Cite this problem as Problem 46.

Problem

Determine the minimal resources required to implement a general Gibbs-preserving map in a one-shot thermodynamic framework whose free operations can reasonably be implemented physically (e.g., thermal operations).

Background

In the resource theory approach to thermodynamics [1][2][3], one establishes a set of rules that an agent has to obey and studies the possible operations that can be achieved according to those rules. These rules specify the set of free operations, i.e., operations that an agent can apply without the use of any further resources. Any operation that is not free requires additional resources, such as thermodynamic work provided in the form of a battery system in a suitable state.

In the resource theory of thermodynamics, a common choice of free operations are thermal operations. A thermal operation consists of a global unitary, applied on the system along with an ancillary system (a heat bath), whereby the ancillary system is initialized in its thermal state, the unitary commutes with the total Hamiltonian of the system and the bath, and the ancillary system is discarded after the application of the unitary. Thermal operations naturally generalize the concept of noisy operations [4] to systems with nontrivial Hamiltonians. For states whose energy off-diagonal terms vanish, a set of necessary and sufficient conditions for transformations under thermal operations are given by thermomajorization [3], a generalization of the concepts of majorization and d-majorization of vectors (see, e.g., [5][6]). General conditions for transformations of fully quantum states remain unclear.

A different possible choice of free operations in the resource theory of thermodynamics are Gibbs-preserving maps [1][7]. A Gibbs-preserving map is a completely positive, trace-preserving map with the property that the thermal state of the system is a fixed point of the map. Gibbs-preserving maps yield a resource theory that is technically convenient to analyze using the toolbox of semidefinite programming [8], given that the condition of being a Gibbs-preserving map is a semidefinite constraint.

As a consequence of its definition, any thermal operation has the Gibbs state as a fixed point and is therefore is a Gibbs-preserving map. However, there are Gibbs-preserving maps that are not thermal operations [7]. There exist, for instance, Gibbs-preserving maps on a single qubit that take the pure excited state of the qubit to a coherent superposition of the ground state and the excited state. Such maps are not time-covariant, as they create off-diagonal terms in the density matrix when written in the energy basis. These maps cannot be thermal operations, because all thermal operations are time-covariant.

This observation begs the question of how thermodynamically resourceful a Gibbs-preserving map is. If an agent is restricted to thermal operations, what additional resources are required in order to implement a given Gibbs-preserving map that is not a thermal operation? Necessary resources are likely to include additional thermodynamic work as well as the ability to break time-translation symmetry.

Accounting for coherence and work resources

Implementing a non-time-covariant map would certainly require some time reference, i.e., access to some system (or process) that is able to break time-translation invariance [9]. The question can be formalized in different ways:

• What is the minimal size (or accuracy) of a quantum clock, provided as an input alongside the input system, such that sufficient time information is provided to achieve the given Gibbs-preserving map? The size (or accuracy) of the clock might be measured for instance in terms of its energy spread, or via the dimension of its Hilbert space [10].

• The amount of coherence resources provided can be measured as a number of copies of a state (), which denotes an even superposition of the ground and excited states of a qubit with some energy gap [11]. How many single resource () states is it necessary to provide in order for it to be possible to implement a given Gibbs-preserving map?

• In addition to the any time-coherence resources, is it necessary to provide pure state ancillas, i.e., additional thermodynamic work to the process?

More generally, it is unclear what the best one-shot thermodynamic framework to analyze this question is. Possible approaches might include:

• Account for thermodynamic work using a doubly-infinite explicit work storage system (battery) in a pure state with a nontrivial energy spread [12][13];

• A thermodynamic framework that allows the output system to remain interacting with any ancillas that are discarded [14];

• Providing the ability to use an arbitrary additional initial resource state, as a source of coherence, limiting only its energy spread [15][16];

• A more “inherently quantum” definition of work [17];

• etc.

Partial results

The present problem has been solved in the special case of a system described by a trivial Hamiltonian. If the Hamiltonian is identically equal to zero, the thermal state is maximally mixed. Gibbs-preserving maps then coincide with unital maps, i.e., ones that have the identity operator as fixed point, while thermal operations reduce to noisy operations [4]. It is known that the set of unital maps is strictly larger than the set of all noisy operations [18][19], a statement related to the failure of Birkhoff’s theorem to generalize to the quantum regime in a straightforward way (see also Problem 30). However, for any accuracy tolerance \(\epsilon>0\), one can implement any given unital map in a one-shot setting with a noisy operation that acts on the system and which consumes an amount of work \(W \sim \log(f(\epsilon))\) independent of the system dimension [20]. The noisy operation is constructed by using decoupling techniques and is based on the protocol of Ref. [21].

Another partial result addresses the special case of a Gibbs-preserving map that is additionally time-covariant and acting on a time-covariant input state, i.e., if the map commutes with time evolution as a superoperator and the input state is block-diagonal in the energy eigenspaces. In Corollary 8.3 of Ref. [15] (arXiv version), an expression for the work cost of any time-covariant map was provided in terms of the hypothesis testing relative entropy and the Stinespring dilation of the map.

References

[1] Dominik Janzing et al., “The thermodynamic cost of reliability and low temperatures: Tightening Landauer’s principle and the Second Law”, quant-ph/0002048.
[2] F. G. S. L. Brandão et al., “Resource Theory of Quantum States Out of Thermal Equilibrium”, Physical Review Letters 111, (2013). 1111.3882 <https://arxiv.org/abs/1111.3882> DOI <https://doi.org/10.1103/physrevlett.111.250404>
[3] M. Horodecki and J. Oppenheim, “Fundamental limitations for quantum and nanoscale thermodynamics”, Nature Communications 4, (2013). 1111.3834 <https://arxiv.org/abs/1111.3834> DOI <https://doi.org/10.1038/ncomms3059>
[4] M. Horodecki, P. Horodecki, and J. Oppenheim, “Reversible transformations from pure to mixed states and the unique measure of information”, Physical Review A 67, (2003). quant-ph/0212019 <https://arxiv.org/abs/quant-ph/0212019> DOI <https://doi.org/10.1103/physreva.67.062104>
[5] R. Bhatia, Matrix Analysis.
[6] A. W. Marshall, I. Olkin, and B. C. Arnold, Inequalities: Theory of Majorization and Its Applications (Springer New York, 2011). DOI <https://doi.org/10.1007/978-0-387-68276-1>
[7] P. Faist, J. Oppenheim, and R. Renner, “Gibbs-preserving maps outperform thermal operations in the quantum regime”, New Journal of Physics 17, 043003 (2015). 1406.3618 <https://arxiv.org/abs/1406.3618> DOI <https://doi.org/10.1088/1367-2630/17/4/043003>
[8] John Watrous, “Semidefinite programs for completely bounded norms”. 0901.4709 <https://arxiv.org/abs/0901.4709>
[9] S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Reference frames, superselection rules, and quantum information”, Reviews of Modern Physics 79, 555 (2007). quant-ph/0610030 <https://arxiv.org/abs/quant-ph/0610030> DOI <https://doi.org/10.1103/revmodphys.79.555>
[10] M. P. Woods, R. Silva, and J. Oppenheim, “Autonomous Quantum Machines and Finite-Sized Clocks”, Annales Henri Poincaré 20, 125 (2018). 1607.04591 <https://arxiv.org/abs/1607.04591> DOI <https://doi.org/10.1007/s00023-018-0736-9>
[11] I. Marvian, “Coherence distillation machines are impossible in quantum thermodynamics”, Nature Communications 11, (2020). 1805.01989 <https://arxiv.org/abs/1805.01989> DOI <https://doi.org/10.1038/s41467-019-13846-3>
[12] L. Masanes and J. Oppenheim, “A general derivation and quantification of the third law of thermodynamics”, Nature Communications 8, (2017). 1412.3828 <https://arxiv.org/abs/1412.3828> DOI <https://doi.org/10.1038/ncomms14538>
[13] Á. M. Alhambra et al., “Fluctuating Work: From Quantum Thermodynamical Identities to a Second Law Equality”, Physical Review X 6, (2016). 1601.05799 <https://arxiv.org/abs/1601.05799> DOI <https://doi.org/10.1103/physrevx.6.041017>
[14] Philippe Faist, “Quantum Coarse-Graining: An Information-Theoretic Approach to Thermodynamics”. 1607.03104 <https://arxiv.org/abs/1607.03104>
[15] P. Faist, M. Berta, and F. G. S. L. Brandao, “Thermodynamic Implementations of Quantum Processes”, Communications in Mathematical Physics 384, 1709 (2021). 1911.05563 <https://arxiv.org/abs/1911.05563> DOI <https://doi.org/10.1007/s00220-021-04107-w>
[16] T. Sagawa et al., “Asymptotic reversibility of thermal operations for interacting quantum spin systems via generalized quantum Stein’s lemma”, Journal of Physics A: Mathematical and Theoretical 54, 495303 (2021). 1907.05650 <https://arxiv.org/abs/1907.05650> DOI <https://doi.org/10.1088/1751-8121/ac333c>
[17] E. H. Mingo and D. Jennings, “Decomposable coherence and quantum fluctuation relations”, Quantum 3, 202 (2019). 1812.08159 <https://arxiv.org/abs/1812.08159> DOI <https://doi.org/10.22331/q-2019-11-11-202>
[18] L. J. Landau and R. F. Streater, “On Birkhoff’s theorem for doubly stochastic completely positive maps of matrix algebras”, Linear Algebra and its Applications 193, 107 (1993). DOI <https://doi.org/10.1016/0024-3795%2893%2990274-r>
[19] U. Haagerup and M. Musat, “Factorization and Dilation Problems for Completely Positive Maps on von Neumann Algebras”, Communications in Mathematical Physics 303, 555 (2011). 1009.0778 <https://arxiv.org/abs/1009.0778> DOI <https://doi.org/10.1007/s00220-011-1216-y>
[20] P. Faist et al., “The minimal work cost of information processing”, Nature Communications 6, (2015). 1211.1037 <https://arxiv.org/abs/1211.1037> DOI <https://doi.org/10.1038/ncomms8669>
[21] L. del . Rio et al., “The thermodynamic meaning of negative entropy”, Nature 474, 61 (2011). 1009.1630 <https://arxiv.org/abs/1009.1630> DOI <https://doi.org/10.1038/nature10123>