Operational models of temperature superpositions

For our first case of a superposition of temperatures we define an interaction between a probe system and a composite purified bath $\rho_{AB}$ comprised of two subsystems in a product state $\rho_{AB}=\rho_{A_{0}B_{0}}\otimes\rho_{A_{1}B_{1}}$. This can arise, for example, if the subsystems are thermally isolated from each other, at least over the time-scale of the interaction with the probe. This model would be relevant to a quantum system probing a Tolman-affected bath, as we discuss further in Section V.

This is called quantum-controlled thermalisation because a quantised control DoF determines which subsystem the probe thermalises with. The probe system (S), baths (AB), and control (C) thus form the initial product state $\rho_{\text{in}}=\rho_{AB}\otimes\rho_{C}\otimes\rho_{S}$. A unitary operator $U^{\prime}_{A^{\prime}B^{\prime}CS}$ for the entire process, by Eq. (5), should have the form $U^{\prime}_{A^{\prime}_{0}B^{\prime}_{0}S}\otimes\ket{0}\bra{0}_{C}+U^{\prime}_{A^{\prime}_{1}B^{\prime}_{1}S}\otimes\ket{1}\bra{1}_{C}$, where the control states $\ket{i}_{C}$, $i=0,1$ are orthonormal and determine that the probe interacts with the corresponding bath $B_{i}$ through unitary $U_{B_{i}S}$.

While this is independent of any specific framework, the MZ interferometer of Figure 1 is certainly a potential realisation of the scenario. In that example, the two states $\ket{i}_{C}$ of the control are identified with the two paths through the interferometer, the probe is an internal DoF of the interfering system (which could be, for example, a particle) and it thermalises with subsystems $B_{0}$ or $B_{1}$ depending on the path taken.

Our $U^{\prime}_{A^{\prime}B^{\prime}CS}$ will include the $w$ isometry as in Eq. (5), but now that we have two subspaces, associated with baths $0$ and $1$, we must introduce an additional isometry, $v:\mathcal{H}_{AB}\rightarrow\mathcal{H}_{A^{\prime}B^{\prime}}$. This ensures that all elements of the channel are embedded in the same Hilbert space.

The unitary operator for the entire process is then:

where the $i=0,1$ subscripts on both isometries identify them with their respective channels.

We take the initial state of the control to be $\rho_{C}=\ket{+}\bra{+}$, where $\ket{+}=\frac{1}{\sqrt{2}}(\ket{0}+\ket{1})$, and measure it in the superposition $\ket{\phi}=\frac{1}{\sqrt{2}}\left(\ket{0}+e^{i\phi}\ket{1}\right)$, with relative phase $\phi$ assumed to be fully controllable.

The sub-normalised output state of the probe is then $\rho_{S}{(\phi)}=\text{Tr}_{A^{\prime}B^{\prime}}\left\{\bra{\phi}_{C}U^{\prime}\rho_{\mathrm{in}}(0)U^{\prime\dagger}\ket{\phi}_{C}\right\}$, which yields four terms:

and H.c. refers to the Hermitian conjugate.

The first two terms in Eq. (9) are each a channel arising from the interaction $U_{B_{i}S}$ between the system and bath $B_{i}$, with the form of a single quantum channel as in Section II. The other two terms are “cross-terms” between channels, with the third term in Eq. (9), for arbitrary pure system state $\rho_{S}=\ket{\eta}_{S}\bra{\eta}_{S}$:

The fourth term is of course its Hermitian conjugate.

As in Section II we take the swap operator $S_{B_{i}S}$, which simply swaps the state of the $i^{\mathrm{th}}$ bath with the state of the system. So the first two terms of Eq. (9) are each thermal states on the probe $\rho^{\beta_{i}}_{S}$ at the respective temperatures of the baths, and Eq. (10) becomes:

with the fourth term again its Hermitian conjugate.

The theta kets and bras form partial inner products, but this is as far as we can reduce these cross-terms without further specifying the involved systems.

Defining:

we can write the final (sub-normalised) state of the probe as:

Note that Eq (12) is also sub-normalised, as $\braket{\Phi_{i}}{\Phi_{i}}\leq 1$.

The first two terms of Eq. (13) taken individually are thermal states at the temperature of each bath. The other two terms are not in general thermal, depending not just on the initial states of both baths and of the system, but also on the isometries associated with each channel dilation, as expected. It is also important to note that the isometries also ensure that we are free to choose any unitary to represent the probe-bath interaction, and that Eq. (13) does not depend on the SWAP operation. See Appendix A for the special case where the isometries are unitaries on the bath, which is similar to the result of Ref. [15] for generic (i.e., not thermal) quantum channels in superposition.

One might have thought that if both baths are at the same temperature, i.e. for $\beta_{0}=\beta_{1}\equiv\beta$, the system simply thermalises to that temperature, however Eq. (13) explicitly demonstrates that this is not the case unless $v^{\dagger}_{i}w_{i}=\mathbb{I}$, or we tune $\phi$ to the particular value that destroys the interference of the control.

In cases where $v$ and $w$ do form an identity, we can further reduce the partial inner products of Eq. (12), and arrive at the corresponding special case of Eq. (13):

So we see that the form of Eq. (13) has crucial implications for the system’s coherence, and thus the extent to which one can speak about any kind of coherence in the final state of the probe (or of the temperature). This result can be understood through the complementarity between quantum coherence and which-path information—via the analogy with the Mach-Zehnder interferometer in Fig. 1. In an interferometric scenario like in Fig. 1 the visibility of the final interference pattern contains relevant information about the dynamics and correlations that develop between the involved systems. Recall that visibility is quantified by the magnitude of the off-diagonal elements of the final state [21, 22]. Here that final state is that of the control, and the visibility is equivalently the contrast of the interference pattern obtained by measuring the control in a superposition basis.

The probability of measuring the control in the state $\ket{\phi}$ is $\text{P}{(\phi)}=\text{Tr}\{\rho_{S}{(\phi)}\}$; explicitly:

where $\psi$ is a phase defined via $\text{Tr}_{S}\{\ket{\Phi_{0}}\bra{\Phi_{1}}\}\equiv|\text{Tr}_{S}\{\ket{\Phi_{0}}\bra{\Phi_{1}}\}|e^{-i\psi}$.

The visibility, therefore, is

Crucially, this visibility depends not just on the temperatures of the two baths, but also the isometries $v$ and $w$:

Again, if $v^{\dagger}_{i}w_{i}=\mathbb{I}$, for $\beta^{0}=\beta_{1}\equiv\beta$ the above reduces further to $\bra{\eta}(\rho^{\beta})^{2}\ket{\eta}=\text{Tr}(\rho^{\beta})^{2}$.

Note that in general the visibility is less than 1. We know by the Cauchy–Schwarz inequality, $|\braket{\Phi_{1}}{\Phi_{0}}|\leq||\Phi_{1}||~{}||\Phi_{0}||$, that visibility can only equal $1$ if the two states are normalised and equal (up to a phase).

Taking Eq. (17):

consider the general structure of this expression: $\bra{\psi}\rho\ket{\psi}$, for $\braket{\psi}{\psi}\leq 1$. We know that since $\rho\geq 1$ and $\text{Tr}\rho=1$, the eigenvalues of $\rho$ cannot be larger than one, so $\bra{\psi}\rho\ket{\psi}=1$ implies that $\rho=\ket{\psi}\bra{\psi}$ and $\braket{\psi}{\psi}=1$.

This implies the condition

and for this to hold the system and bath must both be in a pure state, which can only be achieved at zero temperature (and with a non-degenerate ground state). So, with $\ket{\theta^{\beta_{0}}}=\ket{0}\ket{0}$ signifying a state with these conditions, Eq. (17) becomes

This expression can only be equal to one when the combined isometries rotate the system to the vacuum state, i.e. $v^{\dagger}_{i}w_{i}\ket{0}\ket{\eta}=\ket{0}\ket{0}$.

Hence, in general, the visibility cannot reach unity, directly showing that the probe–bath(s) interactions do affect the coherence of the control even for equal bath temperatures. In other words, even for identical isometries and baths at the same temperature, the bath subsystems still store which-way information [22] about the control, except in the above special case of both baths at zero temperature.

The above also holds for the special case where the isometries are unitaries, as detailed in Appendix A. In that case the visibility is

where $u^{i}$ are unitaries associated with each bath. The maximum visibility is

for an initial pure state of the probe. The intuition is again that visibility is maximised when the local unitaries effectively rotate probe states $\ket{s}$ to the energy ground state (up to a phase) as it has the highest overlap with a thermal state at any finite temperature. Thus, for an arbitrary mixed state, the maximum visibility reads

Without loss of generality, the probabilities $p_{s}$ are ordered decreasingly: $p_{s}\geq p_{s+1}$. This is because thermal weights are likewise ordered decreasingly, i.e. $c_{k}^{\beta_{i}}>c_{k+1}^{\beta_{i}}$, and due to the orthogonality of the states $\ket{s}$, a unitary $u^{i}$ can only rotate one of the states from an orthogonal set to the energy ground state. That being said, the same strategy can be iterated to find a unitary mapping of the eigenstates of $\rho_{S}$ to progressively higher energy eigenstates.

The previous case used quantum control of thermal channels associated with two separate bath subsystems as an operational model of a temperature superposition. Here, we explore the other model identified at the beginning of this section—where the bath is a single system in a superposition of purifications each corresponding to a different temperature. Figure 2 provides a sketch of such a situation. This scenario is highly relevant to accelerating quantum systems which experience the Unruh effect, as we discuss in Section V. The procedure also has some similarity with the “superposition of thermal states” in optics, considered in ref. [23], however no superpositions of different temperatures were considered therein.

Purifications were defined in Section II, and Eq. (3). We now take a more general form

where $x$ signifies that the purification gives rise to a thermal state at temperature $\beta_{x}$, and where $\{\ket{a(b,x)}\}_{b}$ is an orthonormal basis for the ancillary system. This basis can in principle be different for the different purifications $x=\{0,1\}$; phase $\phi_{x}\in\mathbb{R}$.

An unnormalised superposition of Eq. (23) then reads $\ket{\psi}=\sum_{x=0,1}\ket{\theta^{\beta_{x}}(x)}$. One can also prepare the superposition of purifications using another DoF as a control, such that $\ket{\widetilde{\psi}}=\frac{1}{\sqrt{2}}\sum_{x=0,1}\ket{\theta^{\beta_{x}}(x)}\ket{x}_{C}$. For Eq. (23) that is

We could also choose to prepare an initial state for the bath by first measuring the control in the superposition $\ket{\phi}_{C}=\frac{1}{\sqrt{2}}\left(\ket{0}_{C}+e^{i\phi_{C}}\ket{1}_{C}\right)$ after tracing out the ancilla: $\rho_{B}^{\mathrm{in}}(\phi)=\bra{\phi}_{C}\text{Tr}_{A(x)}\{\ket{\widetilde{\psi}}\bra{\widetilde{\psi}}\}\ket{\phi}_{C}$. This would yield a reduced state of the bath such as:

where $\widetilde{\phi}=\phi_{0}-\phi_{1}-\phi_{C}$, and we define $V^{xx^{\prime}}_{bb^{\prime}}:=\braket{a(b^{\prime},x^{\prime})}{a(b,x)}$. $V^{xx^{\prime}}_{bb^{\prime}}$ is a matrix element of a special unitary operator $V^{xx^{\prime}}$ transforming between ancilla bases associated with control states $x$ and $x^{\prime}$. It satisfies $V^{xx}_{bb^{\prime}}=V^{x^{\prime}x^{\prime}}_{bb^{\prime}}=\delta_{bb^{\prime}}$, and $V^{xx^{\prime}}=(V^{x^{\prime}x})^{\dagger}$. If the purifications are the same for both ancilla states, $V^{xx^{\prime}}$, for all $x,x^{\prime}$, reduces to a delta function $\delta_{bb^{\prime}}$.

For the analysis here, however, we do not begin with the reduced state of the bath, instead using the Eq. (24) joint state of bath, ancillary system, and control:

where the entire initial state is $\rho^{\mathrm{in}}=\rho_{ABC}\otimes\rho_{S}$. Again, we have a generic probe-bath unitary interaction $U_{BS}$. To generalise to arbitrary Kraus decompositions we simply introduce additional unitary operators $u^{x}_{AB}$ to the local interaction with the bath, which depend on the control and also act on both the bath and the ancillary system. Hence, the entire interaction, extended to include these “local unitaries”, takes the form

This $\widetilde{U}_{ABCS}$ is already an arbitrary dilation as the ancillae themselves can be of arbitrary dimensions, so we do not need to employ the isometries as in the two-bath scenario.

The final probe state, conditioned on the control $\ket{\phi}_{C}=\frac{1}{\sqrt{2}}\left(\ket{0}_{C}+e^{i\phi_{C}}\ket{1}_{C}\right)$, is

Taking the SWAP operator once more to be our $U_{BS}$ this is, explicitly,

where $W^{xx^{\prime}}_{bb^{\prime}}:=\bra{a(b^{\prime},x^{\prime})}\text{Tr}_{S}\{u^{1\dagger}_{AS}u^{0}_{AS}\rho_{S}\}\ket{a(b,x)}$, similar to the $V^{xx}_{bb^{\prime}}$ discussed above, but now post-interaction, and so dependent on the extra unitaries and the state of the probe.

We immediately see that, in this model too, the final probe state still depends on the temperatures of both baths, the initial state of the probe, and the interaction dynamics as represented by the local unitaries. Again the first two terms correspond to thermal states at each of the involved temperatures while the second two “cross terms” do not, however it is in these two cross terms that there are marked differences to the model of Section III.1.

Unlike in the two-bath case, here the probe can thermalise. Thermalisation can occur when $\beta_{0}=\beta_{1}$ and $W^{01}_{bb^{\prime}}\propto\delta_{bb^{\prime}}$. These criteria are met when: 1) both amplitudes of the purification (or superposed purifications) correspond to the same thermal state of the bath, 2) the purification basis is the same for both amplitudes, and 3) when $u^{1\dagger}_{AS}u^{0}_{AS}=e^{i\psi}\mathbb{I}$ (in which case the matrix $W^{01}_{bb^{\prime}}$ is the identity).

Looking at the visibility of the interference between the amplitudes of the control DoF, we find the probability to measure the control in $\ket{\phi}_{C}$ is

which results in the visibility

There always exist $u_{AS}^{x}$ for which the visibility vanishes. Considering eigenstates $\ket{s}$ of $\rho_{S}$, it suffices to take $u_{AS}^{i}$ which map each $\ket{s}$ to different energy eigenstates of the probe, i.e. such that $u_{AS}^{1}\ket{s}$ is orthogonal to $u_{AS}^{0}\ket{s}$ for every $s$. The same argument applies to the two-bath case under the conditions discussed in Section III.1 and Appendix A.

Considering maximum visibility, on the other hand, firstly, it is easy to see that

where the inequality is saturated—thus maximising the visibility—when $u_{AS}^{0\dagger}u_{AS}^{1}=e^{i\varphi}\mathbb{I}$, for arbitrary $\varphi\in\mathbb{R}$. This means the maximum visibility is, therefore,

Compare this with the analogous scenario in the two-bath case where the isometries are assumed to be unitaries, as discussed in the previous section, and in Appendix A, where the maximum visibility, Equation (22), is $\mathcal{V}_{\mathrm{max}}=\sum_{s}p_{s}c^{\beta_{0}}_{s}c^{\beta_{1}}_{s}$. The terms $p_{s}$ are probabilities arising from the diagonal form of the density matrix representation of the initial probe state. This is markedly different to the above one-bath case as it explicitly depends on the initial state of the probe.

Interestingly, Eq. (33) coincides with the fidelity between the two thermal states, $\widetilde{\mathcal{V}}_{\mathrm{max}}=\textrm{Tr}\sqrt{\sqrt{\rho^{\beta_{0}}}\rho^{\beta_{1}}\sqrt{\rho^{\beta_{0}}}}=\textrm{Tr}\sqrt{\rho^{\beta_{0}}\rho^{\beta_{1}}}$, where the latter simplification is possible because the two states commute [24].

When $\beta_{0}=\beta_{1}=\beta$, independent of the actual value of $\beta$, $\widetilde{V}_{\mathrm{max}}=1$. However, even when $\beta_{0}=\beta_{1}$, the visibility in Eq. (31) will be less than 1 when $W^{01}_{bb}\neq 1$, that is, when the purification bases are not the same.

A further particular case of interest is probe-bath interactions which are independent of the control. This renders $u^{x}_{AB}$ independent of $x$, and the visibility is simply: $\tilde{\mathcal{V}}=\sum_{b}\sqrt{c_{b}^{\beta_{0}}c_{b}^{\beta_{1}}}\big{|}\braket{a(b,1)}{a(b,0)}\big{|}$. This corresponds to a situation where the probe interacts with the bath after it has been prepared in a superposition, that is, after a particular state of the control has been detected post interference.

In summary, we generally find that in the two-bath scenario, full thermalisation in general cannot be attained, while in this single-bath scenario it can, with particular conditions discussed above.

In this model of gradual thermalisation, the state of a complete bath is of the form $|\theta^{\beta}\rangle^{\otimes\mathcal{M}}~{},$ where $\mathcal{M}$ is the number of bath subsystems, and we have considered each of the bath subsystems to be in a purified state (for both one- and two-bath scenarios). We call an individual application of the unitary $U_{BS}$ (on the probe and purified bath subsystem) a collision and assign a “collision number” to each such operation, see Fig. (3).

Such modelling of the thermalisation process enables us to be in the pre-thermalisation regime by controlling the strength and/or the number of collisions. Fig. (3), shows the trace distance between the state of the probe, initialized as $|0\rangle$ (the ground state), and a thermal state at a temperature $T=1$, as a function of the number of collisions. This distance approaches zero as the number of collisions increase. The number of collisions after which the trace norm falls below a fixed threshold $||\rho_{S}-\rho^{\beta=1}||<\epsilon$, signalling that the probe approaches thermalisation, depends on both the interaction strength and the initial state of the probe.

In the following subsection, we use the above model to illustrate the drop in visibility in the pre/post thermalisation regime and the key differences while focusing on the two scenarios identified in the previous sections—the one-bath and two-bath cases. In all instances shown here, we consider any local unitaries or isometries acting on the bath(s) to be the identity. This in particular implies that the bath does not have its own intrinsic dynamics and that the purifications considered in the single-bath case are likewise independent of the control.

In the two-bath case of Sec. III.1 the probe interacts with two separate bath subsystems—for example, localised in the arms of a MZ interferometer Fig. 1. For the study of pre-thermalisation in this model it will be convenient to use purified states of the baths, by which the initial state of both baths is given as

where each state $\ket{\theta^{\beta_{i}}}$ is defined as in Eq. (3).

The interaction between the probe, the control, and the baths takes the form of Eq. (8) where the probe-bath unitaries are now additionally split into a tensor product of terms acting at subsequent collisions. Hence, for the $r^{\mathrm{th}}$ collision we have:

where $U_{BS}$ denotes the unitary operator in Eq. (39), superscript $r$ indicates the unitary being confined to the $r^{\mathrm{th}}$ subsystem, and $\bar{r}$ refers to the remaining subsystems of $\mathcal{M}$. The subscript 2 on the identity operator reminds us that we are modelling each of the $\mathcal{M}$-subsystems of either of the baths as a qubit.

We compare the pre- and post-thermalisation visibilities for the two-bath case (cf Sec. III.1) by changing the number of collisions (no. of subsystems within the bath). The results are presented in Fig. 4. The interaction strength is kept constant and we take the energy ground state as the probe’s initial state. Notably, we see a very good agreement between full- and pre-thermalisation scenarios. In particular, we find that in the pre-thermalisation case, even if the temperatures of the two baths are equal, the visibility is significantly reduced, with a stronger reduction for higher temperatures. This is in contrast with the one-bath case in Fig. 5, to be discussed in the next section, where for equal temperatures the visibility is maximal. Note also that the visibility generally falls with the number of collisions while the high-visibility parameter region shrinks. These results are in agreement with the analytical results for full thermalisation in Sec. III.1.

As outlined in Sec. III.2, in the one bath case the purified state of the bath is entangled with the control DoF, and its initial state is

Since here we assume that the purification of the bath is in the same basis, the operator $V^{01}_{bb^{\prime}}$ is an identity. Note that in the $r^{\mathrm{th}}$ collision the probe interacts with the $r^{\mathrm{th}}$ subsystem of the bath and thus the unitary operator is of the form $(U_{BS})^{r}\otimes(\mathbb{I}_{2}\otimes\mathbb{I}_{2})^{\otimes\bar{r}}$, as in Eq. (41).

The comparison between the visibilities obtained for partially thermalising maps and completely thermalising maps for the one-bath case is shown in Fig. 5. Once again, these results have been obtained by changing the number of collisions, keeping the interaction strength constant, and taking the probe in its energy ground state as the initial state. We find that the visibility is the greatest for similar temperatures of the baths and decreases when the temperatures are different. This decrease is steeper for higher numbers of collisions and therefore depends on the fact that the probe is not fully thermalised. Furthermore, the parameter space of $T_{0}$ and $T_{1}$, where the visibility is close to its maximum, shrinks as the number of collisions increases, signalling an increase in the distinguishability and the resulting loss of coherence.