Thermodynamic resource theories, non-commutativity and maximum entropy principles

Our derivation [3] of the generalised Landauer's bound is based on the following assumptions, that follow the framework introduced in [4]:

• 1.  
  The initial state S of the system, i.e. the memory to be erased, and the reservoir R are initially in a product state ${\rho }_{{SR}}={\rho }_{S}\otimes {\rho }_{R}$. This is the most natural framework, because it models what happens in a typical erasure process. Indeed, allowing initial correlations between memory and reservoirs implies that we could erase a memory while extracting work at the same time, by exploiting the work value of correlations [5–7].
• 2.  
  The reservoirs have the form of a Generalised Gibbs Ensemble,

  $$\begin{eqnarray}&&{\rho }_{R}=\displaystyle \frac{{{\rm{e}}}^{-{\sum }_{i}{\mu }_{i}{C}_{i}}}{{\mathrm{Tr}}_{}\left[{{\rm{e}}}^{-{\sum }_{i}{\mu }_{i}{C}_{i}}\right]},\end{eqnarray} \tag{ 1 }$$

  for observables C_i (we set ${C}_{0}:= H$, the Hamiltonian of the system and ${\mu }_{0}:= \beta =1/({kT})$ the inverse temperature). This bath may factorise in the product of baths if all C_i commute, but this is not necessary for the following.
• 3.  
  The total system of memory and reservoir is isolated, so it undergoes a general unitary evolution U:

  $$\begin{eqnarray*}&&{\rho }_{{SR}}\ \mathop{\longmapsto }\limits^{U}\ {\rho }_{{SR}}^{\prime }.\end{eqnarray*}$$

To fix the notation,

$$\begin{eqnarray*}{\rho }_{S}^{\prime } & := & {\mathrm{Tr}}_{R}[{\rho }_{{SR}}^{\prime }],\,{\rho }_{R}^{\prime }:= {\mathrm{Tr}}_{S}[{\rho }_{{SR}}^{\prime }],\\ {\rm{\Delta }}{C}_{i} & = & {\mathrm{Tr}}_{}[{C}_{i}({\rho }_{R}^{\prime }-{\rho }_{R})],\quad S(\rho | | \sigma ):= -S(\rho )-{\mathrm{Tr}}_{}[\rho \mathrm{log}\sigma ],\end{eqnarray*}$$

with $S(\rho )=-{\mathrm{Tr}}_{}[\rho \mathrm{log}\rho ]$ the von Neumann entropy. From the notation above, ${\rm{\Delta }}{C}_{0}:= {\rm{\Delta }}H$ is the heat flow towards the bath. Then,

Result 1 Under the assumptions 1, 2 and 3,

$$\begin{eqnarray}&&\beta {\rm{\Delta }}H+\displaystyle \sum _{i\geqslant 1}{\mu }_{i}{\rm{\Delta }}{C}_{i}\geqslant -{\rm{\Delta }}{S}_{S},\end{eqnarray} \tag{ 2 }$$

where ${\rm{\Delta }}{S}_{S}=S({\rho }_{S}^{\prime })-S({\rho }_{S})$.

Proof. The proof of [4] (theorem 3) goes through independently of the non-commutativity of the C_i. Let ${\rm{\Delta }}{S}_{X}=S({\rho }_{X}^{\prime })-S({\rho }_{X})$, with $X=S,R$, and denote by $I(S^{\prime} \,:R^{\prime} ):= S({\rho }_{S}^{\prime })+S({\rho }_{R}^{\prime })-S({\rho }_{{SR}}^{\prime })$ the mutual information in the final state. Unitary invariance of the von Neumann entropy gives $S({\rho }_{{SR}})=S({\rho }_{{SR}}^{\prime })$. Then,

$$\begin{eqnarray}&&{\rm{\Delta }}{S}_{S}+{\rm{\Delta }}{S}_{R}=I(S^{\prime} \,:\,R^{\prime} ),\end{eqnarray} \tag{ 3 }$$

i.e. the sum of the changes of the local entropies equals the correlations created in the transformation, as measured by the mutual information (this is a refinement of ${\rm{\Delta }}{S}_{S}+{\rm{\Delta }}{S}_{R}\geqslant 0$). Substituting equation (1) in the expression for ${\rm{\Delta }}{S}_{R}$ one finds

$$\begin{eqnarray}&&{\rm{\Delta }}{S}_{R}=\beta {\rm{\Delta }}H+\displaystyle \sum _{i\geqslant 1}{\mu }_{i}{\rm{\Delta }}{C}_{i}-S({\rho }_{R}^{\prime }| | {\rho }_{R}).\end{eqnarray} \tag{ 4 }$$

This equation reduces to Clausius relation ${\rm{\Delta }}H={kT}{\rm{\Delta }}{S}_{R}$ when we can make the assumption $S({\rho }_{R}^{\prime }| | {\rho }_{R})\approx 0$ (which is typically the case for a macroscopic bath) and when only energy flows are involved. Together with equation (3) this gives

$$\begin{eqnarray*}&&-{\rm{\Delta }}{S}_{S}+I(S^{\prime} \,:R^{\prime} )=\beta {\rm{\Delta }}H+\displaystyle \sum _{i\geqslant 1}{\mu }_{i}{\rm{\Delta }}{C}_{i}-S({\rho }_{R}^{\prime }| | {\rho }_{R}).\end{eqnarray*}$$

From the non-negativity of mutual information and relative entropy, we obtain equation (2).□

The above analysis indicates that the resultant trade-off is not particularly sensitive to the fact that the C_i may not commute. The only impact the non-commutativity has is that throughout the sequence of bath interactions one obtains a 'time-dependent' pairs of expectations $(\langle {C}_{i}(t)\rangle ,\langle {C}_{j}(t)\rangle )$ subject to the uncertainty in the two observables.

The simplest system with multiple conserved quantities is one with Hamiltonian H and a single conserved charge Q, e.g, angular momentum in some fixed direction. As discussed the reservoir is assumed to have the form of equation (1),

$$\begin{eqnarray}&&{\gamma }_{R}=\displaystyle \frac{{{\rm{e}}}^{-\beta {H}_{R}-\alpha {Q}_{R}}}{{\mathrm{Tr}}_{}[{{\rm{e}}}^{-\beta {H}_{R}-\alpha {Q}_{R}}]}.\end{eqnarray} \tag{ 5 }$$

We take here $[H,Q]=[{H}_{R},{Q}_{R}]=0$. H_R, Q_R are observables on the bath Hilbert space physically corresponding to H and Q on the system. β and α are fixed inverse temperatures. Since we assume $[{H}_{R},{Q}_{R}]=0$, one can formally factor the quantum state of equation (5). We will view it as if we have two independent baths at our disposal (a thermal bath and a 'Q-bath') and we are able to put the system in contact with each of them separately.

Consider for simplicity the system Hilbert space ${ \mathcal H }={{\mathbb{C}}}^{2}\otimes {{\mathbb{C}}}^{2}$, spanned by the eigenstates of two commuting observables $H\otimes {\mathbb{1}}$ and ${\mathbb{1}}\otimes Q$. Define the states $\{| 00\rangle ,| 01\rangle ,| 10\rangle ,| 11\rangle \}$, where $| {hq}\rangle := | h\rangle \otimes | q\rangle$ and $| h\rangle$, $| q\rangle$ are eigenstates of H and Q, respectively. Assume these four states to be initially degenerate in energy H and charge Q. Landauer's principle is usually stated as a fundamental lower bound on the energy cost of the process of information erasure [2]. We now show explicitly that this is not necessarily the case.

One can erase a qubit system in the angular momentum degree of freedom using a spin bath, with the erasure cost respecting a bound of the form ${\rm{\Delta }}Q\geqslant {\alpha }^{-1}\mathrm{log}2$ and zero energy cost [8] (here α is the inverse 'temperature' of the spin bath). Going beyond this, we illustrate here the tradeoff between costs in different charges proven in equation (2).

Let us encode classical bits in the states $| 00\rangle$ and $| 10\rangle$ in ${ \mathcal H }$. We now develop a one-parameter family of optimal protocols (see figure 1):

• 1.  
  Let us start with the levels $| 00\rangle$ and $| 01\rangle$ in a maximally mixed state (which describes the single bit memory to be erased):

  $$\begin{eqnarray*}&&{\rho }_{1}=1/2| 00\rangle \langle 00| +1/2| 10\rangle \langle 10| .\end{eqnarray*}$$

  Also notice that, since $| 00\rangle$ and $| 10\rangle$ are degenerate in energy, ${\rho }_{1}$ is initially in thermal equilibrium with the thermal bath.
• 2.  
  Changing the Hamiltonian of the system, map the state $| 01\rangle$ with energy $\bar{\epsilon }$ to a new state (still denoted by $| 01\rangle$) with energy $\bar{\epsilon }+{\rm{\Delta }}\bar{\epsilon }$, at an average cost $p(\bar{\epsilon }){\rm{\Delta }}\bar{\epsilon }$. Then put the system in contact to the heat bath and completely thermalise it with respect to the current Hamiltonian. Repeating these operations in a sequence of $N\to \infty$ steps with ${\rm{\Delta }}\bar{\epsilon }\to 0$, the cost of raising the energy level from 0 to ${\epsilon }$ is given by

  $$\begin{eqnarray*}&&{\rm{\Delta }}H={\int }_{0}^{\epsilon }\displaystyle \frac{{{\rm{e}}}^{-\beta \bar{\epsilon }}}{1+{{\rm{e}}}^{-\beta \bar{\epsilon }}}{\rm{d}}\bar{\epsilon }=\displaystyle \frac{1}{\beta }\mathrm{ln}\left(\displaystyle \frac{2}{1+{{\rm{e}}}^{-\beta \epsilon }}\right).\end{eqnarray*}$$

  The final state of the memory to be erased is

  $$\begin{eqnarray*}&&{\rho }_{2}=\displaystyle \frac{1}{1+{{\rm{e}}}^{-\beta \epsilon }}| 00\rangle \langle 00| +\displaystyle \frac{{{\rm{e}}}^{-\beta \epsilon }}{1+{{\rm{e}}}^{-\beta \epsilon }}| 10\rangle \langle 10| .\end{eqnarray*}$$
• 3.  
  Now we make use of the charge degree of freedom. The two states $| 00\rangle$ and $| 01\rangle$ are degenerate in charge Q. Following the same procedure as before we induce a level-raising $| 01\rangle$ from 0 to q, where we choose

  $$\begin{eqnarray}&&q=\beta \epsilon /\alpha .\end{eqnarray} \tag{ 6 }$$

  The reason for this choice will be clear in a moment. This can be done at no cost, because there is no population in $| 01\rangle$, i.e. $p(q)=0$ throughout the process³ .
• 4.  
  We apply a unitary that performs the swap $| 00\rangle \to | 00\rangle$, $| 01\rangle \leftrightarrow | 10\rangle$, $| 11\rangle \to | 11\rangle$. The costs associated to this unitary are

  $$\begin{eqnarray*}&&{\rm{\Delta }}H=-\displaystyle \frac{{{\rm{e}}}^{-\beta \epsilon }}{1+{{\rm{e}}}^{-\beta \epsilon }}\epsilon ,\quad {\rm{\Delta }}Q=\displaystyle \frac{{{\rm{e}}}^{-\beta \epsilon }}{1+{{\rm{e}}}^{-\beta \epsilon }}q,\end{eqnarray*}$$

  and the final state, using equation (6), is

  $$\begin{eqnarray*}&&{\rho }_{3}=\displaystyle \frac{1}{1+{{\rm{e}}}^{-\alpha q}}| 00\rangle \langle 00| +\displaystyle \frac{{{\rm{e}}}^{-\alpha q}}{1+{{\rm{e}}}^{-\alpha q}}| 01\rangle \langle 01| .\end{eqnarray*}$$
• 5.  
  We can complete the erasure leaving the system in contact with the Q-bath and raising the level $| 01\rangle$ from q to $\infty$. Thanks to the choice of the initial q (equation (6)), ${\rho }_{3}$ is initially in equilibrium with the Q-bath. Hence this part of the protocol is formally analogous to steps 1–2, but is achieved using a physically different bath (e.g. a spin bath). The charge cost of the partial erasure in the charge basis is

  $$\begin{eqnarray*}&&{\rm{\Delta }}Q={\int }_{\tfrac{\beta \epsilon }{\alpha }}^{+\infty }\displaystyle \frac{{{\rm{e}}}^{-\alpha \bar{q}}}{1+{{\rm{e}}}^{-\alpha \bar{q}}}{\rm{d}}\bar{q}=\displaystyle \frac{1}{\alpha }\mathrm{ln}(1+{{\rm{e}}}^{-\beta \epsilon }).\end{eqnarray*}$$

  The final (erased) state of the memory is $| 00\rangle$. This completes the protocol.

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The total cost of erasure splits into an energy contribution and a charge contribution; these can be expressed in terms of the single parameter ${\epsilon }$, the energy at which we decide to swap basis:

$$\begin{eqnarray}&&{\rm{\Delta }}{H}_{{\rm{tot}}}(\epsilon )=\displaystyle \frac{1}{\beta }\mathrm{ln}\left(\displaystyle \frac{2}{1+{{\rm{e}}}^{-\beta \epsilon }}\right)-\displaystyle \frac{{{\rm{e}}}^{-\beta \epsilon }}{1+{{\rm{e}}}^{-\beta \epsilon }}\epsilon ,\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}{Q}_{{\rm{tot}}}(\epsilon )=\displaystyle \frac{1}{\alpha }\mathrm{ln}(1+{{\rm{e}}}^{-\beta \epsilon })+\displaystyle \frac{{{\rm{e}}}^{-\beta \epsilon }}{1+{{\rm{e}}}^{-\beta \epsilon }}\displaystyle \frac{\beta }{\alpha }\epsilon .\end{eqnarray} \tag{ 8 }$$

The result is shown in figure 2. Each value of defines a different protocol, corresponding to a point in the 'energy-cost' versus 'charge cost' graph. The protocols described yield a curve (${\rm{\Delta }}{H}_{{\rm{tot}}}(\epsilon ),{\rm{\Delta }}{Q}_{{\rm{tot}}}(\epsilon )$) for fixed inverse temperatures β and α.

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As expected, we recover the usual Landauer erasure bound, ${\rm{\Delta }}{H}_{{\rm{tot}}}={\beta }^{-1}\mathrm{ln}2$, in the $\epsilon \to \infty$ limit and the erasure at no energy of [8], ${\rm{\Delta }}{Q}_{{\rm{tot}}}={\alpha }^{-1}\mathrm{ln}2$, in the $\epsilon \to 0$ limit. The curves in figure 2 interpolating between these two limits satisfy tightly the bound of equation (2), as can be seen combining equations (7) and (8):

$$\begin{eqnarray}&&\beta {\rm{\Delta }}{H}_{{\rm{tot}}}(\epsilon )+\alpha {\rm{\Delta }}{Q}_{{\rm{tot}}}(\epsilon )=\mathrm{ln}2,\quad \forall \epsilon \in [0,+\infty ].\end{eqnarray} \tag{ 9 }$$

This shows how simple trade-offs exist for erasure in the presence of multiple conserved charges. One could also compare with [9], where the authors show that if erasure is performed with respect to angular momentum degrees of freedom that are not degenerate in energy, equation (9) cannot be achieved.

Note that in a real experiment it might not be physically possible to modify the level structure arbitrarily, as was assumed here. This is the case for a Q-bath being a spin bath. One wishes to change the level structure (in angular momentum) by introducing multiple aligned spins and performing CNOT operations [8]. However Nature places a physical lower bound on the discrete steps, $\sim {\hslash }$. This does not change the general picture presented in the optimal protocol above, but introduces non-optimalities, relevant at low temperatures, which might be worth exploring (see [8] and appendix A)⁴ .

Why do such tradeoffs emerge? We now develop a broader perspective, based on the clash between different notions of equilibrium in the presence of multiple conserved quantities.

A typical way to proceed is to invoke the maximum entropy principle [15]. Let ${ \mathcal C }:= \{{C}_{i}\}$ denote the set of conserved quantities and ${\bar{c}}_{i}$ is their average value at equilibrium (by convention, ${C}_{0}:= H$, the Hamiltonian of the system being trivially a conserved change). The principle says that the equilibrium state is the solution to the problem [16] ⁵

$$\begin{eqnarray}&&{\rm{Maximize}}\,{\rm{}}S(\rho )=-{\mathrm{Tr}}_{}[\rho \mathrm{log}\rho ],\\ &&{\rm{subject}}\,{\rm{to:}}{\mathrm{Tr}}_{}[\rho \,{C}_{i}]={\bar{c}}_{i},\ k=1,\ldots ,n.\end{eqnarray} \tag{ 10 }$$

The solution is a so-called Generalised Gibbs Ensemble with respect to the conserved quantities, i.e.

$$\begin{eqnarray}&&{\rho }_{{ \mathcal C }}={{\rm{e}}}^{-{\displaystyle \sum }_{i}{\mu }_{i}{C}_{i}/}{\mathrm{Tr}}_{}\left[{{\rm{e}}}^{-{\displaystyle \sum }_{i}{\mu }_{i}{C}_{i}}\right],\end{eqnarray} \tag{ 11 }$$

where ${\mu }_{i}$ are generalised chemical potential that can be easily computed as functions of the given values ${\bar{c}}_{i}$ (by the convention chosen, ${\mu }_{0}:= \beta =1/({kT})$).⁶ If ${ \mathcal C }$ consists of only the Hamiltonian, one recovers the standard Gibbs ensemble. ${ \mathcal C }=\{H,N\}$, where N is the particle number operator, gives the grand canonical ensemble. Of course, as N is by assumption a conserved quantity, $[H,N]=0$; hence, non-commutativity cannot appear in this problem. Non-commutativity, on the other hand, appears in the examples given at the beginning of this section.

A brief comment is necessary regarding the use of the maximum entropy principle in the presence of non-commuting quantities. In the commuting case, this can be justified as an essentially unique inference method satisfying consistency axioms in the handling of information [18]. In general, the principle can be given a microcanonical derivation. This is based on 1. Identifying expectation values with ensemble averages 2. Assigning equal probability to every element of an ensemble characterised by a total C_i peaked around ${\bar{c}}_{i}$. Technical challenges arise in the non-commutativity case from the fact that the total C_is are only approximately mutually commuting for any finite ensemble, so they cannot all have strictly well-defined values. These issues were tackled in [19], and recently in [11], through the notion of an approximate microcanonical ensemble (Def. 2 section 3 of [11] and equation (2.24) of [19]). This leads to the recovery of equation (11).

Once the maximum entropy principle is accepted, the important observation is the following: the construction works equally well for commuting or non-commuting observables, In every case, it gives a state of the form of equation (11). However, there are subtler differences. Consider the projection $\rho \mapsto {\rho }_{{ \mathcal C }}$, where ${\rho }_{{ \mathcal C }}$ is the maximum entropy state among all states σ satisfying ${\mathrm{Tr}}_{}[\sigma {C}_{i}]={\mathrm{Tr}}_{}[\rho {C}_{i}]$ for all ${C}_{i}\in { \mathcal C }$. This map is not necessarily completely positive when ${ \mathcal C }$ contains mutually non-commuting observables (see appendix B). This agrees with the fact that the maximum entropy projection entails an inherently inferential procedure and, strictly speaking, cannot be captured purely dynamically without invoking some notion of coarse-graining or state-dependent processes. Another difference regards discontinuities induced by non-commutativity, as we mention later.

We now compare this construction to another standard thermodynamic approach based on the notions of passivity and complete passivity. We show that a disagreement emerges in the presence of multiple conserved quantities and discuss its significance.

Among various ways of describing the content of the second law of thermodynamics, one is that no work can be extracted from a system in thermal equilibrium by means of an adiabatic process [20]. Adiabatic here means a process where some external controls are varied for some finite time, inducing a unitary evolution generated by a time-dependent Hamiltonian H(t). The total work done on the system is then traditionally defined as

$$\begin{eqnarray}&&W(t)={\int }_{0}^{t}{\mathrm{Tr}}_{}\left[\rho (t)\displaystyle \frac{{\rm{d}}{H}(t)}{{\rm{d}}{t}}\right]{\rm{d}}{t},\end{eqnarray} \tag{ 12 }$$

where $\rho (t)$ evolves under the Schrödinger equation, ${\rm{d}}\rho (t)/{\rm{d}}{t}=-{\rm{i}}[\rho (t),H(t)]$. Equivalently, one can easily check that if U is the unitary evolution generated by H(t) up to the final interaction time t_F,

$$\begin{eqnarray}&&W({t}_{F})={\mathrm{Tr}}_{}[U\rho {U}^{\dagger }H]-{\mathrm{Tr}}_{}[\rho H]:= {W}_{H}(U,\rho ),\end{eqnarray} \tag{ 13 }$$

where ρ and H denote, respectively, the state and the Hamiltonian at the initial time [20]. The second law in the form stated above imposes that ρ, when in equilibrium, should take a form such that ${W}_{H}(U,\rho )\geqslant 0$ for all unitaries U. This can be interpreted as the fact that one can never displace a weight system upwards using the state ρ. If this holds for every unitary evolution, ρ is called a passive state. A relatively straightforward computation shows that a state ρ is passive if and only if $[\rho ,H]=0$ and its eigenvalues are a non-increasing function of energy (i.e. there are no 'population-inversions') [20–22].

More importantly, in 1978 an answer was formulated for the following question: what states are completely passive, in the sense that arbitrarily many copies cannot raise a weight? More precisely, for what states ρ do we have ${W}_{H}(U,{\rho }^{\otimes n})\geqslant 0$ for all $n\in {\mathbb{N}}$ and all unitaries U? The answer was given first from an algebraic perspective [21] and later by Lenard using finite-dimensional methods [20]. One consequence of these seminal works is that, for finite-dimensional systems, the Gibbs state is the only functional form (modulo limiting cases of ground states and 'infinite temperature' systems) that does not trivialise every work process when it is assumed to be freely available⁷ .

Extending equation (13) to other conserved quantities beyond energy, passivity and complete passivity can be defined in an analogous way:

Definition 1. Given an observable C, a state ρ is called C-passive if ${W}_{C}(U,\rho )\geqslant 0$ for every unitary U. ρ is C-completely passive if ${\rho }^{\otimes n}$ is C-passive for every $n\in {\mathbb{N}}$.

In the case of a single conserved quantity, there is agreement between the maximum entropy and the complete passivity point of views of equilibrium. For example, a Z-spin bath with degenerate Hamiltonian has a maximum entropy state $\propto {{\rm{e}}}^{-\alpha {L}_{z}}$, which is completely passive with respect to L_z, so that angular momentum in the Z-direction cannot be extracted from any number of copies of it by any unitary interaction.

On the other hand, let us analyse the multiple observables case. We then have

Result 2 Let ${ \mathcal C }=\{H,{C}_{1},\ldots ,{C}_{n}\}$. Then, for $n\geqslant 1$, the maximum entropy state $\propto \exp \left[{\sum }_{i=0}^{n}{\mu }_{i}{C}_{i}\right]$ is $C({\boldsymbol{\mu }})$-completely passive, where $C({\boldsymbol{\mu }}):= {\sum }_{i=0}^{n}{\mu }_{i}{C}_{i}$, but in general it is not ${C}_{i}$-completely passive for each $i$.

Proof. Consider first the case where C_i are mutually commuting. It then suffices to focus on the case where there is a single extra conserved charge C₁ beyond energy (with ${C}_{1}\ne H$). In this case the maximum entropy state ${\rho }_{{ \mathcal C }}$ reads, due to $[H,{C}_{1}]=0$,

$$\begin{eqnarray*}&&{\rho }_{{ \mathcal C }}\propto {{\rm{e}}}^{-\beta H}{{\rm{e}}}^{-{\mu }_{1}{C}_{1}}:= {\gamma }_{0}{\gamma }_{1}.\end{eqnarray*}$$

A necessary condition for ${\gamma }_{0}{\gamma }_{1}$ to be H-completely passive is to be H-passive. Passivity is equivalent to 1. $[{\gamma }_{0}{\gamma }_{1},H]=0$ (which is satisfied) and 2. The eigenvalues of ${\gamma }_{0}{\gamma }_{1}$ are monotonically decreasing for strictly increasing eigenvalues of H. However fix ${\epsilon }$ and $\tilde{\epsilon }$ with $\epsilon \lt \tilde{\epsilon }$. One in general can find ${\ell }\gt \tilde{{\ell }}$ such that ${{\rm{e}}}^{-\beta \epsilon -{\mu }_{1}{\ell }}\gt {{\rm{e}}}^{-\beta \tilde{\epsilon }-{\mu }_{1}\tilde{{\ell }}}$. Hence H-passivity is generically violated.

Consider now the non-commuting case. Given a conserved quantity C, a necessary condition for passivity is $[\rho ,C]=0$. A useful equivalent way of thinking about this condition is that ρ must be symmetric under the U(1) group generated by C, i.e. ${{\rm{e}}}^{-{\rm{i}}Ct}\rho {{\rm{e}}}^{{\rm{i}}Ct}=\rho$ for all t. Consider then the simple example ${ \mathcal C }=\{H,{L}_{x},{L}_{y},{L}_{z}\}$, with H spherically symmetric. The maximum entropy state of equation (11) reads

$$\begin{eqnarray}&&{\rho }_{{ \mathcal C }}\propto {{\rm{e}}}^{-\beta H}{{\rm{e}}}^{-{\mu }_{x}{L}_{x}-{\mu }_{y}{L}_{y}-{\mu }_{z}{L}_{z}}:= {{\rm{e}}}^{-\beta H}{{\rm{e}}}^{-{\boldsymbol{\mu }}\cdot {\boldsymbol{L}}},\end{eqnarray} \tag{ 14 }$$

where ${\boldsymbol{\mu }}:= ({\mu }_{x},{\mu }_{y},{\mu }_{z})$ and ${\boldsymbol{L}}=({L}_{x},{L}_{y},{L}_{z})$. Assuming ${\boldsymbol{\mu }}\ne {\bf{0}}$, one can quickly check that this generalised Gibbs ensemble is not spherically symmetric, i.e. it is not invariant under the SU(2) symmetry group generated by L_x, L_y, L_z. In fact, ${\rho }_{{ \mathcal C }}$ is only invariant under a U(1) subgroup representing rotations about the direction ${\boldsymbol{\mu }}$. This implies that in general the state is not passive with respect any of the conserved quantities L_i, and so it is not L_i-completely passive.

Finally, showing that the maximum entropy state is $C({\boldsymbol{\mu }})$-completely passive is based on a standard argument, that we give in appendix C.□

Notice that we cannot strengthen result 2 and say that ${\rho }_{{ \mathcal C }}$ is not C_i-completely passive for any i, even under the additional assumption ${\mu }_{i}\ne 0$ for all i. Here is a counterexample: let ${C}_{1}=| +\rangle \langle +|$, ${C}_{2}=| 0\rangle \langle 0|$, ${C}_{3}=| 1\rangle \langle 1|$ be observables on an effective qubit degree of freedom, with ${\mu }_{1}={\mu }_{2}={\mu }_{3}:= \mu \ne 0$ (H commutes so we can leave it out of this discussion). One can check that $[{C}_{1},{C}_{2}]\ne 0$, $[{C}_{1},{C}_{3}]\ne 0$. But ${\rho }_{C}\propto {{\rm{e}}}^{-\mu S}$, $S={C}_{1}+{C}_{2}+{C}_{3}={\mathbb{1}}+{C}_{1}$, so ${\rho }_{{ \mathcal C }}$ is completely passive with respect to C₁.

What result 2 implies for the example of angular momentum presented above is that from arbitrarily many copies of the maximum entropy state of a system with rotationally invariant Hamiltonian we can generically extract an unbounded amount of angular momentum in every direction excluding ${\boldsymbol{\mu }}$. We can now understand the tradeoffs of result 1 from result 2, noticing that the maximum entropy state used in the derivation is $\beta {\rm{\Delta }}H+{\sum }_{i\geqslant 1}{\mu }_{i}{C}_{i}$-completely passive, but not C_i-completely passive for each i.

Let us come back to the above example of a spherically symmetric Hamiltonian H, with ${ \mathcal C }=\{H,{L}_{x},{L}_{y},{L}_{z}\}$, and ${\rho }_{{ \mathcal C }}$ given by equation (14). Now notice that ${\rho }_{{ \mathcal C }}={\rho }_{\tilde{{ \mathcal C }}}$, where $\tilde{{ \mathcal C }}=\{\tilde{H}({\boldsymbol{\mu }})\}$ and $\tilde{H}({\boldsymbol{\mu }})=H+{\beta }^{-1}{\boldsymbol{\mu }}\cdot {\boldsymbol{L}}$. In fact, this is true in general, taking $\tilde{{ \mathcal C }}=\{\tilde{H}({\boldsymbol{\mu }})\}$, $\tilde{H}({\boldsymbol{\mu }})=H+{\beta }^{-1}C({\boldsymbol{\mu }})$.

What this means is that the maximum entropy construction gives the same state for a system with multiple observables that it would assign to a system with a single Hamiltonian $\tilde{H}({\boldsymbol{\mu }})$, where the Lagrange multipliers of the extremum problem play the role of coupling strengths ${\mu }_{i}$ to the various charges. In other words, we can think of $C({\boldsymbol{\mu }})$-complete passivity in result 2 as a constrained form of passivity, that coincides with the standard one applied only to a particular 'direction' ${\boldsymbol{\mu }}$.

In the example ${ \mathcal C }=\{H,{L}_{x},{L}_{y},{L}_{z}\}$, the generalised Gibbs state coincides with the equilibrium state of a system with the same Hamiltonian and subject to a magnetic field in the direction of ${\boldsymbol{\mu }}$. In this dual description the Lagrange multipliers are naturally interpreted as constraint parameters defined by the physics; the residual U(1) symmetry of ${\rho }_{{ \mathcal C }}$ has an obvious interpretation as a special direction singled out by the physical problem.

There are good reasons why it may be sensible to frame the problem in this way. It is well-established that the low temperature states of interacting spin systems (such as Ising models with transverse magnetic fields) display thermodynamic phase transitions depending on the particular external field parameters, and are intrinsically quantum-mechanical in origin. We may argue that a thermodynamic approach to non-commuting conserved charges benefits from starting with this weakened form of passivity, computing its properties, and then ascertaining if subsequent variation of the parameters displays discontinuities. In fact, it is known that the maximum entropy inference can have discontinuities [23] if some of the conserved quantities are mutually non-commuting (whereas it is continuous in the commuting case [24]), and that these are related to quantum phase transitions [25].

These considerations seem to favour an approach based on constrained passivity and in agreement with the maximum entropy principle. On the other hand, as we now clarify, result 2 also shows that the maximum entropy state effectively acts as a (quantum) reference frame [26]. We discuss why this challenges a proper inclusion of non-commutativity within the so-called resource theory approach to thermodynamics.

Recently a resource theory formulation of thermodynamics has been put forward, initiated in [27, 28]. Every resource theory is based on two notions: a subset of all quantum maps defines the set of allowed operations, and a subset of all preparations defines the set of free states ${ \mathcal F }$ (these are generically assumed to be available in any number). For example, in the theory of entanglement, the free operations are local operations and classical communication and ${ \mathcal F }$ is given by the set of separable states. Resource states are all entangled states.

In the resource theory of thermal operations the allowed transformations are defined through a conservation law: they are all unitaries preserving energy; and ${ \mathcal F }$ is given by all thermal states, i.e. ${\gamma }_{R}={{\rm{e}}}^{-\beta {H}_{R}}/{\mathrm{Tr}}_{}[{{\rm{e}}}^{-\beta {H}_{R}}]$, for arbitrary H_R and fixed β. We are also allowed to trace away some degrees of freedom. Combining the above, a general thermal operation can be written as

$$\begin{eqnarray}&&{ \mathcal E }(\rho )={\mathrm{Tr}}_{R^{\prime} }[U(\rho \otimes {\gamma }_{R}){U}^{\dagger }],\end{eqnarray} \tag{ 15 }$$

with U satisfying $[U,H+{H}_{R}]=0$, H the Hamiltonian of the system and ${\gamma }_{R}\in { \mathcal F }$. In general $R\ne R^{\prime}$. A detailed discussion of these assumptions and their connection to other approaches is given in appendix D.

Here we extend the conservation law on which thermal operations are defined to multiple, and in general non-commuting, conserved quantities. For simplicity, we can limit ourselves to two conserved quantities, the extension to more charges being obvious. We define $(H,{C}_{1},{C}_{2})$-thermal operations as the set of transformations whose Stinespring dilation reads

$$\begin{eqnarray}&&{ \mathcal E }(\rho )={\mathrm{Tr}}_{R^{\prime} }[V(\rho \otimes {\gamma }_{R}){V}^{\dagger }],\end{eqnarray} \tag{ 16 }$$

with V obeying

$$\begin{eqnarray*}&&[V,X\otimes {{\mathbb{1}}}_{R}+{\mathbb{1}}\otimes {X}_{R}]=0,\quad X\,=\,H,{C}_{1},{C}_{2}\end{eqnarray*}$$

and ${\gamma }_{R}\in {{ \mathcal F }}_{M}$. Here ${{ \mathcal F }}_{M}$ is an appropriate generalisation of the set of free states to multiple and generally non-commuting conserved quantities. How do we choose ${{ \mathcal F }}_{M}$?

In the case of a single conservation law the choice is based on the notion of non-trivial work transformations. A resource theory is called non-trivial when some transitions $\rho \to \sigma$ are not allowed by means of free operations. An infinite set of possible non-trivial resource theories can be built from different choices of ${ \mathcal F }$, given the conservation law on energy. These include the resource theory of thermal operations, U(1)-asymmetry theory (where ${ \mathcal F }$ is given by all states ρ invariant with respect to time translations [29], ${{\rm{e}}}^{-{\rm{i}}{H}{t}}\rho {{\rm{e}}}^{-{\rm{i}}{H}{t}}=\rho$), and theories where ${ \mathcal F }$ includes states with some specific modes of U(1)-asymmetry (as defined in [30]).

A natural question is what singles out ${ \mathcal F }$ in the resource theory of thermodynamics among all non-trivial theories. An answer was given in [31], where it was shown that a state $\propto {{\rm{e}}}^{-\beta H}$ is the only form ensuring non-trivial work processes. Specifically, this is the only choice of free state that does not allow to increase arbitrarily the average energy of a battery system (theorem 8 of [31]). This is linked to the fact that the Gibbs state is, modulo limiting cases, the only completely passive state, and of course it also agrees with a choice of ${ \mathcal F }$ based on the maximum entropy principle. All guiding principles suggest one and the same choice of ${ \mathcal F }$.

We are interested here in the generalisation of the resource theory of thermal operations in the presence of multiple conserved quantities. Due to the incompatibility between different approaches, it seems there are at least two distinct possibilities to choose ${{ \mathcal F }}_{M}$:

• 1.  
  ${{ \mathcal F }}_{M}$ should be given by states that are completely passive with respect to all thermodynamic variables.
• 2.  
  ${{ \mathcal F }}_{M}$ should be given by the maximum entropy construction, or equivalently it should respect a constrained form of complete passivity, as established by result 2. This is the choice made in [11].

As we will now see, both these approaches have their limitations, highlighting current issues in the resource theory approach to thermodynamics.

If we follow the complete passivity approach, we find that the free states γ should satisfy $[\gamma ,{C}_{i}]=0$ for all C_i. In the generic case this implies

$$\begin{eqnarray}&&\gamma =\displaystyle \sum _{k}{a}_{k}{O}_{k},{\rm{with}}\,{a}_{k}\in {\mathbb{R}},\end{eqnarray} \tag{ 17 }$$

for some collection of observables ${O}_{k}\in {\cap }_{i}\mathrm{Com}({C}_{i})$, where Com $({C}_{i})$ is the commutant of the operator C_i. The significance of this is that a $({C}_{1},\ldots ,{C}_{n})$-passive state cannot contain any component of C_i in it. To explain what this means, for simplicity consider the case of a trivial Hamiltonian, and two non-commuting operators A and B such that the only operator that commutes with both is one proportional to the identity (as in the qubit example with the Pauli observables A = X and B = Y). In this case the only passive or completely passive state with respect to both observables is the maximally mixed state ${\mathbb{1}}/d$. In particular, it is impossible to reproduce the maximum entropy Gibbsian distribution in the A and B degrees of freedom. This is not to say that a resource theory is impossible, but the bath states act as random noise in the non-commuting charge degrees of freedom, in disagreement with the maximum entropy principle and equilibration based on typicality arguments.

Notice that the choice of free states based on complete passivity is in agreement with asymmetry theory, as γ is required to be symmetric with respect to the smallest group G generated by the non-commuting observables at hand. More specifically, ${U}_{g}\rho {U}_{g}^{\dagger }=\rho$ for all $g\in G$, where $g\mapsto {U}_{g}$ is an appropriate unitary representation of G. Any state not invariant under the action of G is called a reference frame, in that it can be used to encode information about the group element $g\in G$ (for $G={SU}(2)$, g is a direction) [26]. For the qubit example with A = X and B = Y, we have $G={SU}(2)$, so the only symmetric state is the maximally mixed one. Any $\rho \ne {\mathbb{1}}/2$ breaks rotational symmetry. From a symmetry perspective, any non symmetric state is a resource, and complete passivity ensures that no reference frame for G is being introduced.

Having a thermodynamic application in mind, and given the strong constraints imposed by complete passivity, one may be tempted to follow the second route and select ${{ \mathcal F }}_{M}$ on the basis of the maximum entropy principle or, equivalently, a constrained notion of complete passivity with respect to $C({\boldsymbol{\mu }})$ (an approach followed in [11]). As the previous discussion implies, however, this choice of ${{ \mathcal F }}_{M}$ allows the free introduction of reference frames for the group G generated by the non-commuting charges. The point is easily understood in the example of equation (14), noticing that the maximum entropy states are not rotationally symmetric. The states in ${{ \mathcal F }}_{M}$ are only invariant under a U(1) subgroup of G. The symmetry constraints imposed by the conservation of multiple conserved quantities are then partially lifted, with exclusion of an abelian (specifically, U(1)) subgroup. Thermodynamically this implies that only the average value of $\beta H+{\sum }_{i=1}^{n}{\mu }_{i}{C}_{i}$ cannot be increased indefinitely acting on a large number of free states.

Hence to recover a clear thermodynamic interpretation we do not fully capture the underlying non-commutativity. To summarise, one has to choose between

• 1.  
  No external, symmetry-breaking axis, but only a limited Gibbsian form.
• 2.  
  A symmetry-broken scenario with a clear thermodynamic interpretation.

Despite some advances [11, 12] we hence believe that it is still an open question how resource theory approaches can capture the core aspects of the role of non-commutativity in thermodynamic processes. One reason at the root of some of these difficulties may be that resource theories do not easily handle external fields [32].