Structure of the resource theory of quantum coherence

TL;DR

This paper advances the understanding of quantum coherence resource theory by providing bounds on incoherent operations and exploring their implications in quantum thermodynamics, especially for single qubits.

Contribution

It offers new bounds on the number of incoherent Kraus operators and introduces Gibbs-preserving strictly incoherent operations for quantum thermodynamics.

Findings

Maximum of 5 incoherent Kraus operators for a single qubit

Open question on reducing the Kraus operators to 4

Solved mixed-state conversion for single qubit in thermodynamics

Abstract

Quantum coherence is an essential feature of quantum mechanics which is responsible for the departure between classical and quantum world. The recently established resource theory of quantum coherence studies possible quantum technological applications of quantum coherence, and limitations which arise if one is lacking the ability to establish superpositions. An important open problem in this context is a simple characterization for incoherent operations, constituted by all possible transformations allowed within the resource theory of coherence. In this work, we contribute to such a characterization by proving several upper bounds on the maximum number of incoherent Kraus operators in a general incoherent operation. For a single qubit, we show that the number of incoherent Kraus operators is not more than 5, and it remains an open question if this number can be reduced to 4. The presented results are also relevant for quantum thermodynamics, as we demonstrate by introducing the class of Gibbs-preserving strictly incoherent operations, and solving the corresponding mixed-state conversion problem for a single qubit.

Figures

Equations

$$\rho =\sum _i{p}_i|i⟩⟨i|,$$

$$\Lambda (\rho )=\sum _j{K}_j\rho K_j^{†},$$

$$s_x^2+s_y^2$$

$$s_z^2$$

$$\Lambda (\tau )=\tau .$$

$${\rho }_{\Lambda }=(\Lambda \otimes \mathrm{id})({\Phi }_d^{+}),$$

$${\Lambda }_{\mathrm{s}}(\rho )=\sum _i\left(A_i\otimes B_i\right)\rho (A_i^{†}\otimes B_i^{†}).$$

$$\left\{\left(\begin{array}{cc}{a}_1& b_1\\ 0& 0\end{array}\right),\left(\begin{array}{cc}0& 0\\ a_2& b_2\end{array}\right),\left(\begin{array}{cc}{a}_3& 0\\ 0& b_3\end{array}\right),\left(\begin{array}{cc}0& b_4\\ a_4& 0\end{array}\right),\left(\begin{array}{cc}{a}_5& 0\\ 0& 0\end{array}\right)\right\},$$

$$\left\{\left(\begin{array}{cc}{a}_1& 0\\ 0& b_1\end{array}\right),\left(\begin{array}{cc}0& b_2\\ a_2& 0\end{array}\right),\left(\begin{array}{cc}{a}_3& 0\\ 0& 0\end{array}\right),\left(\begin{array}{cc}0& 0\\ a_4& 0\end{array}\right)\right\},$$

$$s_x^2+s_y^2$$

$$s_z^2$$

$$s_x^2$$

$$s_z^2$$

$$K_1$$

$$K_3$$

$$s_x$$

$$s_z$$

$${\mathbf{a}}^{\prime }$$

$${\mathbf{b}}^{\prime }$$

$$a_1$$

$$b_1$$

$$s_x$$

$$s_z$$

$$\varphi$$

$$s_z=({\cos }^2\theta +r_z^2{\sin }^2\theta {)}^{1/2}.$$

$$\Lambda (\tau )=\tau ,$$

$$\tau =p|0⟩⟨0|+(1-p)|1⟩⟨1|$$

$$p=\frac{e^{-\beta E_0}}{e^{-\beta E_0}+e^{-\beta E_1}},$$

$$(1+t_z)a_2^2=(1-t_z)(1-|b_1{|}^2).$$

$$0\le a_2^2\le \min \left\{1,\frac{1-t_z}{1+t_z}\right\}.$$

$$s_x^2\le r_x^2,s_z^2\le r_z^2.$$

$$|b_1{|}^2=1-a_2^2.$$

$$s_z$$

$$a_1$$

Full text

Quantum resource theories Horodecki and Oppenheim (2013); Coecke et al. (2016) provide a strong framework for studying fundamental properties of quantum systems and their applications for quantum technology. The basis of any quantum resource theory is the definition of free states and free operations. Free states are quantum states which can be prepared at no additional cost, while free operations capture those physical transformations which can be implemented without consumption of resources. Having identified these two main features, one can study the basic properties of the corresponding theory, such as possibility of state conversion, resource distillation, and quantification. An important example is the resource theory of entanglement, where free states are separable states, and free operations are local operations and classical communication Vedral et al. (1997); Horodecki et al. (2009).

In the resource theory of quantum coherence Åberg (2006); Baumgratz et al. (2014); Winter and Yang (2016); Streltsov et al. (2016a); Ben Dana et al. (2017), free states are identified as incoherent states

[TABLE]

i.e., states which are diagonal in a fixed specified basis $\{\ket{i}\}$. The choice of this basis depends on the particular problem under study, and in many relevant scenarios such a basis is naturally singled out by the unavoidable decoherence Zurek (2003).

The definition of free operations within the theory of coherence is not unique, and several approaches have been discussed in the literature, based on different physical (or mathematical) considerations Streltsov et al. (2016a). Two important frameworks are known as incoherent Baumgratz et al. (2014) and strictly incoherent operations Winter and Yang (2016); Yadin et al. (2016), which will be denoted by IO and SIO, respectively. The characterizing feature of IO is the fact that they admit an incoherent Kraus decomposition, i.e., they can be written as Baumgratz et al. (2014)

[TABLE]

where each of the Kraus operators $K_{j}$ cannot create coherence individually, $K_{j}\ket{m}\sim\ket{n}$ for suitable integers $n$ and $m$. This approach is motivated by the fact that any quantum operation can be interpreted as a selective measurement in which outcome $j$ occurs with probability $p_{j}=\mathrm{Tr}[K_{j}\rho K_{j}^{\dagger}]$, and the state after the measurement is given by $K_{j}\rho K_{j}^{\dagger}/p_{j}$. An IO can then be interpreted as a measurement which cannot create coherence even if one applies post-selection on the measurement outcomes.

Strictly incoherent operations are incoherent operations with the additional property that all $K_{i}^{\dagger}$ are also incoherent Winter and Yang (2016); Yadin et al. (2016). These operations have several desirable properties which distinguish them from the larger class IO. In particular, it has been shown that SIO is the most general class of operations which do not use coherence Yadin et al. (2016).

Other important frameworks which are discussed in the recent literature include maximally incoherent operations (MIO) Åberg (2006): this is the most general class of operations which cannot create coherence from incoherent states. It has recently been shown that this framework has maximally coherent mixed states, i.e., quantum states which are the optimal resource among all states with a given spectrum Streltsov et al. (2016b). Another important class are translationally invariant operations (TIO) Gour and Spekkens (2008), these are quantum operations which commute with time translations $e^{-iHt}$ induced by a given Hamiltonian $H$. The set IO is strictly larger than TIO for nondegenerate Hamiltonians Marvian et al. (2016). Moreover, the class TIO has found several applications in the literature, including the resource theory of asymmetry Gour and Spekkens (2008); Marvian et al. (2016); Vaccaro et al. (2008) and quantum thermodynamics Lostaglio et al. (2015, 2015). Further approaches, also going beyond incoherent states, have been investigated recently in Refs. Chitambar and Gour (2016a, b, 2017); Killoran et al. (2016); Theurer et al. (2017); Regula et al. (2017).

The quantification of coherence is another important research direction. Postulates for coherence quantifiers have been presented Åberg (2006); Baumgratz et al. (2014); Levi and Mintert (2014); Streltsov et al. (2016a), based on earlier approaches in entanglement theory Vedral et al. (1997); Plenio and Virmani (2007); Horodecki et al. (2009). Operational coherence measures include distillable coherence and coherence cost Yuan et al. (2015); Winter and Yang (2016), which quantify optimal rates for asymptotic coherence distillation and dilution via the corresponding set of free operations. Another operational quantifier is the robustness of coherence Napoli et al. (2016); Piani et al. (2016), which is also closely related to coherence quantifiers based on interferometric visibility Biswas et al. (2017). Distance-based coherence quantifiers have also been investigated Baumgratz et al. (2014); Streltsov et al. (2015); Rana et al. (2016), and it was shown that the relative entropy of coherence is equal to the distillable coherence for the sets IO and MIO Winter and Yang (2016). For the latter set, distillable coherence coincides with the coherence cost, which implies that the resource theory of coherence based on MIO is reversible Winter and Yang (2016). Dynamics of coherence quantifiers under noisy evolution has also been investigated Bromley et al. (2015); Silva et al. (2016); Mani and Karimipour (2015); García-Díaz et al. (2016); Puchała et al. (2016); Zanardi et al. (2017).

While initially formulated for a single particle, the framework of coherence has recently been extended to distributed scenarios Chitambar et al. (2016); Chitambar and Hsieh (2016); Matera et al. (2016); Streltsov et al. (2017). This extension has found several applications in remote quantum protocols, including the tasks of quantum state merging Streltsov et al. (2016); Horodecki et al. (2005) and assisted coherence distillation Chitambar et al. (2016); Streltsov et al. (2017), which has also been demonstrated experimentally very recently Xiang (2017). Coherence in multipartite systems has also been studied with respect to other types of nonclassicality such as entanglement Streltsov et al. (2015); Killoran et al. (2016); Zhu et al. (2017); Regula et al. (2017) and quantum discord Ma et al. (2016); Modi et al. (2012).

Having identified relevant classes of free operations, it is now important to ask about a description of those operation efficient in the physical dimension. Such an efficient description seems crucial for a rigorous investigation of the corresponding resource theory. In entanglement theory, it is known that the set of local operations and classical communication is notoriously difficult to capture mathematically Chitambar et al. (2014). In this work, we address this question for the resource theory of coherence, focusing on the classes IO and SIO. We show that both classes admit a minimal standard form, and use these results to give a full solution for the mixed-state conversion problem via SIO, IO, and MIO. We further introduce the set of Gibbs-preserving SIO, and solve the mixed-state conversion problem for a single qubit also for these operations.

Summary of results. A general quantum operation, acting on a Hilbert space of dimension $d$, admits a decomposition with at most $d^{2}$ Kraus operators Nielsen and Chuang (2010) – this is the maximum Kraus rank. However, since (strictly) incoherent Kraus operators have a very specific structure, it is unclear if this result also transfers to IO and SIO 111Clearly, any IO and SIO admits a decomposition into (at most) $d^{2}$ Kraus operators, where $d$ is the dimension of the Hilbert space. However, it is unclear if such a minimal decomposition will have the desired (strictly) incoherent structure.. In the following two statements, we provide upper bounds for the number of (strictly) incoherent Kraus operators for these operations. We refer to this minimal number as the (strictly) incoherent Kraus rank.

This theorem is a combination of Propositions 3, 4, and 5, which will be presented and discussed below. The corresponding bound for SIO is given in the following statement.

This theorem follows by combining Propositions 3 and 6. In general, it remains an open question if the bounds in Theorems 1 and 2 are tight. However, as we prove in Proposition 7, there exist SIO which require $d^{2}$ Kraus operators. This implies that the bound in Theorem 2 is tight for SIO on a single qubit.

The above theorems assist in a rigorous investigation of the resource theory of coherence, since they significantly reduce the number of free parameters for the sets IO and SIO. For a single qubit, any IO admits a decomposition into 5 incoherent Kraus operators, and for SIO this number further reduces to 4.

We will now demonstrate the power of these results by providing a full characterization for all possible state transformations via single-qubit SIO, IO, and MIO. As we show in Appendix A, a single-qubit state $\rho$ with Bloch vector $(r_{x},r_{y},r_{z})^{T}$ can be converted into another single-qubit state $\sigma$ with Bloch vector $(s_{x},s_{y},s_{z})^{T}$ via SIO, IO, or MIO if and only if the following inequalities are fulfilled:

[TABLE]

For a given initial Bloch vector $(r_{x},r_{y},r_{z})^{T}$, these inequalities completely characterize the achievable region for the final Bloch vectors $(s_{x},s_{y},s_{z})^{T}$. The achievable region is symmetric under rotations around the $z$-axis and corresponds to a cylinder with radius $(r_{x}^{2}+r_{y}^{2})^{1/2}$ and height $2r_{z}$ with ellipsoids attached at the top and the bottom. In Fig. 1 we show the projection of the achievable region into the $x$-$z$ plane for four initial states. The proof of Eqs. (3) in Appendix A makes use of our result that any single-qubit SIO can be decomposed into four strictly incoherent Kraus operators, see Theorem 2 and discussion below Proposition 4 for the general form of these operators. Alternatively, the form of the achievable region can be proven using results in Chitambar and Gour (2016a, b, 2017). We also note that for pure states the conversion problem has been completely solved for SIO Winter and Yang (2016) and IO Zhu et al. (2017).

As a second application for our results, we investigate strictly incoherent operations which preserve a given incoherent state $\tau$, i.e.,

[TABLE]

The motivation for this constraint originates from quantum thermodynamics. In particular, if $\tau=e^{-\beta H}/\mathrm{Tr}[e^{-\beta H}]$ is the Gibbs state of a system with Hamiltonian $H$, then the condition (4) is known to hold for thermal operations Janzing et al. (2000); Lostaglio et al. (2015, 2015). For a non-degenerate Hamitonian $H$ thermal operations cannot create coherence in the eigenbasis of $H$, and conditions for state transformations under these operations and the role of coherence therein have been extensively studied in Refs. Horodecki and Oppenheim (2013); Brandão et al. (2015); Ćwikliński et al. (2015). The most general quantum operations that fulfill Eq. (4) are known as Gibbs-preserving operations, and it has been shown that such operations can create coherence Faist et al. (2015). Here, we contribute to this discussion by introducing the set of Gibbs-preserving SIO, and giving a full solution for the mixed-state conversion problem on a single qubit. As an example, in Fig. 2 we show the achievable region in the $x$-$z$ plane for an initial state $\rho$ with Bloch vector $\boldsymbol{r}=(0.5,0,0.5)^{T}$, and the preserved Gibbs state $\tau$ has the Bloch vector $\boldsymbol{t}=(0,0,-0.2)^{T}$. Note that the achievable region is convex and symmetric under rotations around the $z$-axis. We refer to Appendix B for more details and further examples.

This discussion clearly demonstrates how Theorems 1 and 2 lead to deep insights on the structure of the resource theory of quantum coherence. In particular, they lead to a full solution of the state conversion problem under single-qubit SIO, IO, and MIO. These results have clear relevance within the resource theory of coherence, and also extend to other related fields, including quantum thermodynamics.

In the remainder of this letter we will present statements which are needed for proving the aforementioned theorems, and provide further detailed discussion of the results.

Bounds from Choi-Jamiołkowski isomorphism. We will now present bounds for IO and SIO arising from the Choi-Jamiołkowski isomorphism between a quantum operation $\Lambda$ and the corresponding Choi state

[TABLE]

where $\Phi_{d}^{+}=d^{-1}\sum_{i,j=0}^{d-1}\ket{i,i}\!\bra{j,j}$ is a maximally entangled state of dimension $d^{2}$. The rank of the Choi state is the Kraus rank, which is the smallest number of (not necessarily incoherent) Kraus operators.

The proof can be found in Appendix C. It is interesting to note that the proof of the above proposition does not used the fact that the Kraus operators are incoherent. Thus, the proposition is not limited to IO, but also applies to SIO. Moreover, it also applies to separable operations, i.e., operations of the form

[TABLE]

Any separable operation can be decomposed into (at most) $d^{4}+1$ product Kraus operators $A_{i}\otimes B_{i}$, where $d=d_{A}d_{B}$ is the dimension of the total system. The same is true for separable quantum-incoherent operations Streltsov et al. (2017), i.e., operations of the form (6) with incoherent operators $B_{i}$, and separable incoherent operations where both $A_{i}$ and $B_{i}$ are incoherent Streltsov et al. (2017). For single-qubit IO and SIO the upper bound in Proposition 3 gives 17 operators. As we will show in the following, this number can be significantly reduced.

**Bounds from the structure of (strictly) incoherent operations. **We will now provide improved bounds which explicitly make use of the structure of IO and SIO. For this, we will first consider incoherent operations on a single-qubit, i.e., the corresponding Hilbert space has dimension $d=2$. The following proposition shows that any single-qubit IO can be decomposed into 5 incoherent Kraus operators with a simple structure.

[TABLE]

where $a_{i}$ can be chosen real, while $b_{i}\in\mathbbm{C}$. Moreover, it holds that $\sum_{i=1}^{5}a_{i}^{2}=\sum_{j=1}^{4}|b_{j}|^{2}=1$ and $a_{1}b_{1}+a_{2}b_{2}=0$.

We refer to Appendix D for the proof. We also note that the same techniques can be applied to single-qubit SIO, in which case the number of strictly incoherent operators reduces to 4. The corresponding general form of strictly incoherent Kraus operators can be given as

[TABLE]

where $a_{i}$ are real and $\sum_{i=1}^{4}a_{i}^{2}=\sum_{j=1}^{2}|b_{j}|^{2}=1$. A more general bound for SIO for arbitrary dimensions will be given below.

It is important to note that the proofs of Propositions 3 and 4 are fundamentally different: while the argument leading to Proposition 3 is based on the Choi-Jamiołkowski isomorphism and Caratheodory’s theorem, the proof of Proposition 4 makes explicit use of the structure of IO. In the next step, we will extend Proposition 4 to arbitrary dimension.

We refer to Appendix E for the proof. For single-qubit IO Proposition 5 gives us 6 incoherent operators as an upper bound. As we have already seen in Proposition 4, this number can be reduced to 5. For qutrits we obtain 39 Kraus operators, while the bound in Proposition 3 gives 82 Kraus operators. For dimensions larger than 3 Proposition 3 always gives a better bound.

We will now see how the tools presented above can be applied to study the structure of SIO. By Proposition 3, any SIO admits a decomposition into (at most) $d^{4}+1$ strictly incoherent Kraus operators. As we will show in the following, for small dimensions this number can be significantly reduced.

The proof of the proposition can be found in Appendix F. Note that the bound in this proposition is below $d^{4}+1$ for $d\leq 5$. For larger dimensions $d^{4}+1$ gives a better bound. For $d=2$ we see that any single-qubit SIO admits a decomposition into 4 strictly incoherent Kraus operators. This was already discussed below Proposition 4. As we will show in the following, this bound is tight.

The proof of the proposition can be found in Appendix G. Note that for $d=2$ the bounds in Propositions 6 and 7 coincide: any single-qubit SIO can be decomposed into 4 strictly incoherent Kraus operators, and some single-qubit SIO require 4 Kraus operators in their decomposition.

Conclusions. In this work we have studied the structure of the resource theory of quantum coherence, focusing in particular on the structure of incoherent and strictly incoherent operations. We have shown that any (strictly) incoherent operation can be written with at most $d^{4}+1$ (strictly) incoherent Kraus operators, where $d$ is the dimension of the Hilbert space under study. For small dimensions this number can be significantly reduced. For a single qubit any IO can be decomposed into 5 incoherent Kraus operators, while any SIO admits a decomposition into 4 strictly incoherent Kraus operators. While the latter bound is tight, the tightness of the other bounds remains an open question.

Our results assist in the systematic investigation of the resource theory of coherence due to the significant reduction of unknown parameters. We have applied our results to solve the mixed-state conversion problem for single-qubit SIO, IO, and MIO. We further introduced the class of Gibbs-preserving strictly incoherent operations and also solved the corresponding mixed-state conversion problem for a single qubit. A natural next step would be to consider single-qubit incoherent operations applied on one subsystem of a bipartite quantum state. Such multipartite scenarios have been previously studied in Refs. Chitambar et al. (2016); Chitambar and Hsieh (2016); Matera et al. (2016); Streltsov et al. (2017), and the results presented in this letter provide a strong framework for their further investigation. Another important question which is left open in this work is the relation of Gibbs-preserving strictly incoherent operations to thermal operations. We expect that further results in this direction will be presented in the near future, exploring Gibbs-preserving strictly incoherent operations in relation to recent works on quantum thermodynamics Lostaglio et al. (2015, 2015); Horodecki and Oppenheim (2013); Brandão et al. (2015); Ćwikliński et al. (2015) and extending them to other notions of quantum coherence Streltsov et al. (2016a).

Finally, our results also transfer to other related concepts, including translationally invariant operations, which are relevant in the resource theory of asymmetry and quantum thermodynamics. Recalling that TIO is a subset of IO, the results presented in this letter also give bounds on decompositions of TIO into incoherent Kraus operators. Thus, numerical simulations and optimizations over all these classes now also become feasible, at least for small dimensions.

Note added. Upon completion of this manuscript, a related work has appeared on the arXiv Shi et al. (2017).

Acknowledgements. We thank Gerardo Adesso, Dario Egloff, Martin Plenio, Thomas Theurer, Henrik Wilming, and especially Andreas Winter for discussions. We acknowledge financial support by the Alexander von Humboldt-Foundation, the National Science Center in Poland (POLONEZ UMO-2016/21/P/ST2/04054), EU grants OSYRIS (ERC-2013-AdG Grant No. 339106), TAQ (ERC CoG Grant No. 307498) and QUIC (H2020-FETPROACT-2014 Grant No. 641122), the Spanish MINECO grant FISICATEAMO (FIS2016-79508-P), the Severo Ochoa Programme (SEV-2015-0522), MINECO CLUSTER (ICFO15-EE-3785), the Generalitat de Catalunya (2014 SGR 874 and CERCA/Program), the Fundació Privada Cellex, the DFG (CRC 183), and Studienstiftung des Deutschen Volkes.

Appendix A Proof of Eqs. (3a)

and (3b)

In the following, we will prove that via SIO it is possible to convert a single-qubit state $\rho$ with Bloch vector $\boldsymbol{r}=(r_{x},r_{y},r_{z})^{T}$ into another single-qubit state $\sigma$ with Bloch vector $\boldsymbol{s}=(s_{x},s_{y},s_{z})^{T}$ if and only if

[TABLE]

Note that by Theorem 3 in Ref. Chitambar and Gour (2016a), this also proves that these conditions are necessary and sufficient for state transformations via single-qubit IO and MIO.

In the first step, we will discuss the symmetries and main properties of the considered problem. For a given initial state $\rho$, the achievable region (i.e., the set of all states which is achievable via SIO) is invariant under rotations around the $z$-axis of the Bloch sphere. This follows from the fact that such rotations are strictly incoherent unitaries. The achievable region is further symmetric under reflection at the $x$-$y$ plane, since such a reflection corresponds to a $\sigma_{x}$ rotation, which is SIO. Moreover, the achievable region is convex and contains the maximally mixed state $\openone/2$.

Due to these properties we can significantly simplify our further analysis by focusing on the positive part of the $x$-$z$ plane, i.e., we set $r_{y}=s_{y}=0$, and all the other coordinates are nonnegative. Our statement then reduces to the following inequalities:

[TABLE]

In a next step, we note that any single-qubit SIO can be decomposed into 4 Kraus operators of the form

[TABLE]

In particular, any SIO is completely determined by the vectors $\boldsymbol{a}=(a_{1},a_{2},a_{3})^{T}$ and $\boldsymbol{b}=(b_{1},b_{2},b_{3})^{T}$. The normalization condition $\sum_{i=1}^{4}K_{i}^{\dagger}K_{i}=\openone$ implies that $|\boldsymbol{a}|=|\boldsymbol{b}|=1$. Moreover, $\boldsymbol{a}$ can be chosen real. The proof for this form of single-qubit SIOs follows the same line of reasoning as the proof of Proposition 4, see also Appendix D for more details. By using the explicit form of the Kraus operators (13), the Bloch coordinates $s_{x}$ and $s_{z}$ of the final state take the form

[TABLE]

where $\mathrm{Re}[b_{i}]=(b_{i}+b_{i}^{*})/2$ denotes the real part of the complex number $b_{i}$.

We will now focus on the boundary of the achievable region in the positive part of the $x$-$z$ plane. In particular, we will show that this boundary can be attained with real vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ with $a_{3}=b_{3}=0$. For this, let $(s_{x},s_{z})$ be a point at the boundary. Assume now that the corresponding vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ do not have the aforementioned properties. Then, we can introduce the following normalized real vectors

[TABLE]

The vectors $\boldsymbol{a}^{\prime}$ and $\boldsymbol{b}^{\prime}$ give rise to a state with Bloch coordinates $s_{x}^{\prime}$ and $s_{z}^{\prime}$, as given in Eqs. (14a) and (14b). It is now straightforward to check that $s_{z}^{\prime}=s_{z}$ and $s_{x}^{\prime}\geq s_{x}$. However, we assumed that $(s_{x},s_{z})$ belongs to the boundary. Since the achievable region is convex, it must be that $s_{x}^{\prime}=s_{x}$. This proves that the boundary can be attained with real vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ with $a_{3}=b_{3}=0$.

In the final step, we introduce the parametrization

[TABLE]

Using these parameters in Eqs. (14a) and (14b) we obtain

[TABLE]

Recall that due to the symmetry of the problem we are considering only nonnegative values of $r_{x}$ and $s_{x}$. Thus, without loss of generality, the parameter $\theta$ can be chosen from the range $0\leq\theta\leq\pi/2$.

From Eq. (19) we can immediately verify Eq. (11), and it remains to prove Eq. (12). For this, we will evaluate the maximal value for $s_{z}$, achievable for given values of $r_{x}$, $r_{z}$, and $s_{x}$. The maximum of $s_{z}$ is determined by $ds_{z}/d\phi=0$. Taking the aforementioned parameter regions into account, we can solve this problem explicitly for $\phi$, with the solution

[TABLE]

Using this solution in Eq. (20) we obtain the explicit form for the maximal $s_{z}$

[TABLE]

In summary, we have shown that the points $(r_{x}\sin\theta,[\cos^{2}\theta+r_{z}^{2}\sin^{2}\theta]^{1/2})$ with parameter $\theta\in[0,\pi/2]$ are located at the boundary of the achievable region in the positive $x$-$z$ plane. Note that all these points fulfill Eq. (12) with equality. Moreover, this result completely characterizes the full achievable region due to convexity, rotational symmetry around the $z$-axis, and reflection symmetry at the $x$-$y$ plane. This completes the proof. We note that the statement can also be proven in an alternative way, by using results in section III of Ref. Chitambar and Gour (2016b).

Appendix B Gibbs-preserving SIO

In the following, we consider single-qubit SIO which preserve a given incoherent state $\tau$, i.e.,

[TABLE]

where the Bloch vector corresponding to $\tau$ is given by $\boldsymbol{t}=(0,0,t_{z})$.

Before presenting the solution for this problem, we note that any single-qubit incoherent state

[TABLE]

can be interpreted as a Gibbs state $\tau=e^{-\beta H}/\mathrm{Tr}[e^{-\beta H}]$ for a suitable inverse temperature $\beta=\frac{1}{kT}$ and Hamiltonian $H$ which is diagonal in the incoherent basis. For $0<p<1$ and $p\neq 1/2$ this can be seen by choosing $\beta$ and $H$ such that

[TABLE]

where $E_{0}$ and $E_{1}$ are the eigenvalues of the Hamiltonian. Such choice of parameters is always possible for $0<p<1$ and $p\neq 1/2$. The cases $p=0$ and $p=1$ can be obtained by taking the limits $E_{0}\rightarrow\infty$ and $E_{1}\rightarrow\infty$, respectively. The case $p=1/2$ is obtained by taking the limit of infinite temperature, i.e., $\beta\rightarrow 0$.

In the following, we will determine the achievable region of these operations. The achievable region is convex and invariant under rotations around the $z$-axis of the Bloch sphere, as such rotations are strictly incoherent unitaries which preserve any incoherent state. We can thus restrict ourselves to the $x$-$z$ plane in the following discussion.

We now recall that any single-qubit SIO admits a decomposition into four Kraus operators of the form (13), and the coordinates $s_{x}$ and $s_{z}$ of the final Bloch vector take the form given in Eqs. (14). Moreover, the condition in Eq. (23) implies the equality

[TABLE]

This equality leads to constraints on the range of the parameter $a_{2}$. In particular, $a_{2}$ can take any real value compatible with

[TABLE]

Before we consider the most general case, let us investigate the situation where the SIO is unital, i.e., $t_{z}=0$. As we will now show, in this case the achievable region in the $x$-$z$ plane is determined by

[TABLE]

For proving this, note that for $t_{z}=0$ we obtain from Eq. (26)

[TABLE]

Moreover, from Eq. (27) we see that $a_{2}$ can take any value between $-1$ and $1$. Using these results in Eq. (14b), we obtain

[TABLE]

and it follows that $s_{z}$ can take any value between $-|r_{z}|$ and $|r_{z}|$. It remains to show that $s_{x}$ can take any value between $-|r_{x}|$ and $|r_{x}|$ for any given value of $s_{z}$. This can be seen by choosing the parameters $a_{i}$ and $b_{i}$ as follows,

[TABLE]

For this choice of parameters Eq. (14a) gives us $s_{x}=r_{x}$. Note that a larger value for $s_{x}$ cannot be achieved via SIO in general, see also the discussion in Appendix A. By symmetry, this completes the proof of Eq. (28). In general (i.e. without restricting to the $x$-$z$ plane), the achievable region for unital single-qubit SIOs is given by

[TABLE]

In the right part of Fig. 3 we show this region for the initial state with the Bloch vector $\boldsymbol{r}=(0.5,0,0.5)^{T}$.

In the next step we will focus on another special case, namely $t_{z}=1$. From Eq. (26) we immediately see that $a_{2}=0$, and $|b_{1}|$ can take any value between 0 and 1. From Eq. (14b) we further obtain

[TABLE]

which means that $s_{z}$ can take any value in the range $r_{z}\leq s_{z}\leq 1$. We now use Eq. (14a), noting that for $a_{2}=0$ and fixed value of $|b_{1}|$ the maximal value of $s_{x}$ is given as $s_{x}=|r_{x}b_{1}|$. Together with Eq. (35), we finally obtain

[TABLE]

for the maximal value of $s_{x}$. By symmetry, the achievable region in the $x$-$z$ plane is thus determined by the inequalities

[TABLE]

In general (i.e., without restricting to the $x$-$z$ plane) the achievable region is given by

[TABLE]

In the left part of Fig. 4 we show this region for the initial state with the Bloch vector $\boldsymbol{r}=(0.5,0,0.5)^{T}$.

We will now consider another special case, namely $t_{z}=-1$. In this situation we see from Eq. (26) that $|b_{1}|=1$, and $a_{2}^{2}$ can take any value between 0 and 1. From Eq. (14b) we get

[TABLE]

which implies that $s_{z}$ can take any value between $-1$ and $r_{z}$. Moreover, for $|b_{1}|=1$ and a fixed value of $|a_{2}|$ the maximal value of $s_{x}$ is given as $s_{x}=|r_{x}|({1-a_{2}^{2}})^{1/2}$, which can be seen directly from Eq. (14a). Together with Eq. (41) we arrive at the following result for the maximal value of $s_{x}$:

[TABLE]

By symmetry, this proves that the achievable region in the $x$-$z$ plane is determined by

[TABLE]

In general, i.e., without restricting to the $x$-$z$ plane, the achievable region is given by

[TABLE]

In the right part of Fig. 4 we show this region for the initial state with the Bloch vector $\boldsymbol{r}=(0.5,0,0.5)^{T}$.

In the final step we consider the remaining case $-1<t_{z}<1$, $t_{z}\neq 0$. By solving Eq. (26) for $|b_{1}|^{2}$ and inserting it in Eq. (14b) we obtain

[TABLE]

Using this result together with Eq. (27) we can determine the range of $s_{z}$, given by

[TABLE]

We will now determine all possible values of $s_{x}$ for a given value of $s_{z}$. Note that due to Eqs. (26) and (47) a fixed value of $s_{z}$ immediately fixes the values of $|a_{2}|$ and $|b_{1}|$ as follows:

[TABLE]

In the next step, we use the explicit expression for $s_{x}$ in Eq. (14a). Recalling that the vectors $\boldsymbol{a}=(a_{1},a_{2},a_{3})$ and $\boldsymbol{b}=(b_{1},b_{2},b_{3})$ are normalized, it is straightforward to see that for fixed values of $|a_{2}|$ and $|b_{1}|$ the maximal value for $s_{x}$ is given by

[TABLE]

By using Eqs. (49) it is straightforward to express $s_{x}$ as a function of $r_{x}$, $r_{z}$, $s_{z}$, and $t_{z}$. We do not display the final expression here. Recalling that the achievable region is symmetric under rotation around the $z$-axis, it is straightforward to obtain the full achievable region from Eq. (50). In the left part of Fig. 3 we show the achievable region for the initial state with the Bloch vector $\boldsymbol{r}=(0.5,0,0.5)^{T}$, and the state $\tau$ has the Bloch vector $\boldsymbol{t}=(0,0,0.7)^{T}$.

Appendix C Proof of Proposition 3

We will present the proof for IO. The proof for SIO follows the same lines of reasoning. In the first step we define the operator

[TABLE]

where $K$ is an arbitrary incoherent operator. The operators $M$ are Hermitian operators of dimension $d^{2}$, i.e., the number of real parameters is $d^{4}$. For any incoherent operation $\Lambda$, the corresponding Choi state $\rho_{\Lambda}$ belongs to the convex hull of the operators $M$. This follows directly from the definition of an incoherent operation in Eq. (2). By Caratheodory’s theorem, any Choi state can thus be written as a convex combination of at most $n=d^{4}+1$ operators of the form (51). That is, there exist $n$ incoherent operators $N_{j}$ and a probability distribution $p_{j}$ such that

[TABLE]

This, together with the Choi-Jamiołkowski isomorphism implies that the incoherent operation $\Lambda$ can be written as

[TABLE]

with $n$ incoherent Kraus operators defined as $L_{j}=\sqrt{p_{j}}N_{j}$. The fact that $L_{j}$ are indeed Kraus operators, i.e., fulfill the completeness relation $\sum_{j=1}^{n}L_{j}^{\dagger}L_{j}=\openone$ can be checked directly from Eq. (52), recalling that $\mathrm{Tr}_{1}(\rho_{\Lambda})=\openone/d$.

Appendix D Proof of Proposition 4

The main ingredient in the proof is the fact that two sets of Kraus operators $\{K_{j}\}_{j=1}^{n}$ and $\{L_{i}\}_{i=1}^{k}$ give rise to the same quantum operation if and only if Nielsen and Chuang (2010)

[TABLE]

where $U_{i,j}$ are the elements of a unitary $U\in U(\max\{n,k\})$. That is to say, the two sets of Kraus operators are connected by an isometry. The smallest number of Kraus operators achievable for a given quantum operation is the Kraus rank, which is identical with the rank of the Choi state.

Before we proceed, we note that any incoherent operation on a finite dimensional Hilbert space admits a finite set of incoherent Kraus operators, as was proven in Proposition 3. For a single qubit, the incoherence condition $K_{j}\ket{m}\sim\ket{n}$ implies that every incoherent Kraus operator belongs to one of the following four types:

[TABLE]

where * denotes an arbitrary complex number.

We will now show that there always exists a set of incoherent Kraus operators $\{L_{i}\}$ where each of the four types occurs at most twice, i.e., the total number of incoherent Kraus operators $L_{i}$ is at most eight. For this, assume that the incoherent Kraus representation $\{K_{j}\}_{j=1}^{n}$ contains at least three nonzero Kraus operators of the first type, i.e., $K_{1},K_{2},K_{3}\in K^{I}$. Note that these three Kraus operators are then linearly dependent: there is a nontrivial choice of numbers $z_{i}\in\mathbbm{C}$ such that $\sum_{i=1}^{3}z_{i}K_{i}=0$. Without loss of any generality, we can assume that the vector $(z_{1},z_{2},z_{3})^{T}$ is normalized, i.e., $\sum_{i=1}^{3}|z_{i}|^{2}=1$. In the next step we introduce a $U\in U(n)$ as

[TABLE]

where $\openone_{n-3}$ is the identity operator acting on dimension $n-3$ and $V\in U(3)$ unitary defined such that its first row corresponds to $(z_{1},z_{2},z_{3})$. With this definition, it is straightforward to check that the Kraus operators $L_{i}=\sum_{j}U_{i,j}K_{j}$ are all incoherent, and moreover $L_{1}=0$. These arguments are not limited to Kraus operators of the first type $K^{I}$, but can be applied in the same way for all types given above. Applying this procedure repeatedly, we see that any incoherent single-qubit quantum operation can be written with at most two incoherent Kraus operators of every type, i.e, with at most $8$ incoherent Kraus operators in total.

In the next step, we show that the number of incoherent Kraus operators can be further reduced to 6. For this, note that without any loss of generality, the respective sets are given by

[TABLE]

Following similar arguments as above, the joint set $K^{I}\cup K^{II}$ can be reduced to three operators, and the same is true for $K^{III}\cup K^{IV}$. This proves that any incoherent single-qubit operation can be written with 6 Kraus operators of the form

[TABLE]

We will now complete the proof, showing that this set of 6 Kraus operators can be reduced to 5 Kraus operators of the form (7). For achieving this, we will rearrange the aforementioned Kraus operators, focusing in particular on the 3 operators

[TABLE]

where $a_{i}$ and $b_{i}$ are complex numbers. The missing 3 Kraus operators will be denoted by $K_{4}$, $K_{5}$, and $K_{6}$. Their elements will not be important in the following discussion, but we note that they have the form

[TABLE]

where $*$ denotes some complex number.

Consider now the following $3\times 3$ unitary matrix

[TABLE]

where the parameters $l$, $m$, and $n$ are nonnegative and chosen as

[TABLE]

We now introduce a new Kraus decomposition $\{L_{i}\}$ as

[TABLE]

where $U_{i,j}$ are elements of the unitary matrix in Eq. (63). It can now be verified by inspection that the operators $L_{1}$, $L_{2}$, and $L_{3}$ have the form

[TABLE]

where $*$ denote some complex numbers which can be written in terms of the parameters $a_{i}$ and $b_{i}$, but the explicit form is not important for the following discussion.

Recall now that the remaining Kraus operators have the same form as in Eq. (62), i.e.,

[TABLE]

and in particular the operator $L_{6}$ has the same form as $L_{3}$. Thus, the set $L_{3}\cup L_{6}$ can be reduced to one Kraus operator. This proves that any incoherent operation on a single qubit can be written with at most 5 Kraus operators as given in Eq. (7).

From the completeness condition $\sum_{i=1}^{5}K_{i}^{\dagger}K_{i}=\openone$, it is straightforward to see that $\sum_{i=1}^{5}|a_{i}|^{2}=\sum_{j=1}^{4}|b_{j}|^{2}=1$ and $a_{1}^{*}b_{1}+a_{2}^{*}b_{2}=0$. Finally, the numbers $a_{j}=|a_{j}|e^{i\phi_{j}}$ can be chosen real by multiplying each of the Kraus operators with the corresponding phase $e^{-i\phi_{j}}$.

Appendix E Proof of Proposition 5

The proof follows similar ideas as the proof of Proposition 4. We start with the fact that every incoherent operation admits a finite decomposition into incoherent Kraus operators, see Proposition 3. Now we classify the incoherent Kraus operators according to their types, i.e., the location of their nonzero elements. In general, there are $d^{d}$ different types. For each type, at most $d$ incoherent Kraus operators can be linearly independent. Following the arguments in the proof of Proposition 4, this implies that any incoherent operation admits a decomposition with at most $d^{d+1}$ incoherent Kraus operators.

To reduce this number further, let $C_{k}$ denote the number of incoherent Kraus operators such that the first $k-1$ columns have all entries zero, and the $k$-th column has one nonzero entry. In particular, $C_{1}$ denotes the number of Kraus operators which have one nonzero element in the first column, and possibly some nonzero elements in the other columns. Moreover, $C_{d}$ denotes the number of Kraus operators which have one nonzero entry in the last column, and all other elements are zero.

As we will show in the following, every incoherent operation admits an incoherent Kraus decomposition such that

[TABLE]

for all $k\in[1,d]$. For proving the statement, note that $d^{d-k+1}$ is exactly the number of different shapes for incoherent Kraus operators which have zero entries in the first $k-1$ columns. Assume now that some Kraus decomposition $\{K_{i}\}$ has $C_{k}>d^{d-k+1}$ . Then, there must be two Kraus operators $K_{1}$ and $K_{2}$ which have the same shape and all entries in the first $k-1$ columns are zero. Then – similar as in the proof of Proposition 4 – we can introduce a new incoherent Kraus decomposition $\{L_{i}\}$ such that $L_{1}$ and $L_{2}$ are linear combinations of $K_{1}$ and $K_{2}$. Moreover, we can choose the new Kraus operators such that $L_{i}=K_{i}$ for $i>2$, and all elements in the $k$-th column of $L_{1}$ are zero. This proves the existence of an incoherent Kraus decomposition fulfilling Eq. (67).

To complete the proof of the proposition we evaluate the sum over all $C_{k}$, giving rise to

[TABLE]

Appendix F Proof of Proposition 6

A strictly incoherent Kraus operator can have at most one nonzero element in each row and column. Similar as in the proof of Proposition 5, we now introduce the number $C_{k}$, which counts the number of strictly incoherent Kraus operators with the first $k-1$ columns having all entries zero, and the $k$-th column having one nonzero entry. We will now show that every SIO admits a strictly incoherent Kraus decomposition such that

[TABLE]

For this, note that ${d!}/{(k-1)!}$ is the number of different shapes of strictly incoherent Kraus operators which have zero entries in the first $k-1$ columns. Assume now that some strictly incoherent Kraus decomposition $\{K_{i}\}$ has $C_{k}>{d!}/{(k-1)!}$. Then, there must be two Kraus operators $K_{1}$ and $K_{2}$ which have the same shape and all entries in the first $k-1$ columns are zero. We can then introduce a new Kraus decomposition $\{L_{i}\}$ such that $L_{1}$ and $L_{2}$ are linear combinations of $K_{1}$ and $K_{2}$. Moreover, we can choose the new Kraus operators such that $L_{i}=K_{i}$ for $i>2$, and all elements in the $k$-th column of $L_{1}$ are zero. This proves the existence of a strictly incoherent Kraus decomposition which fulfills Eq. (69). The proof of the proposition is complete by noticing that the sum $\sum_{k=1}^{d}C_{k}$ is an upper bound on the total number of strictly incoherent Kraus operators, and thus

[TABLE]

Appendix G Proof of Proposition 7

Consider a quantum operation defined via the Kraus operators

[TABLE]

with $0\leq i,j\leq d-1$. We will now prove that the strictly incoherent operation $\Lambda(\rho)=\sum_{i,j}K_{i,j}\rho K_{i,j}$ cannot be implemented with fewer than $d^{2}$ Kraus operators. For this, it is enough to show that the corresponding Choi state $\rho_{\Lambda}$ has rank $d^{2}$. This can be seen by applying the operators $K_{i,j}\otimes\openone$ to the maximally entangled state vector $\ket{\Phi_{d}^{+}}=\sum_{k=0}^{d-1}\ket{k,k}/\sqrt{d}$:

[TABLE]

The proof is complete by observing that all these $d^{2}$ (unnormalized) states are linearly independent.