, choose $\tau\in
J^{\bot}_{1}\cap\partial_{e}X^{\ast}$ such that $\tau(x)=1$ . Since
$\tau\in\partial_{x}$ by Milman’s converse of the Krein-Milman theorem,
$\tau\in\overline{\\{f\in\partial_{e}J^{\ast}_{1}:f(x)=1\\}}$, closure in the
weak∗-topology of $X^{\ast}$. Let $\\{f_{\alpha}\\}$ be a net from the set in
RHS such that $f_{\alpha}\rightarrow\tau$ in the weak∗-topology of $X^{\ast}$.
Since $\tau(J)=0$, we get a contradiction. So $d(x,J)<1$.

Now suppose we have 2). Since $\tau\in\overline{\partial_{e}J^{\ast}_{1}}$,
for the net $\\{f_{\alpha}\\}$ as in the proof of 1), we get by hypothesis in
2), $\tau=\lambda h$ for $h\in\partial_{e}J^{\ast}_{1}$, $\lambda\in[0,1]$.
Since $\|\tau\|=1=\|h\|$, we get $\lambda=1$. Hence our claim follows as
before. ∎

######  Remark 8.

Examples of spaces that satisfy above extremal conditions include spaces of
the form $\\{f\in
C(\Omega):f(t_{\alpha})=\lambda_{\alpha}f(s_{\alpha})~{}for~{}all~{}\alpha\in\Delta\\}$.
Here $\Omega$ is a compact Hausdorff space, $\Delta$ is any index set,

$\\{t_{\alpha}\\}_{\alpha\in\Delta}~{},~{}\\{s_{\alpha}\\}_{\alpha\in\Delta}\subset\Omega~{}and~{}\\{\lambda_{\alpha}\\}_{\alpha\in\Delta}\subset[-1,1].$

We next consider conditions specific to spaces of compact operators. We recall
that in the case of $\ell^{p}$-spaces, $0$ is a weak-accumulation point of the
unit sphere.

######  Proposition 9.

Suppose $J\subset{\mathcal{K}}(X,Y)\subset{\mathcal{L}}(X,Y)$ is a $M$-ideal
in ${\mathcal{L}}(X,Y)$ and it contains all finite rank operators. Assume
further, $0$ is not a weak∗ extreme point of $\partial_{e}X^{\ast\ast}_{1}$
and $\partial_{e}Y^{\ast}_{1}$. Let $A\in{\mathcal{L}}(X,Y),~{}\|A\|=1$ and
$\partial_{A}=\overline{CO}\\{x^{\ast\ast}\otimes
y^{\ast}:x^{\ast\ast}(A^{\ast}(y^{\ast}))=1~{},~{}x^{\ast\ast}\in\partial_{e}X^{\ast\ast}_{1},~{}y^{\ast}\in\partial_{e}Y^{\ast}_{1}\\}$.
Then $d(A,J)<1$.

###### Proof.

We will show that $0\notin\overline{\partial_{e}J^{\ast}_{1}}$. If not, in
view of the description of the extreme points, we get two nets indexed by the
same set, $\\{x^{\ast\ast}_{\alpha}\\}\subset\partial_{e}X^{\ast\ast}_{1}$ and
$\\{y^{\ast}_{\alpha}\\}\subset\partial_{e}Y^{\ast}_{1}$ such that the
functionals $x^{\ast\ast}_{\alpha}\otimes y^{\ast}_{\alpha}\rightarrow 0$ in
the weak∗-topology of $J^{\ast}$. Using weak∗-compactness, by going through
appropriate subnets, if necessary, we may and do assume that
$x^{\ast\ast}_{\alpha}\rightarrow x^{\ast\ast}$ and
$y^{\ast}_{\alpha}\rightarrow y^{\ast}$ in the weak∗-topology, for some
$x^{\ast\ast}\neq 0\neq y^{\ast}$. For any $T\in J$, since $T$ is compact,
$T^{\ast}(y^{\ast}_{\alpha})\rightarrow T^{\ast}(y^{\ast})$ in the norm.
Consequently $x^{\ast\ast}_{\alpha}(T^{\ast}(y^{\ast}_{\alpha}))\rightarrow
x^{\ast\ast}(T^{\ast}(y^{\ast}))$. Therefore $x^{\ast\ast}_{\alpha}\otimes
y^{\ast}_{\alpha}\rightarrow x^{\ast\ast}\otimes y^{\ast}$ in the
weak∗-topology of $J^{\ast}$. So $x^{\ast\ast}\otimes y^{\ast}=0$ on $J$ . A
contradiction since $J$ contains all operators of rank one. ∎

We next apply this to the situation when $X$ is a $M$-ideal in the canonical
embedding, in its bidual $X^{\ast\ast}$, the so called $M$-embedded spaces.
See Chapter III of [4], for examples among spaces of operators, particularly
when for reflexive Banach spaces $X,Y$ ${\mathcal{L}}(X,Y)$ is the canonical
bidual of ${\mathcal{K}}(X,Y)$ (like in the case of Hilbert spaces and
$\ell^{p}$-spaces).

######  Proposition 10.

Suppose $X$ is a $M$-ideal in $X^{\ast\ast}$. Let $\tau\in X^{\ast\ast}$ be
such that $d(\tau,X)<\|\tau\|$, then $\tau$ attains its norm. The set
$\partial\tau$ has the same description in all higher odd ordered duals of
$X$.

###### Proof.

Since $X$ is a $M$-ideal in $X^{\ast\ast}$, the canonical projection
$\Lambda\rightarrow\Lambda|X$ is a $L$-projection, thus
$X^{\ast\ast\ast}=X^{\ast}\bigoplus_{1}X^{\bot}$ for the canonical embedding
of $X^{\ast}$ in $X^{\ast\ast\ast}$. We see that the hypothesis implies there
is a $g\in\partial_{e}X^{\ast}_{1}$ such that $\tau(g)=1$. The main result of
[8] says that under the hypothesis, $X$ continues to be a $M$-ideal of all
higher even ordered duals of $X$. Hence the conclusion follows. ∎

Let $X$ be canonically embedded in its bidual, $X^{\ast\ast}$. Another
interesting question is to consider when the equality,

$lim_{t\rightarrow
0^{+}}\frac{\|x+t\tau\|-\|x\|}{t}=sup\\{re(\tau(f)):f\in\partial_{x}\\}$

holds for any $\tau\in X^{\ast\ast}$.

Our next proposition answers this question using the notion of the preduality
mapping on $X^{\ast\ast}$ (see [3]). For $x\in X$ considered as an element of
$X^{\ast\ast}$, we let $\partial^{\prime}_{x}=\\{\tau\in
X^{\ast\ast\ast}_{1}:\tau(x)=\|\tau\|\\}$.

We recall that for a norm attaining functional $\Lambda\in X^{\ast\ast}$,
attaining its norm at $f_{0}\in X^{\ast}_{1}$ is a point of norm-weak upper
semi-continuity for the preduality map, if for any weak neighbourhood $V$ of
$0$ in $X^{\ast}$, there is a $\delta>0$ such that for any
$\Lambda^{\prime}\in X^{\ast\ast}$, $\|\Lambda^{\prime}-\Lambda\|<\delta$,
$\\{g\in
X^{\ast}_{1}:\Lambda^{\prime}(g)=\|\Lambda^{\prime}\|\\}\subset\\{h\in
X^{\ast}_{1}:\Lambda(h)=\|\Lambda\|\\}+V$. Part of our next proposition is
perhaps folklore, as we are unable to find a reference, we include its easy
proof. It is well known that any unitary in a $C^{\ast}$\- algebra satisfies
the condition assumed in the following proposition.

######  Proposition 11.

Let $x\in X$ be a point of norm-weak upper semi-continuity for the preduality
mapping of $X^{\ast\ast}$. Then

$lim_{t\rightarrow
0^{+}}\frac{\|x+t\tau\|-\|x\|}{t}=sup\\{re(\tau(f)):f\in\partial_{x}\\}$

for any $\tau\in X^{\ast\ast}$.

###### Proof.

It follows from Lemma 2.2 in [3] that the hypothesis implies $\partial_{x}$ is
weak∗-dense (in the weak∗-topology of $X^{\ast\ast\ast}$) in
$\partial^{\prime}_{x}$. Since $\tau$ is weak∗-continuous on
$X^{\ast\ast\ast}$, the conclusion follows.

∎

######  Remark 12.

We note that if $J\subset X$ is a $M$-ideal, then $J$ need not be a $M$-ideal
in $X^{\ast\ast}$ ( it is easy to see that in $C([0,1])$, the ideal $J=\\{f\in
C([0,1]):f([0,\frac{1}{2}])=0\\}$ is not an ideal in $C([0,1])^{\ast\ast}$.

##  3\. Application

As an application of the previous set of ideas, we formulate the compact
operator version of the operator norm subdifferential problem and provide a
partial solution.

Let $X$ be a reflexive Banach spaces and suppose $Y$ has the metric
approximation property. Suppose $J\subset Y$ be a $M$-ideal and let
$T\in{\mathcal{K}}(X,Y)$ be such that $\partial_{T}=\overline{CO}\\{x\otimes
y^{\ast}:y^{\ast}(T(x))=(x\otimes
y^{\ast})(T)=\|T\|,~{}~{}x\in\partial_{e}X_{1},~{}y^{\ast}\in\partial_{e}J^{\ast}_{1}\\}$.
An interesting question that arises from the previous section is that, when
can one get a $x_{0}\in\partial_{e}X_{1}$ such that $\|T\|=\|T(x_{0})\|$ and
$\partial_{T(x_{0})}=\overline{CO}\\{y^{\ast}\in\partial_{e}J^{\ast}_{1}:y^{\ast}(T(x_{0}))=\|T(x_{0})\|\\}$.

Since $Y$ and hence $J$ have the metric approximation property, we have that
${\mathcal{K}}(X,J)$ is a $M$-ideal in ${\mathcal{K}}(X,Y)$ (see [4]
Proposition VI.3.1). Thus from the structure of extreme points of the dual
unit ball of spaces of compact operators, we see that the description of
$\partial_{T}$, as before, is in terms of the extreme points of the dual unit
ball of the $M$-ideal ${\mathcal{K}}(X,J)$.

In what follows we will be using a minimax formula for compact operators. This
author has recently proved ([9]), Theorem 1) the validity of this when $J$ is
a $L^{1}$-predual space (i.e., $J^{\ast}$ is isometric to a
$L^{1}(\mu)$-space, for example, the spaces considered in Remark 8).

######  Theorem 13.

Let $X,Y,J$ be as above. Assume that $X$ is a separable space, $0$ is not a
weak accumulation point of $\partial_{e}X_{1}$ and $0$ is not a
weak∗-accumulation point of $\partial_{e}Y^{\ast}_{1}$ and suppose the compact
operator $T$ satisfies the minimax formula,
$d(T,{\mathcal{K}}(X,J))=sup\\{d(T(x),J):x\in X_{1}\\}$. If
$\partial_{T}=\overline{CO}\\{x\otimes y^{\ast}:(x\otimes
y^{\ast})(T)=\|T\|,~{}x\in\partial_{e}X_{1}~{},~{}y^{\ast}\in\partial_{e}J^{\ast}_{1}\\}$,
then there is a $x_{0}\in\partial_{e}X_{1}$ such that, $\|T\|=\|T(x_{0})\|$,
$\partial_{T(x_{0})}=\overline{CO}\\{y^{\ast}\in\partial_{e}Y^{\ast}_{1}:y^{\ast}(T(x_{0}))=\|T(x_{0})\|\\}$.

###### Proof.

Since $X$ is reflexive and $T$ is compact, let $x_{0}\in\partial_{e}X_{1}$ be
such that $\|T\|=\|T(x_{0})\|$. Since ${\mathcal{K}}(X,J)$ is a $M$-ideal in
${\mathcal{K}}(X,Y)$, by our arguments given in the proof of Proposition 9,
and our hypothesis implies, $d(T,{\mathcal{K}}(X,J))<\|T\|$. Now by the
minimax formula, $d(T(x_{0}),J)<\|T\|=\|T(x_{0})\|$. Now since $J$ is a
$M$-ideal, we get the description for $\partial_{T(x_{0})}$. ∎

For a compact Hausdorff space $\Omega$, let $C(\Omega,Y)$ be the space of
$Y$-valued continuous functions, equipped with the supremum norm. The minimax
formula above was inspired by a classical result of Light and Cheney (see
[6]), for a closed subspace $J\subset Y$ and $f\in C(\Omega,Y)$,

$d(f,C(\Omega,J))=sup\\{d(f(w),J):w\in\Omega\\}.$

Analogous to the above theorem, if $J$ is a $M$-ideal in $Y$, then
$C(\Omega,J)$ is a $M$-ideal in $C(\Omega,Y)$ (see [4]). The following
corollary, which assumes these conditions, is now easy to prove. For
$\omega\in\Omega$, $\delta(\omega)$ denotes the Dirac measure at $\omega$.

######  Corollary 14.

Suppose $0$ is not a weak∗-accumulation point of $\partial_{e}Y^{\ast}_{1}$.
Let $f\in C(\Omega,Y)$ and suppose,
$\partial_{f}=\overline{CO}\\{\delta(\omega)\otimes
y^{\ast}:y^{\ast}(f(\omega))=\|f\|~{},~{}\omega\in\Omega~{},~{}y^{\ast}\in\partial_{e}J^{\ast}_{1}\\}$.
There exists a $\omega_{0}\in\Omega$ such that $\|f\|=\|f(\omega_{0})\|$,
$\partial_{f(\omega_{0})}=\overline{CO}\\{y^{\ast}\in\partial_{e}J^{\ast}_{1}:y^{\ast}(f(\omega_{0}))=\|f(\omega_{0})\|\\}$.

Let $WC(\Omega,Y)$ denote the space of continuous functions when $Y$ is
considered with the weak topology, equipped with the supremum norm. We recall
that for $f\in WC(\Omega,Y)$,
$\|f\|=Sup_{\omega\in\Omega,y^{\ast}\in\partial_{e}Y^{\ast}_{1}}|y^{\ast}(f(\omega))|$.
Thus by an application of the separation theorem,
$WC(\Omega,Y)^{\ast}_{1}=\overline{CO}\\{\delta(\omega)\otimes
y^{\ast}:\omega\in\Omega~{},~{}y^{\ast}\in\partial_{e}Y^{\ast}_{1}\\}$. Now
suppose $0$ is not a weak∗-accumulation point of $\partial_{e}Y^{\ast}_{1}$,
then as before we can conclude that for $f\in WC(\Omega,Y)$, if
$\Lambda\in\partial_{e}(\partial_{f})$, then
$\Lambda=\delta(\omega_{0})\otimes y_{0}^{\ast}$ for some
$\omega_{0}\in\Omega$, $y_{0}^{\ast}\in\partial_{e}Y^{\ast}_{1}$. In
particular, $\|f\|=\|f(\omega_{0})\|$. We do not know how to prove a version
of Corollary 14 in this set up.

The author thanks the referee for his/her remarks and the Journal
administration for their efficient handling of this article during the
pandemic.

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# Steady-state Quantum Thermodynamics with Synthetic Negative Temperatures

Mohit Lal Bera  ICFO - Institut de Ciències Fotòniques, The Barcelona
Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain.
Tanmoy Pandit<EMAIL_ADDRESS>Fritz Haber Research Center for
Molecular Dynamics, Hebrew University of Jerusalem, Jerusalem 9190401, Israel
Kaustav Chatterjee  Faculty of Physics, Arnold Sommerfeld Centre for
Theoretical Physics (ASC), Ludwig-Maximilians-Universität München,
Theresienstr. 37, 80333 München, Germany  Varinder Singh  Center for
Theoretical Physics of Complex Systems, Institute for Basic Science (IBS),
Daejeon 34126, Korea  Maciej Lewenstein  ICFO - Institut de Ciències
Fotòniques, The Barcelona Institute of Science and Technology, 08860
Castelldefels (Barcelona), Spain.  ICREA, Pg. Lluis Companys 23, ES-08010
Barcelona, Spain.  Utso Bhattacharya  ICFO - Institut de Ciències Fotòniques,
The Barcelona Institute of Science and Technology, 08860 Castelldefels
(Barcelona), Spain.  Manabendra Nath Bera<EMAIL_ADDRESS>Department of
Physical Sciences, Indian Institute of Science Education and Research (IISER),
Mohali, Punjab 140306, India

###### Abstract

A bath with a negative temperature is a subject of intense debate in recent
times. It raises fundamental questions not only on our understanding of
negative temperature of a bath in connection with thermodynamics but also on
the possibilities of constructing devices using such baths. In this work, we
study steady-state quantum thermodynamics involving baths with negative
temperatures. A bath with a negative temperature is created synthetically
using two baths of positive temperatures and weakly coupling these with a
qutrit system. These baths are then coupled to each other via a working
system. At steady-state, the laws of thermodynamics are analyzed. We find that
whenever the temperatures of these synthetic baths are identical, there is no
heat flow, which reaffirms the zeroth law. There is always a spontaneous heat
flow for different temperatures. In particular, heat flows from a bath with a
negative temperature to a bath with a positive temperature which, in turn,
implies that a bath with a negative temperature is ‘hotter’ than a bath with a
positive temperature. This warrants an amendment in the Kelvin-Planck
statement of the second law, as suggested in earlier studies. In all these
processes, the overall entropy production is positive, as required by the
Clausius statement of the second law. We construct continuous heat engines
operating between positive and negative temperature baths. These engines yield
maximum possible heat-to-work conversion efficiency, that is, unity. We also
study the thermodynamic nature of heat from a bath with a negative temperature
and find that it is thermodynamic work but with negative entropy.

##  I Introduction

Thermodynamics constitutes a fundamental building block of the modern
understanding of nature. With the advent of quantum mechanics, there have been
numerous efforts to extend the framework to systems composed of a finite or
large number of quantum particles while each particle has a discrete energy
spectrum and the states are in a superposition of different energy levels, see
for example Binder  _et al._ (2018). One of the possibilities for quantum
systems with bounded energy is that, in certain situations, they can assume
‘negative’ temperatures. This arises when the population distribution of a
system or bath becomes an inverted Boltzmann distribution, i.e., states with
higher energy are populated more than the ones with lower energy. It was first
pointed out by Purcell and Pound in the context of nuclear spin systems
Purcell and Pound (1951). Subsequently, Ramsey comprehensively discussed the
thermodynamic implications of such negative temperatures and the inter-
relation between negative and positive temperatures Ramsey (1956). He
advocates for an amendment to the Kelvin-Planck statement of the second law to
incorporate that heat flows spontaneously from a bath with a negative
temperature to one with a positive temperature. In this sense, the negative
temperature is ‘hotter’ than a positive temperature.

Initially, Schöpf raised some foundational questions regarding the dynamics of
negative temperature Schöpf (1962). He claimed that it is impossible to
transform a thermodynamic system adiabatically: from a positive finite
temperature to the positive infinite temperature, then from there to a
negative infinite (Boltzmann) temperature, and then subsequently to a negative
finite (Boltzmann) temperature Schöpf (1962). Tykodi and Tremblay Tykodi
(1975); Tremblay (1976); Tykodi (1978) disagreed and showed that the arguments
used by Schöpf are thermodynamically inconsistent as these violate the second
law of thermodynamics. Nevertheless, the debates on thermodynamics with
negative temperatures are not settled. Recent theoretical Mosk (2005); Rapp
_et al._ (2010) and experimental Braun  _et al._ (2013); Carr (2013); Muniz
_et al._ (2023) studies with cold atoms have brought the debate on negative
temperatures back again into the spotlight. The works in Dunkel and Hilbert
(2013); Hilbert  _et al._ (2014); Hänggi  _et al._ (2016) claim that “all
previous negative temperature claims and their implications are invalid as
they arise from the use of an entropy definition that is inconsistent both
mathematically and thermodynamically." Another study in Struchtrup (2018)
states that thermodynamic equilibrium at negative temperatures would be
unstable but can be used for work storage or battery. Several researchers have
come forward and systematically explained that identification of the
thermodynamic entropy exclusively with the volume entropy proposed by Gibbs is
the root of all doubts Cerino  _et al._ (2015); Buonsante  _et al._ (2016);
Poulter (2016); Abraham and Penrose (2017); Swendsen (2018), and it is
inconsistent with the postulates of thermodynamics Wang (2015); Swendsen and
Wang (2016). Using Boltzmann entropy as the thermodynamic entropy, they argue
that negative temperature is a valid extension of thermodynamics.

Apart from these foundational issues, there are questions on whether a
negative temperature bath can be used to construct thermal devices, such as
heat engines, refrigerators, heat pumps, etc. Initially, a study on Carnot
engines was made by Geusic, et al. Geusic  _et al._ (1967) and, later on, by
Landsberg and Nakagomi Landsberg (1977); Nakagomi (1980) in this context.
There are also some studies about how the Carnot cycle should be modified in
the presence of negative temperatures Dunning-Davies (1976, 1978); Landsberg
_et al._ (1980). Further, some propositions are made to construct a quantum
Otto engine Xi and Quan (2017); de Assis  _et al._ (2019) and refrigerators
Damas  _et al._ (2022) using a bath with effective negative temperature. It is
shown that the heat-to-work conversion efficiency of an engine operating
between negative and positive temperatures would be greater than unity Braun
_et al._ (2013); Carr (2013); de Assis  _et al._ (2019). However, much of
these models of thermal devices either utilize already existing negative
temperature baths without caring how it may be created or effectively prepare
one by inverting the populations using some external means.

In this article, we outline how to create a thermal bath with arbitrary
temperature, including negative temperature, and study steady-state quantum
thermodynamics. The bath is synthesized by letting a quantum system interact
simultaneously with two thermal baths at different positive temperatures
without external driving. We study various laws of steady-state quantum
thermodynamics with these synthetic baths and construct continuous heat
engines. We start by proving the zeroth law and show that there is no net heat
flow whenever two such baths with identical temperatures are brought in
contact with each other. This, in turn, legitimizes the notion of the
temperature of a synthetic bath (namely, the synthetic temperature). In the
case of two different temperatures, we prove the Kelvin-Planck statement of
the second law and demonstrate that there is a spontaneous heat flow from a
bath with a negative temperature to one with a positive temperature. This
corroborates with the finding of Ramsey Ramsey (1956) \- baths with negative
temperatures are ‘hotter’ than the ones with positive temperatures.
Interestingly, in such cases, the entropy flow is opposite to the direction of
heat flow which is again expected for the baths with negative temperatures. We
also construct Carnot engines involving synthetic baths and find that engines
operating between positive and negative temperature baths can yield unit
engine efficiency. This leads us to question the physical meaning of heat flow
in the presence of a bath with negative temperatures. With a systematic
analysis, we show that the heat associated with a bath with a negative
temperature is equivalent to work but with a negative entropy flow.

##  II Synthetic baths and Negative temperatures

In general, naturally occurring thermal equilibrium results in non-negative
temperatures. Only in certain situations, as discussed earlier, can the
temperatures be negative. Below, we introduce a method through which a bath
can be synthesized. The temperatures of these synthetic baths can assume
arbitrary values, including negative ones.

The method utilizes a qutrit system, a hot bath ($H$) with inverse temperature
$\beta_{H}$, and a cold bath ($C$) at inverse temperature $\beta_{C}$. The
energy levels of the qutrit are denoted by $|1\rangle$, $|2\rangle$,
$|3\rangle$, with the corresponding Hamiltonian $H=(E_{H}-E_{C})\
|2\rangle\\!\langle 2|+E_{H}\ |3\rangle\\!\langle 3|$. As shown in Fig. 1, the
hot (cold) bath weakly interacts with the levels $|1\rangle$ and $|3\rangle$
(levels $|2\rangle$ and $|3\rangle$). As convention, we consider
$\beta_{H}<\beta_{C}$, Planck constant $\hbar=1$, and the Boltzmann constant
$k_{B}=1$ throughout the article. The levels $|1\rangle$ and $|2\rangle$ are
not directly coupled. However, they are indirectly linked through the level
$|3\rangle$. When the couplings between the qutrit and the baths are weak and
satisfy the Markov condition, the overall dynamics is expressed in terms of
the Lindblad or Lindblad-Gorini-Kossakowski-Sudarshan (LGKS) form of master
equation Breuer and Petruccione (2007)

$\displaystyle\dot{\rho}=\mathcal{L}_{U}(\rho)+\mathcal{L}_{H}(\rho)+\mathcal{L}_{C}(\rho).$
(1)

Here $\rho$ represents the density matrix corresponding to a state of the
qutrit. The first term on the right-hand side of Eq. (1)
$\mathcal{L}_{U}(\rho)=i\ [\rho,H]$ takes care of the unitary part of the
evolution due to the system Hamiltonian $H$. The second and third terms, with
the Lindblad super-operators (LSOs) $\mathcal{L}_{H}(\cdot)$ and
$\mathcal{L}_{C}(\cdot)$, represent the contributions due to the dissipative
part of the evolution induced by the hot and cold thermal baths, respectively.
The LSOs are expressed (for $X=H,\ C$) as

$\displaystyle\mathcal{L}_{X}(\rho)=\ $
$\displaystyle\Gamma_{X}(N_{X}+1)\left(A_{X}\rho A_{X}^{\dagger}-1/2\
\\{A_{X}^{\dagger}A_{X},\rho\\}\right)$
$\displaystyle+\Gamma_{X}N_{X}\left(A_{X}^{\dagger}\rho A_{X}-1/2\
\\{A_{X}A_{X}^{\dagger},\rho\\}\right),$ (2)

where $A_{H}=|1\rangle\langle 3|$, $A_{C}=|2\rangle\langle 3|$, anti-
commutator $\\{Y,Z\\}=YZ+ZY$, and $N_{X}=1/(e^{\beta_{X}E_{X}}-1)$. The
coefficient $\Gamma_{X}$ is the Weiskopf-Wigner decay constant. The overall
dynamics leads to heat exchange between the baths and the system. The heat
fluxes are quantified as
$\dot{Q}_{X}=\operatorname{\mathrm{Tr}}[\mathcal{L}_{X}(\rho)\ H]$ due to
interaction with the bath $X$ Boukobza and Tannor (2006a, b, 2007). Heat flux
$\dot{Q}_{X}>0$ implies that heat is flowing into the qutrit system from the
bath with inverse temperature $\beta_{X}$.

Figure 1: A synthetic bath is created using two baths with different
temperatures and letting them interact with a qutrit system. In particular,
the hot bath ($H$) with inverse temperature $\beta_{H}$ is weakly coupled to
the energy eigenstates $|1\rangle$ and $|3\rangle$. The cold bath ($C$) with
inverse temperature $\beta_{C}$ weakly interacts with the energy eigenstates
$|2\rangle$ and $|3\rangle$. As a result, the populations of the states
$|1\rangle$ and $|2\rangle$ reach an equilibrium corresponding to a synthetic
temperature $\beta_{S}$. By tuning the temperature of the baths and the energy
spacing between the states, an arbitrary synthetic temperature can be
obtained, including negative temperatures. See text for more details.

This dynamics always leads to a steady state, say $\sigma$, which is diagonal
in the energy eigenstates. The the populations $\\{p_{i}\\}$ of the states
$\\{|i\rangle\\}$ satisfy

$\displaystyle~~\frac{p_{1}}{p_{3}}=e^{\beta_{H}E_{H}},\ \ \mbox{and}\ \
\frac{p_{2}}{p_{3}}=e^{\beta_{C}E_{C}}.$

The populations corresponding to states $|1\rangle$ and $|3\rangle$ attain
thermal equilibrium with the hot bath, and similarly, the populations of
$|2\rangle$ and $|3\rangle$ attain thermal equilibrium with the cold bath. In
fact, the dynamics drives the overall system to reach thermal equilibrium,
albeit in interactions with two baths at different temperatures. This is
justified because the heat flux and entropy production vanish, i.e.,
$\dot{Q}_{X}=0$ for $X=H,C$ and $-\beta_{H}\dot{Q}_{H}-\beta_{C}\dot{Q}_{C}=0$
respectively. As the entire system is in thermodynamic equilibrium, so are the
populations of the states $|1\rangle$ and $|2\rangle$.

In general, if one introduces an interaction between the levels $|1\rangle$
and $|2\rangle$, be it time-dependent or time-independent, the heat and
entropy fluxes become non-zero Boukobza and Tannor (2006b, 2007) and the
corresponding populations change. But once the interaction is switched off,
the populations revert to their equilibrium values. This is as if the levels
$|1\rangle$ and $|2\rangle$, or the subspace spanned by these two levels, are
interacting with a synthetic bath at inverse temperature $\beta_{S}$, defined
as

$\displaystyle\beta_{S}=\frac{1}{E_{S}}\ln\left(\frac{p_{1}}{p_{2}}\right)=\frac{1}{E_{S}}\ln\left(\frac{p_{1}}{p_{3}}\
\frac{p_{3}}{p_{2}}\right)=\frac{\beta_{H}E_{H}-\beta_{C}E_{C}}{E_{S}},$ (3)

where $E_{S}=E_{H}-E_{C}$. We note that a Lindblad super-operator (LSO) cannot
be given exclusively for the equilibration dynamics due to the synthetic bath.
However, as we discuss in the later sections, this is a legitimate thermal
bath. We call $\beta_{S}$ the ’synthetic’ inverse temperature because it can
be tuned to assume arbitrary values, including negative values, by changing
the energy-level spacings and the $\beta_{H}$ and $\beta_{C}$. In literature,
there are debates on whether the temperature of a system can be continuously
changed from a positive to a negative equilibrium temperature Struchtrup
(2018). However, in this setup, the inverse temperature of the synthetic bath
can be tuned continuously, e.g., from $\beta_{S}>0$ to $\beta_{S}<0$,
including $\beta_{S}=0$ (infinite temperature).

##  III Thermodynamics with synthetic baths

To study thermodynamics with synthetic temperatures, we consider two different
qutrit systems $L$ and $R$ with the corresponding Hamiltonians
$H_{X}=(E_{XH}-E_{XC})\ |2\rangle\\!\langle 2|+E_{XH}\ |3\rangle\\!\langle
3|$, with $X=L,R$. We assume $E_{XH}-E_{XC}=E_{S}$ for both systems, i.e., the
energy spacing between $|1\rangle$ and $|2\rangle$ for both $L$ and $R$ are
same. For brevity, we denote $H_{L}\equiv H_{L}\otimes\mathbb{I}$ and
$H_{R}\equiv\mathbb{I}\otimes H_{R}$. Each system couples a hot and a cold
bath with inverse temperatures $\beta_{H}$ and $\beta_{C}$ respectively (see
Fig. 2) and reaches an equilibrium state. Without an interaction in between,
the equilibrium state of the composite $LR$ becomes

$\displaystyle~~\rho_{L}\otimes\rho_{R}=\sum_{m,n=1}^{3}p_{m}q_{n}\ ~|m\
n\rangle\\!\langle m\ n|,$ (4)

where $p_{1}/p_{3}=e^{\beta_{H}E_{LH}}$, $p_{2}/p_{3}=e^{\beta_{C}E_{LC}}$,
$q_{1}/q_{3}=e^{\beta_{H}E_{RH}}$, and $q_{2}/q_{3}=e^{\beta_{C}E_{RC}}$. The
population ratio between the degenerate energy states $|21\rangle$ and
$|12\rangle$ is

$\displaystyle~p_{1}q_{2}/p_{2}q_{1}=e^{(\beta_{LS}-\beta_{RS})E_{S}},~$ (5)

where $\beta_{LS}$ and $\beta_{RS}$ are the synthetic temperatures
corresponding to qutrit $L$ and $R$ respectively.

Figure 2:  Two synthetic baths are engineered with the help of a hot and a
cold bath at inverse temperatures $\beta_{H}$ and $\beta_{C}$, respectively.
Each synthetic bath is created by letting the baths weakly interact with one
qutrit, as shown in Fig. 1. Different synthetic temperatures are engineered by
tuning energy spacings between the states $|1\rangle$, $|2\rangle$, and
$|3\rangle$. In addition, an interaction is introduced between the synthetic
baths to study the heat and entropy flow. See text for more details.

An interaction is introduced that only couples subspace spanned by the energy
levels belonging to $|1\rangle$ and $|2\rangle$ in each qutrit, ensuring an
energy exchange between $L$ and $R$ only through these subspaces. The most
general interaction Hamiltonian that drives an energy exchange between these
subspaces is given by

$\displaystyle H_{in}=(\lambda+i\ \gamma)\ |12\rangle\langle 21|+(\lambda-i\
\gamma)\ |21\rangle\langle 12|,$ (6)

where $\lambda,\gamma\in\mathbb{R}$. This interaction also strictly conserves
energy, as $[H_{in},H_{L}+H_{R}]=0$. The overall dynamics of $LR$ is expressed
as

$\displaystyle\dot{\rho}_{LR}=i\
~[\rho_{LR},H_{T}]+\mathcal{L}_{L}(\rho_{LR})+\mathcal{L}_{R}(\rho_{LR}),$ (7)

where $H_{T}=H_{L}+H_{R}+H_{in}$,
$\mathcal{L}_{L}(\cdot)=\mathcal{L}_{LH}(\cdot)+\mathcal{L}_{LC}(\cdot)$, and
$\mathcal{L}_{R}(\cdot)=\mathcal{L}_{RH}(\cdot)+\mathcal{L}_{RC}(\cdot)$.
Here, $\mathcal{L}_{XH}(\cdot)$ and $\mathcal{L}_{XC}(\cdot)$ are the LSOs
taking into account the dissipative part of the dynamics due to the coupling
with hot and cold baths, respectively, with the qutrit $X$. Under this
dynamics, the composite system $LR$ reaches a steady state, say $\sigma_{LR}$.
Then, the heat flux and the entropy flux, respectively, are

$\displaystyle~~\dot{Q}_{X}=~\operatorname{\mathrm{Tr}}[\mathcal{L}_{X}(\sigma_{LR})\
H_{X}],\ \ \ ~~~\dot{S}_{X}=\beta_{XS}\dot{Q}_{X},$ (8)

with $X=L,R$, and $\dot{Q}_{X}=\dot{Q}_{XH}+\dot{Q}_{XC}$.

In absence of any interaction between the $L$ and $R$, the steady (or
equilibrium) state is $\sigma_{LR}=\rho_{L}\otimes\rho_{R}$ (see Eq. (4)).
Then, the heat flux from $L$ is $\dot{Q}_{L}=0$, as
$\dot{Q}_{LH}=\dot{Q}_{LC}=0$. However in presence of interaction via
$H_{in}$, the steady state becomes $\sigma_{LR}\neq\rho_{L}\otimes\rho_{R}$,
and then the $\dot{Q}_{L}=\dot{Q}_{LH}+\dot{Q}_{LC}\neq 0$. This means that
there is heat flux through the subspace spanned by $\\{|1\rangle,|2\rangle\\}$
of $L$, which we may consider as the heat flux due to the synthetic bath
associated with $L$. By convention, $\dot{Q}_{L}>0$ implies a heat flux from
the synthetic bath to $L$, which is then passed to $R$. With these tools at
hand, we now set out to explore steady-state quantum thermodynamics with
synthetic temperatures. Note that the first law is always respected at steady
state as $\dot{Q}_{L}+\dot{Q}_{R}=0$. Thus, our emphasis would be on studying
the zeroth and second laws.

Figure 3: The figures represent heat and entropy fluxes, $\dot{Q}_{L}$ and
$\dot{S}_{L}$, respectively, through the qutrit $L$. The numerical calculation
is carried out with the parameters: $\Gamma_{H}=\Gamma_{C}=0.001$,
$\beta_{H}=0.05$, $\beta_{C}=1$, $\lambda=1$, $\gamma=0$, $E_{S}=9.5$. The
different synthetic inverse temperatures are obtained by tuning the energy
$E_{H}$ of $|3\rangle$ for the qutrits $L$ and $R$. (a) The density plot
represents the Kelvin-Planck statement of the second law in terms of the heat
flux through $L$, i.e., $\dot{Q}_{L}$. The plot shows no heat flux for
$\beta_{LS}=\beta_{RS}$. This corroborates with the zeroth law. However, there
is a positive (negative) heat flux $\dot{Q}_{L}$, i.e., heat flows from $L$ to
$R$ (from $R$ to $L$), whenever $-\beta_{LS}>-\beta_{RS}$
($-\beta_{LS}<-\beta_{RS}$) implying bath with $\beta_{LS}$ ($\beta_{RS}$) is
hotter than $\beta_{RS}$ ($\beta_{LS}$). Clearly, a bath with a negative
temperature is always ‘hotter’ than any bath with a positive temperature. (b)
The density plot represents the variation of entropy flux $\dot{S}_{L}$ for
different synthetic temperatures. As seen from the plot, for
$\beta_{LS}=\beta_{RS}$, $\dot{S}_{L}=0$. For $\beta_{LS}>0$ and
$\beta_{RS}>0$, the direction of entropy flux is same with the heat flux as
expected for the baths with positive temperatures. However, the direction of
heat flow is opposite to the direction of entropy flow, in general, for baths
with negative inverse temperatures. See text for details.

Zeroth law \- In thermodynamics, the zeroth law interlinks the notion of
temperature with equilibrium. It states that if two systems are in thermal
equilibrium, they must have the same temperature and vice versa. Again,
thermal equilibrium implies that when two systems are in contact, there is no
net flux in any thermodynamic quantities (such as heat and entropy) between
systems. In such situations, the overall entropy production also vanishes.
Below we show that, in the two-qutrit scenario discussed above, there is no
net flux of any thermodynamic quantity whenever the synthetic temperatures are
identical for both qutrits.

Recall the setup we consider in Fig. 2. Without any interaction, the steady
state of $LR$ is $\rho_{L}\otimes\rho_{R}$, which is diagonal in the energy
eigenstates (see Eq. (4)). With same synthetic inverse temperatures
$\beta_{LS}=\beta_{RS}$, the populations of the states $|12\rangle$ and
$|21\rangle$ satisfy $p_{1}q_{2}=p_{2}q_{1}$ (see Eq. (5)). It means, the
matrix corresponding to the state $\rho_{L}\otimes\rho_{R}$ in the subspace
spanned by $\\{|12\rangle,\ |21\rangle\\}$ is proportional to an identity
operator. Now an interaction between $L$ and $R$ introduced by $H_{in}$, as in
Eq. (6). It is easily seen that $[H_{in},\rho_{L}\otimes\rho_{R}]=0$ for
$\beta_{LS}=\beta_{RS}$. Thus, even after the interaction is switched on, the
steady state remains unaltered, i.e., $\sigma_{LS}=\rho_{L}\otimes\rho_{R}$
and $\dot{\sigma}_{LR}=\mathbb{0}$. Hence, there is no exchange of heat and
entropy between $L$ and $R$, as $\dot{Q}_{X}=\dot{S}_{X}=0$ for $X=L,R$. This
implies that the synthetic baths are in thermal equilibrium whenever the
synthetic temperatures are identical, irrespective of whether the temperatures
are positive or negative. A numerical analysis also confirms this. See Fig.
3(a).

Second law \- For $\beta_{LS}\neq\beta_{RS}$, heat and entropy flow is
possible from one qutrit to the other. However, the flow cannot be arbitrary.
The second law dictates the physically allowed processes given that zeroth and
first laws are respected. There are various statements of the second law.
Below, we analyze the Kelvin-Planck statement to deal with the directionality
of heat flow and the Clausius statement regarding entropy production.

Let us consider the case for which $-\beta_{LS}>-\beta_{RS}$. When there is no
interaction between the qutrits, the equilibrium state of the composite $LS$
is $\rho_{L}\otimes\rho_{R}$. The state is expressed in the block-diagonal
form as

$\displaystyle\rho_{L}\otimes\rho_{R}=\Pi_{0}\rho_{L}\otimes\rho_{R}\Pi_{0}+\Pi_{1}\rho_{L}\otimes\rho_{R}\Pi_{1},$

where $\Pi_{0}=|12\rangle\\!\langle 12|+|21\rangle\\!\langle 21|$ and
$\Pi_{1}=\mathbb{I}-\Pi_{0}$. The populations corresponding to the energy
eigenstates $|12\rangle$ and $|21\rangle$ satisfy $p_{1}q_{2}<p_{2}q_{1}$ (see
Eq. (5)).

Now let us disconnect the thermal baths and introduce an interaction driven by
$H_{in}$. This, or the total Hamiltonian $H_{T}$, evolves the composite and
induces a rotation onto the subspace spanned by $|12\rangle$ and $|21\rangle$
only. At the same time, the other part of the density matrix remains
unchanged. As a result, there appear off-diagonal elements in this subspace.
Say, the state of $LR$ after any evolution becomes

$\displaystyle\rho_{LR}^{\prime}=a\ |12\rangle\langle 12|$ $\displaystyle+b\
|12\rangle\langle 21|+c\ |21\rangle\langle 12|+d\ |21\rangle\langle 21|$
$\displaystyle+\Pi_{1}\rho_{L}\otimes\rho_{R}\Pi_{1}.$

The unitary nature of the evolution in the subspace $|12\rangle$ and
$|21\rangle$ guarantees that $a>p_{1}q_{2}$ and $d<p_{2}q_{1}$. For this
reason and as the off-diagonal elements do not contribute to the populations
of the reduced state of $L$, i.e.,
$\rho_{L}^{\prime}=\operatorname{\mathrm{Tr}}_{R}\rho_{LR}^{\prime}$, we find
$p^{\prime}_{1}=\langle 1|\rho_{L}^{\prime}|1\rangle>p_{1}$ and
$p^{\prime}_{2}=\langle 2|\rho_{L}^{\prime}|2\rangle<p_{2}$. Similarly, for
the reduced state of
$\rho_{R}^{\prime}=\operatorname{\mathrm{Tr}}_{L}\rho_{LR}^{\prime}$, the
modified populations becomes $q^{\prime}_{1}=\langle
1|\rho_{R}^{\prime}|1\rangle<q_{1}$ and $q^{\prime}_{2}=\langle
2|\rho_{R}^{\prime}|2\rangle>q_{2}$. Note, the populations corresponding to
level $|3\rangle$ for both $L$ and $R$ remain unchanged, i.e., $\langle
3|\rho_{L}^{\prime}|3\rangle=p_{3}$ and $\langle
3|\rho_{R}^{\prime}|3\rangle=q_{3}$. Clearly, the qutrit $L$ loses some
energy. As the evolution respects strict energy conservation, the qutrit $R$
gains the same amount of energy. Thus, any evolution due to $H_{in}$ ensures
that there is an energy flow from $L$ to $R$ for $-\beta_{LS}>-\beta_{RS}$.
After this modification, if $L$ is now exposed to its baths, the dissipative
dynamics due to $\mathcal{L}_{L}(\cdot)$ forces the qutrit to restore its
equilibrium state, $p_{1}^{\prime}\to p_{1}$ and $p_{2}^{\prime}\to p_{2}$.
This, in turn, increases the energy of $L$ by absorbing some heat from the hot
and cold baths or, equivalently, from the synthetic bath. Similarly, if $R$ is
exposed to its baths, some of its energy is released to its synthetic bath in
the form of heat and thereby attains its equilibrium. Note in this process, to
reach the equilibrium, the populations of $|3\rangle$ do not remain constant
throughout in both $L$ and $R$.

From the above arguments, we see that the unitary evolution with the
interaction between $L$ and $R$ drives the composite out of equilibrium
leading to a spontaneous heat flow from $L$ to $R$ for
$-\beta_{LS}>-\beta_{RS}$. While, the dissipative evolution due to thermal
interactions with the baths tries to restore the composite back to the initial
equilibrium state ($\rho_{L}\otimes\rho_{R}$) by pumping some heat into $L$
and absorbing some heat from $R$. When both unitary and dissipative evolutions
occur simultaneously, as in Eq. (7), the opposing tendencies balance each
other and result in a steady state, say $\sigma_{LR}$. This steady state is
again block-diagonal in total energy eigenstates and has off-diagonal elements
in the eigenstates $|12\rangle$ and $|21\rangle$. Nevertheless, the steady-
state dynamics generate a heat flux from $L$ to $R$,

$\displaystyle~\dot{Q}_{L}=\operatorname{\mathrm{Tr}}[\mathcal{L}_{L}(\sigma_{LR})\
H_{L}]=\operatorname{\mathrm{Tr}}[\mathcal{L}_{L}(\sigma_{L})\ H_{L}]>0,~$ (9)

and $\dot{Q}_{R}=-\dot{Q}_{L}$. The expression of $\dot{Q}_{L}$ can be given
analytically, and it has complicated dependencies with all the parameters.
Rather, a numerical analysis is more illuminating; we have done so in Fig.
3(a). Again for $-\beta_{LS}<-\beta_{RS}$, we find that $\dot{Q}_{L}<0$. This
means that there is a spontaneous heat flow from $R$ to $L$.

In thermodynamics, the Kelvin-Planck statement of the second law states that
heat can only flow from a hot bath to a cold bath when no external work is
performed. As we see above for $-\beta_{LS}>-\beta_{RS}$, there is a heat flow
from $L$ to $R$. This means that: (1) baths with negative inverse temperatures
are ’hotter’ than the baths with positive inverse temperatures; (2) baths with
larger negative inverse temperatures are ’hotter’ than those with smaller
negative inverse temperatures. This is in conformation with the findings of
Ramsey Ramsey (1956). However, one important point to be noted here is that
although there is a heat flow from a negative to a positive temperature bath,
the entropy flow is the opposite. This is indeed a signature of a bath having
a negative temperature.

The Clausius statement, another formulation of the second law, states that the
overall entropy production is always positive in a thermodynamical process.
For steady-state thermodynamics with synthetic baths, the overall entropy
production rate is given by

$\displaystyle\Sigma=\dot{S}_{LR}-\beta_{LS}\dot{Q}_{L}-\beta_{RS}\dot{Q}_{R},$
(10)

where $\dot{S}_{LR}=\partial S_{LR}/\partial t$ is the rate of change in von
Neumann entropy
$S_{LR}=-\operatorname{\mathrm{Tr}}[\sigma_{LR}\log\sigma_{LR}]$, and
$\beta_{LS}$ ($\beta_{RS}$) is the synthetic inverse temperature of $L$ ($R$)
and $\dot{Q}_{L}$ ($\dot{Q}_{R}$) is the heat flux from $L$ ($R$). Note at
steady state, the $\partial S_{LR}/\partial t=0$ and
$\Sigma=(\beta_{RS}-\beta_{LS})\ \dot{Q}_{L}$, as $\dot{Q}_{L}=-\dot{Q}_{R}$.
For $-\beta_{LS}>-\beta_{RS}$, the heat flux from $L$ is positive,
$\dot{Q}_{L}>0$. Consequently, $\Sigma>0$. Similarly, $\dot{Q}_{L}<0$ for
$-\beta_{LS}<-\beta_{RS}$, and thus $\Sigma>0$. For $\beta_{LS}=\beta_{RS}$,
we have $\Sigma=0$. Thus, the Clausius inequality in the differential form is

$\displaystyle\Sigma\geq 0,$

and it is always satisfied as long as the Kelvin-Planck statement is
respected.

At steady state, the entropy production rate is positive, $\Sigma\geq 0$. This
is mainly due to the dissipative interaction between the baths and the system.
However, one may find out an entropy flow through the system $LR$, as

$\displaystyle\dot{S}_{X}=-\operatorname{\mathrm{Tr}}[\mathcal{L}_{X}(\sigma_{LR})\log\sigma_{LR}],$

for $X=L,R$ (see Appendix A). At steady state, $\dot{S}_{L}+\dot{S}_{R}=0$ as
the state does not evolve over time. $\dot{S}_{L}>0$ implies that there is an
entropy flux from bath with inverse temperature $\beta_{LS}$ to $R$ via $L$,
and similarly for $\dot{S}_{R}>0$. In general, for a bath with positive
temperature, an outflow of heat is associated to a decrease in entropy. One
striking feature we must note here is that, although there is a spontaneous
heat flow from a bath with negative temperature to a bath with positive
temperature, the entropy flow is just opposite to that (see Fig. 3(b)). This
is also true when both baths are of negative temperatures. For a bath with
negative temperature, an outflow of heat is associated with an increase in
entropy of the bath. Thus, a bath with negative temperature in general acts as
an entropy sink.

##  IV Quantum heat engines with a bath at negative temperature

Now we discuss heat engines operating with a bath at synthetic temperatures,
particularly at negative temperatures. A device acting as a heat engine aims
to transform heat into work. A generic heat engine consists of three primary
parts: two separate heat baths with different temperatures and a working
system. It operates by absorbing heat from the hot bath. The working system
transforms part of this heat into work, dumping the rest into the cold bath.
The model of a quantum heat engine (QHE) we are concerned with utilizes a
synthetic bath with negative temperature and a heat bath with positive
temperature (as depicted in Fig. 4). The working system is composed of a
qutrit ($L$) and a qubit ($W$). The synthetic bath with inverse temperature
$\beta_{LS}$ is created using two baths at different temperatures $\beta_{H}$
and $\beta_{C}$ and letting these weakly interact with $L$, similar to the one
considered in Fig. 1. The Hamiltonian of $L$ is given by
$H_{L}=(E_{H}-E_{C})|2\rangle\\!\langle 2|+E_{H}|3\rangle\\!\langle 3|$. The
qubit $W$ is weakly coupled to a bath at inverse temperature $\beta_{W}\geq
0$, and its Hamiltonian is $H_{W}=E_{W}|2\rangle\\!\langle 2|$, where
$E_{W}=E_{H}-E_{C}$. To operate the device as a heat engine, a time-dependent
interaction is introduced between $L$ and $W$ driven by the interaction
Hamiltonian

$\displaystyle~H_{in}^{E}(t)=\delta\ (|11\rangle\langle 22|\ ~e^{i\omega
t}+|22\rangle\langle 11|\ e^{-i\omega t}).~$ (11)

The total Hamiltonian is then $H_{T}^{E}=H_{0}+H_{in}^{E}(t)$, where
$H_{0}=H_{L}+H_{W}$. The overall dynamics will never lead to a steady state
for a time-dependent interaction. However, for a periodic time dependence,
there is a rotating frame in which the interaction becomes time-independent.
For instance, to move from the laboratory frame to a rotating frame to we may
introduce a rotation, given by the unitary $U=\exp[iH_{R}t]$ which satisfies
$[H_{R},H_{0}]=0$. For a suitable $H_{R}$, the interaction becomes time-
independent, i.e., $V_{in}=UH_{in}^{E}(t)U^{\dagger}$. In this rotating frame,
the overall dynamics comprising the unitary and the dissipative evolution is
given by (see Appendix A)

$\displaystyle~\dot{\rho}_{LW}^{R}=i[\rho_{LW}^{R},\bar{H}_{T}^{E}]+\mathcal{L}_{L}(\rho_{LW}^{R})+\mathcal{L}_{W}(\rho_{LW}^{R}),~~$
(12)

for a state $\rho_{LW}$, with ${\rho}_{LW}^{R}=U\rho_{LW}U^{\dagger}$ and
$\bar{H}_{T}^{E}=H_{0}-H_{R}+V_{in}$. Note the LSOs remain unchanged in the
rotating frame. Now that the time-dependence in the Hamiltonian is lifted, the
dynamics attains a steady state $\sigma_{LW}^{R}$ in the rotating frame.

Figure 4: A schematic of a quantum heat engine operating with two baths; one
with negative inverse temperature $\beta_{SL}$ and the other with positive
temperature $\beta_{W}$. The negative temperature is synthesized by weakly
coupling a qutrit $L$ with a hot ($H$) and a cold ($C$) bath, as discussed in
Fig. 1. This is as if the energy levels $|1\rangle$ and $|2\rangle$ are
coupled to a synthetic bath at inverse temperature $\beta_{SL}$. A qubit $W$
is weakly coupled to the bath with $\beta_{W}$. In the engine, $L$ and $W$ are
coupled through a time-dependent interaction as in Eq. (11). The arrows
represent the direction of heat ($\dot{Q}_{X}$) and entropy ($\dot{S}_{X}$)
fluxes for $X=L,W$, and $P$ represents the power of the engine. See text for
more details.

With this, the heat flux, entropy flux, and the power in the laboratory frame
are quantified as (see Appendix A)

$\displaystyle\dot{Q}_{X}={\operatorname{Tr\,}}[\mathcal{L}_{X}$
$\displaystyle(\sigma_{LW}^{R})\ H_{0}],\ \
\dot{S}_{X}=-\operatorname{\mathrm{Tr}}[\mathcal{L}_{X}(\rho_{LW}^{R})\log\sigma_{LW}^{R}],$
and $\displaystyle P=i\ {\operatorname{Tr\,}}[\sigma_{LW}^{R}\
[V_{in},H_{0}]].$ (13)

for $X=L,W$, and $[A,B]=AB-BA$. Here $\dot{Q}_{X}$ and $\dot{S}_{X}$ represent
the heat and entropy fluxes through system $X$ respectively, and $P$
represents the power. The condition $\dot{Q}_{L}+\dot{Q}_{W}+P=0$ is always
satisfied as required by the first law Alicki (1979); Kosloff and Levy (2014)
and at steady state, $\dot{S}_{L}+\dot{S}_{W}=0$. For any negative inverse
temperature $\beta_{LS}<0$, we have, as confirmed by numerical analysis,
$\dot{Q}_{L}>0$, $\dot{Q}_{W}>0$, and $P<0$. This means the device draws heat
from both the synthetic bath and the bath with inverse temperature
$\beta_{W}$. For traditional engines, the efficiency is calculated as the
ratio of work extracted and the heat absorbed by the engine from the hot bath.
Here, the heat is absorbed from both baths. For each bath, the corresponding
engine efficiency may be defined as

$\displaystyle\eta_{L}=\frac{-P}{\dot{Q}_{L}},\ \ \mbox{and}\ \
\eta_{W}=\frac{-P}{\dot{Q}_{W}}.$

It can be easily checked that $\eta_{L}>1$ and $\eta_{W}>1$. Thus, the
efficiency exceeds unity for an engine operating between baths with positive
and negative temperatures. This is what is also claimed in literature Geusic
_et al._ (1967). We, however, find this conclusion incomplete. The heat-to-
work conversion efficiency should always be defined with respect to the total
amount of heat entering the engine and the amount of work produced out of
that. In that sense, the total heat flux entering the engine is
$\dot{Q}_{L}+\dot{Q}_{W}$, and this entire heat is converted into work. As a
result, we find

$\displaystyle\eta=\frac{-P}{\dot{Q}_{L}+\dot{Q}_{W}}=1,$

i.e., the engine efficiency becomes unity. Thus, the efficiency of an engine
can never exceed unity in any circumstance as long as the first law, i.e., the
overall energy conservation, is respected.

Figure 5: Heat and entropy fluxes in the engine. The red-solid and blue-solid
traces represent the change in heat flux ($\dot{Q}_{L}$) and entropy flux
($\dot{S}_{L}$) through $L$ in the engine with respect to $\beta_{SL}$
respectively. The calculation is done with the parameters: $\beta_{H}=0.01$,
$\beta_{W}=0.1$, $\beta_{C}=10$, $\delta=5$, $\omega=1$, and $E_{S}=5$. As
clearly seen, $\dot{Q}_{L}>0$ and $\dot{S}_{L}<0$ for all $\beta_{SL}<0$. See
text for more details.

This is, nevertheless, different from what we see in traditional heat engines.
It raises a question on the physical meaning of the heat released or absorbed
by a bath with a negative temperature. By definition, heat is a form of energy
that is always associated with a change in the entropy of the corresponding
bath. Heat flow, thus, occurs with an entropy flow. The work, on the other
hand, is a pure form of energy and is not associated with any flow of entropy.
In traditional engines operating with baths at positive temperatures, the
direction of heat flow and the direction of entropy flow is the same. That is
how a bath gets cooled down when it releases some heat. For a bath with a
negative temperature, this is not true (see Fig. 3(b)). There the bath’s
entropy increases as it releases heat. For the engine we have considered
above, the direction of heat flux $\dot{Q}_{L}>0$ is opposite to the direction
of entropy flux $\dot{S}_{L}<0$ in $L$ which is coupled to the negative bath
(see Fig. 5). While for $W$, coupled to positive temperature bath, the
direction of heat flux $\dot{Q}_{W}>0$ is the same as the direction of entropy
flux $\dot{S}_{W}>0$. In fact, while heat is entering the engine from both
baths, there is an entropy flow from the bath with inverse temperature
$\beta_{W}$ to the bath with $\beta_{SL}$ where the latter acts as an entropy
sink.

In order to understand the thermodynamic nature of this heat, let us make a
closer inspection of the process that is happening on $L$ and $W$ alone in the
rotating frame. To create a synthetic bath with negative temperature, $L$
interacts with hot and cold baths with inverse temperatures $\beta_{H}$ and
$\beta_{C}$. The corresponding heat fluxes are
$\dot{Q}_{LH}=\operatorname{\mathrm{Tr}}[\mathcal{L}_{LH}(\sigma_{LW}^{R})\
H_{L}]$ and
$\dot{Q}_{LC}=\operatorname{\mathrm{Tr}}[\mathcal{L}_{LC}(\sigma_{LW}^{R})\
H_{L}]$. Recall that
$\mathcal{L}_{L}(\cdot)=\mathcal{L}_{LH}(\cdot)+\mathcal{L}_{LC}(\cdot)$. We
can also quantify the power as $P_{L}=i\operatorname{\mathrm{Tr}}[\sigma_{LW}\
[V_{in},H_{L}]]$ which is produced in $L$. For the interaction $V_{in}$, we
always find $\dot{Q}_{LH}>0$, $\dot{Q}_{LC}<0$, and $P_{L}>0$. For $W$, the
heat flux and power can be similarly calculated, and they are $\dot{Q}_{W}>0$
and $P_{W}<0$. At steady state, we have

$\displaystyle\dot{Q}_{LH}+\dot{Q}_{LC}+P_{L}=0,$
$\displaystyle\dot{Q}_{W}+P_{W}=0,$ $\displaystyle P=P_{L}+P_{W}.$

Note $\dot{Q}_{L}=\dot{Q}_{LH}+\dot{Q}_{LC}$ and $\dot{Q}_{L}=-P_{L}$. This
may imply that the heat flux from the synthetic bath is quantitatively equal
to the power extracted from $L$. But, as mentioned above, this cannot be just
power, as power is not associated with any entropy flux. For justification, we
may remove the interaction between qutrit $L$ and qubit $W$ and drive $L$ with
an oscillating external field instead. This is exactly what is considered in
Boukobza and Tannor (2007), where power is quantified as the flux of pure
energy being stored in the external field and is not associated with entropy
flow (see Appendix B). Here, on the contrary, we see an entropy flux opposite
to the power extracted (heat flux) in $L$. This leads us to conclude that heat
from a bath with a negative temperature is thermodynamic work but with
negative entropy.

##  V Conclusion

In this article, we have studied steady-state quantum thermodynamics with
negative temperatures. For that, we have engineered synthetic baths by
utilizing two baths at different positive temperatures and letting them weakly
interact with qutrit systems in a particular fashion. These synthetic baths
can assume arbitrary temperatures, including negative ones. These baths with
negative temperatures are exploited to study steady-state thermodynamics. We
have explored the thermodynamic laws, particularly the zeroth and second laws.
We have shown that whenever two synthetic baths with identical temperatures
are brought in contact, there is no heat flow in between. This, in turn,
legitimizes the notion of temperatures in synthetic baths. On the other hand,
for non-identical temperatures, there is a heat flow. Further, heat always
flows spontaneously from a bath with a negative temperature to a positive one.
This implies that a bath with a negative temperature is ‘hotter’ than a bath
with a positive temperature. Further, there is a heat flow from a bath with a
less negative temperature to a bath with a more negative temperature. We have
then studied the Clausius statement of the second law in case of negative
temperatures and found that there is always non-negative entropy production in
the steady-state thermal processes. Then, we have introduced a heat engine
model that operates between two baths, one with negative temperature and the
other with positive temperature. Unlike traditional engines, these engines
always yield unit heat-to-work conversion efficiency. A systematic analysis
has revealed that the heat flow from a bath with a negative temperature is
equivalent to an injection of work into the working system by an equal amount.
This is exactly the reason why these engines yield unit efficiency.

The proposed engines utilizes qutrit and qubit systems and two or three baths
at different temperatures. Thus, these engines can be realized experimentally,
for example, using atom-optical Rossnagel  _et al._ (2016); Klatzow  _et al._
(2019); Wang  _et al._ (2022) and NMR Peterson  _et al._ (2019) systems.
Finally, we conclude that:

  * •

A thermal bath with a negative temperature can be synthesized with two baths
with different positive temperatures.

  * •

In steady-state thermodynamics with negative temperatures, the zeroth law and
the Clausius statement of the second law remain unchanged. However, the
Kelvin-Plack statement is to be appended to incorporate that there is a
spontaneous heat flow from a bath with a negative temperature to a bath with a
positive temperature and from a bath with smaller negative temperature to a
bath with a larger negative temperature.

  * •

A continuous heat engine can be constructed using baths with negative and
positive temperatures. In such engines, the heat-to-work conversion efficiency
is always unity. This is maximum for any device that respects first law, i.e.,
conservation of total energy.

  * •

On the fundamental level, the thermodynamic nature of heat from a bath with a
positive temperature is qualitatively different from the one with a negative
temperature. For the former, heat flows in the same direction as entropy flow.
For the latter, heat flows in the opposite direction of entropy flow. Further,
heat from a bath with a negative temperature is thermodynamic work but with
negative entropy.

###### Acknowledgements.

M.L.B., U.B., and M.L. thankfully acknowledge support from: ERC AdG NOQIA;
Ministerio de Ciencia y Innovation Agencia Estatal de Investigaciones
(PGC2018-097027-B-I00/10.13039/501100011033,
CEX2019-000910-S/10.13039/501100011033, Plan National FIDEUA
PID2019-106901GB-I00, FPI, QUANTERA MAQS PCI2019-111828-2, QUANTERA DYNAMITE
PCI2022-132919, Proyectos de I+D+I “Retos Colaboración" QUSPIN
RTC2019-007196-7); MICIIN with funding from European Union
NextGenerationEU(PRTR-C17.I1) and by Generalitat de Catalunya; Fundació
Cellex; Fundació Mir-Puig; Generalitat de Catalunya (European Social Fund
FEDER and CERCA program, AGAUR Grant No. 2021 SGR 01452, QuantumCAT
U16-011424, co-funded by ERDF Operational Program of Catalonia 2014-2020);
Barcelona Supercomputing Center MareNostrum (FI-2022-1-0042); EU (PASQuanS2.1,
101113690); EU Horizon 2020 FET-OPEN OPTOlogic (Grant No 899794); EU Horizon
Europe Program (Grant Agreement 101080086 - NeQST), National Science Centre,
Poland (Symfonia Grant No. 2016/20/W/ST4/00314); ICFO Internal “QuantumGaud"
project; European Union’s Horizon 2020 research and innovation program under
the Marie-Sklodowska-Curie grant agreement No 101029393 (STREDCH) and No
847648 (“La Caixa" Junior Leaders fellowships ID100010434:LCF/BQ/PI19/
11690013,LCF/ BQ/PI20/ 11760031, LCF/BQ/PR20/11770012,LCF/BQ/ PR21/11840013).
M.L.B acknowledges the financial support from MCIN/AEI/10.13039/5011000 11033.
V.S. acknowledges the financial support from the Institute for Basic Science
(IBS) in the Republic of Korea through the project IBS-R024-D1. M.N.B.
gratefully acknowledges financial support from SERB-DST (CRG/2019/002199),
Government of India.

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##  Appendix A Rotating frame and steady-state thermodynamics

Let us consider the setup discussed in Section IV of the main text. The
Hamiltonian of the working systems are

$\displaystyle H_{L}=(E_{H}-E_{C})|2\rangle\\!\langle
2|+E_{H}|3\rangle\\!\langle 3|,$ $\displaystyle H_{W}=E_{W}|2\rangle\\!\langle
2|,$ $\displaystyle H_{in}^{E}(t)=\delta\ (|11\rangle\langle 22|\ ~e^{i\omega
t}+|22\rangle\langle 11|\ e^{-i\omega t}),$ $\displaystyle
H_{T}^{E}(t)=H_{L}+H_{W}+H_{in}^{E}(t),$

where $E_{W}=E_{H}-E_{C}$. The Hamiltonian $H_{L}$ corresponds to a qutrit
$L$. It weakly interacts with a hot ($H$) and a cold ($C$) bath at inverse
temperatures $\beta_{H}$ and $\beta_{C}$ (as described in Section IV of the
main text) to synthesize a bath with negative inverse temperature
$\beta_{LS}$. The qubit $W$, with Hamiltonian $H_{W}$, weakly interacts with a
bath with positive inverse temperature $\beta_{W}$. The $L$ and $W$ interact
between them with a time-dependent interaction Hamiltonian given by
$H_{in}^{E}(t)$. After having all these interactions, the overall dynamics of
the composite $LW$ is

$\displaystyle~\frac{\partial\rho_{LW}(t)}{\partial
t}=i[\rho_{LW}(t),H_{T}^{E}(t)]+\mathcal{L}_{L}(\rho_{LW}(t))+\mathcal{L}_{W}(\rho_{LW}(t)),~~$
(14)

for a state $\rho_{LW}$, where
$\mathcal{L}_{L}(\cdot)=\mathcal{L}_{LH}(\cdot)+\mathcal{L}_{LC}(\cdot)$ is
the Lindbld super-operator (LOS) representing the dissipative dynamics due to
baths $H$ and $C$, and $\mathcal{L}_{W}(\cdot)$ is the LOS for the bath with
inverse temperature $\beta_{W}$. For this dynamics, the heat flux and power
are defined as Alicki (1979); Boukobza and Tannor (2006a)

$\displaystyle\dot{Q}$
$\displaystyle=\operatorname{\mathrm{Tr}}\left[\frac{\partial\rho_{LW}(t)}{\partial
t}\ H_{T}^{E}(t)\right],$ (15) $\displaystyle P$
$\displaystyle=\operatorname{\mathrm{Tr}}\left[\rho_{LW}\ \frac{\partial
H_{T}^{E}(t)}{\partial t}\right].$ (16)

Note the heat flux $\dot{Q}$ and the power $P$ may have time dependence.

For time-dependent Hamiltonians, the dynamics generally never leads to a
steady state. However, for a periodic time-dependence, as in $H_{in}^{E}(t)$,
there is a rotating frame in which the Hamiltonian can be made time-
independent. For that, a counter-rotation is applied on the laboratory frame
by $U=e^{iH_{R}t}$ with $[H_{R},H_{0}]$, where $H_{0}=H_{L}+H_{W}$. In the
rotating frame, an operator $A$ in the laboratory frame transforms as
$A\to(A)_{R}=UAU^{\dagger}$. Further, there exists a Hamiltonian $H_{R}$ for
which the interaction Hamiltonian reduces to a time-independent one, given by
$V_{in}=UH_{in}^{E}(t)U^{\dagger}$. Accordingly, the overall Hamiltonian
becomes time-independent, and it is $\bar{H}^{E}_{T}=H_{0}-H_{R}+V_{in}$. In
this rotating frame, the overall dynamics is recast as

$\displaystyle~\frac{\partial\rho_{LW}^{R}(t)}{\partial
t}=\mathcal{L}_{U}(\rho_{LW}^{R})+\mathcal{L}_{L}(\rho_{LW}^{R})+\mathcal{L}_{W}(\rho_{LW}^{R}),~~$
(17)

for a state $\rho_{LW}$, with ${\rho}_{LW}^{R}=U\rho_{LW}U^{\dagger}$. Here we
denote $\mathcal{L}_{U}(\rho_{LW}^{R})=i[\rho_{LW}^{R},\bar{H}_{T}^{E}]$ which
is the unitary controbution to the dynamics. This dynamics can lead to a
steady state, say $\sigma^{R}_{LW}$. It can be easily checked that the LOSs
remain unchanged in this rotating frame. Given that
$\operatorname{\mathrm{Tr}}[AB]=\operatorname{\mathrm{Tr}}[(A)_{R}(B)_{R}]$
for two arbitrary operators $A$ and $B$, we may re-express the heat flux and
power as Boukobza and Tannor (2006a)

$\displaystyle\dot{Q}=\operatorname{\mathrm{Tr}}\left[\left(\frac{\partial\rho_{LW}(t)}{\partial
t}\right)_{R}\
\left(H_{T}^{E}(t)\right)_{R}\right]=\operatorname{\mathrm{Tr}}\left[\mathcal{L}(\rho_{LW}^{R})\
H_{0}\right],$ (18)

where
$\mathcal{L}(\rho_{LW}^{R})=\mathcal{L}_{L}(\rho_{LW}^{R})+\mathcal{L}_{W}(\rho_{LW}^{R})$,
and the power as

$\displaystyle P=\operatorname{\mathrm{Tr}}\left[\left(\rho_{LW}\right)_{R}\
\left(\frac{\partial H_{T}^{E}(t)}{\partial
t}\right)_{R}\right]=-i\operatorname{\mathrm{Tr}}\left[\rho_{LW}^{R}\
[H_{0},V_{in}]\right],$ (19)

where $[A,B]=AB-BA$. The heat flux $\dot{Q}$ can be divided into two parts.
One contribution comes from interaction of $L$ with baths $H$ and $C$, i.e.,
$\dot{Q}_{L}=\operatorname{\mathrm{Tr}}\left[\mathcal{L}_{L}(\rho_{LW}^{R})\
H_{0}\right]$ and the other due to interaction between $W$ with its bath,
i.e.,
$\dot{Q}_{W}=\operatorname{\mathrm{Tr}}\left[\mathcal{L}_{W}(\rho_{LW}^{R})\
H_{0}\right]$.

Now we study the entropy flux through $LW$. Note the rate of change in von
Neumann entropy is given by

$\displaystyle\dot{S}$
$\displaystyle=-\operatorname{\mathrm{Tr}}\left[\frac{\partial\rho_{LW}(t)}{\partial
t}\log\rho_{LW}(t)\right]$
$\displaystyle=-\operatorname{\mathrm{Tr}}\left[\left(\frac{\partial\rho_{LW}(t)}{\partial
t}\right)_{R}\log\left(\rho_{LW}(t)\right)_{R}\right].$

In the rotating frame and at steady-state, it reduces to

$\displaystyle\dot{S}=-\operatorname{\mathrm{Tr}}\left[\frac{\partial\sigma_{LW}^{R}}{\partial
t}\log\sigma_{LW}^{R}\right]=0.$

The rate of change in entropy vanishes because the state does not change over
time. Nevertheless, there still can be a non-vanishing flux of entropy passing
through $L$ and $W$. Given that unitary dynamics does not contribute to the
entropy flux, i.e.,
$-\operatorname{\mathrm{Tr}}[\mathcal{L}_{U}(\rho_{LW}^{R})\log\sigma_{LW}^{R}]=0$,
we can calculate the entropy flux through $L$ and $W$ respectively as

$\displaystyle\dot{S}_{L}=-\operatorname{\mathrm{Tr}}[\mathcal{L}_{L}(\rho_{LW}^{R})\log\sigma_{LW}^{R}],$
(20)
$\displaystyle\dot{S}_{W}=-\operatorname{\mathrm{Tr}}[\mathcal{L}_{W}(\rho_{LW}^{R})\log\sigma_{LW}^{R}],$
(21)

where $\dot{S}_{L}+\dot{S}_{R}=0$.

##  Appendix B Power and associated entropy flux in traditional steady-state
engines

In traditional engines, unlike heat, work is a form of energy not associated
with entropy. To understand that, we may reconsider the heat engine composed
of a qutrit and two baths at different temperatures, which is also studied in
Boukobza and Tannor (2007). Consider the qutrit system outlined in Section II
where the Hamiltonian of the qutrit is $H_{0}=(E_{H}-E_{C})|2\rangle\\!\langle
2|+E_{H}|3\rangle\\!\langle 3|$, and it is weakly interacting with a hot ($H$)
and cold ($C$) with inverse temperatures $\beta_{H}$ and $\beta_{C}$
respectively. In addition, an external driving

$\displaystyle H_{d}(t)=\alpha\ (|1\rangle\langle 2|e^{i\omega
t}+|2\rangle\langle 1|e^{-i\omega t})$ (22)

is introduced. As a result, the reduced dynamics becomes

$\displaystyle\dot{\rho}=i\
~[\rho,H_{T}(t)]+\mathcal{L}_{H}(\rho)+\mathcal{L}_{H}(\rho),$ (23)

where $H_{T}(t)=H_{0}+H_{d}(t)$, and $\mathcal{L}_{H}(\cdot)$ and
$\mathcal{L}_{C}(\cdot)$ are LSOs representing the dissipation due to
interactions with hot and cold baths. Again, as discussed in Appendix A, this
dynamics leads to a steady-state, say $\sigma^{R}$, in a rotating frame. The
heat fluxes, entropy fluxes, and power are calculated respectively as

$\displaystyle\dot{Q}_{H}$
$\displaystyle={\operatorname{Tr\,}}\left[\mathcal{L}_{H}(\sigma^{R})H_{0}\right],$
$\displaystyle\dot{Q}_{C}$
$\displaystyle={\operatorname{Tr\,}}\left[\mathcal{L}_{C}(\sigma^{R})H_{0}\right],$
$\displaystyle\dot{S}_{H}$
$\displaystyle=-{\operatorname{Tr\,}}\left[\mathcal{L}_{H}(\sigma^{R})\log\sigma^{R}\right],$
$\displaystyle\dot{S}_{C}$
$\displaystyle=-{\operatorname{Tr\,}}\left[\mathcal{L}_{C}(\sigma^{R})\log\sigma^{R}\right],$
$\displaystyle P$ $\displaystyle=-i{\operatorname{Tr\,}}\left[\sigma^{R}\
[H_{0},V]\right],$

where $V=\alpha\ (|1\rangle\langle 2|+|2\rangle\langle 1|)$. Here
$\dot{Q}_{X}$ and $\dot{S}_{X}$ represent the heat and entropy fluxes from
bath $X=H,C$, and $P$ is the power. As required by first law,
$\dot{Q}_{H}+\dot{Q}_{C}+P=0$ and, at steady-state,
$\dot{S}_{H}=-\dot{S}_{C}$. When the device operates as an engine, we have
$\dot{Q}_{H}>0$, $\dot{Q}_{C}<0$, and $P<0$. In this case, heat enters from
the hot bath. Part of that heat is converted into work which is stored in the
driving field, and the rest is dumped into the cold bath. Further, in this
case, $\dot{S}_{H}>0$ and $\dot{S}_{C}<0$. Clearly, at steady-state, whatever
amount of entropy enters the system from the hot bath is released to the cold
bath. Thus the work-flux, i.e., power $P$, does not carry any entropy.