Wigner-Yanase skew information wigner1963information wigner1964positive was introduced originally to express the amount of information in density operator not commuting with the observables. For now there are a lot of generalizations and applications of skew information in quantum information sciences luo2003wigner luo2007skew luo2018coherence . In the other hand, Quantum resource theories (QRTs) chitambar2018quantum offer a highly versatile and powerful framework for studying different phenomena in quantum physics, typical examples are quantum entanglement horodecki2009quantum , quantum coherence winter2016operational , quantum reference frames gour2008resource and Quantum Thermodynamics brandao2015reversible . A general quantum resource theory consists of a class of “free” states along with a class of “free” or allowable operations coecke2016mathematical . The essential resource theoretic condition is that the set of free states is closed under the set of free operations. Hence, any state that is not free is a resource since it cannot be obtained using the allowable operations chitambar2016comparison . In the quantum coherence theory, for example, there are various free operations such as incoherent operation (IO), dephasing- covariant operation (DIO) and strictly-incoherent operation (SIO). In conjunction with each of the operational classes, one can define different measures of coherence. From a resource theoretic perspective, the crucial property of these measures is that they are monotonic under the specified class of operations. To give the measures physical meaning, one seeks to find some operational interpretation of the measure, thereby enabling the measure to quantify some particular physical property or process. Various resource measures and resource monotones had been formulated and some operational interpretations had bee provided chitambar2018quantum winter2016operational .
Among various type of resources, one can specify the resource using the resource destroying maps liu2017resource . Kollas Nikolaos showed a type of optimization-free measures based on resource destroying maps kollas2018optimization . Recently, Shunlong Luo luo2018coherence formulated a generalization of skew information luo2003wigner from the viewpoint of state-channel interaction. So it is naturally to consider whether we can use this state-channel interaction to quantifying the resource given the resource destroying channel.
In this paper we first give an alternative proof of the monotonicity of skew information via operator algebra approach, and then prove that the skew information is of strong monotonicity, which is an open problem in luo2007skew . Second we formulate a family of generalized skew information acted as proper resource measures in the framework of resource destroying map, the free operation should be also modified in the sense that it does not disturb the resource destroying map.
Ii Monotonicity of skew information
for and Dyson suggested extending this to . Recently Shunlong Luo luo2018coherence gave some new generalizations. Given a Hilbert space , let denote the set of all operator on . For any operator , which needs not to be Hermitian, the skew skew information of with respect to is defined as luo2018coherence :
where for all and is the norm induced by the Hilbert-Schmidt inner product . And if is a Trace-Preserving and Completely Positive (TPCP) map with Kraus operators , i.e., , it is naturally to define the state-channel interaction luo2018coherence by
The quantity enjoys some pleasant properties such as non-negativity, convexity, monotonicity, and is independent of the choice of Kraus operators of , see luo2018coherence for more details. Among various properties of , it is crucial that is monotone under some TPCP map , i.e.,
The original proof of monotonicity in luo2018coherence is based on Landau-von Neumann equation. In this paper we give an alternative proof of the monotonicity of the skew information via the operator algebra method. And furthermore, we show that is of strong monotonicity, i.e.,
where and . To this end, we first recall some lemmas.
Lemma 1 stormer2012positive : If is a unital positive map, then for every normal element in its domain, we have and .
Lemma 2 hiai2012quasi : Assume that is an operator monotone function with and is a unital Schwarz mapping, that is, for all , then
for and for invertible density operators , where
and is the linear mapping defined by
Lemma 3: For a TPCP map acted as and an operator , if and only if and ; if and only if and , where is the adjoint of in the sense that .
Proof.—The necessity is very simple since if for all then and then , the conclusion follows.
For the sufficiency we only need to consider the equality
The second statement is followed similarly.
We then give an proof of the monotonicity as well as strong monotonicity of .
Theorem 4: For a TPCP map , is well defined:
If TPCP map satisfies and as well as , then is monotone and strong monotone under the action of :
where and .
Proof.— Since is a TPCP map, then by Lemma 1 is a unital completely positive map and satisfies the Schwarz inequality: . For a fixed Kraus operator of , and , consider the function , it is easy to see that is operator concave as well as operator monotone bhatia2013matrix , by Lemma 2, we have
Since and , by taking inner product with respect to , we have
Since these two terms are both linear on , we can conclude that
by summing up all it must holds that
Thus we reproduce the monotonicity result of in luo2018coherence via the operator algebra viewpoint. Next we consider the strong monotonicity of .
For the free operation , consider the channel
where is a unitary operator and is the number of Kraus operators . It is easy to verify that
by the monotonicity of , therefore,
This finished the proof.
In luo2007skew , the authors showed that for Hermitian operator , is monotone under a TPCP map :
as long as satisfies: and , or equivalently, for all , but they only proved that satisfies the strong monotonicity in two-dimensional case. In fact, by same line reasoning in Theorem 4, we can show that satisfies the strong monotonicity in general. Virtually,
Hence we solve the open problem leaved in luo2007skew .
We can even do further. In jenvcova2010unified the authors formulated a family of functions
where and . For strictly positive they define jenvcova2010unified
One can see that this definition generalizes the relative entropy as well as the skew information. In fact, when and , reduces to the usual relative entropy, i.e.,
And when , and ,
which yields the original skew information up to a constant.
Since enjoys some pleasant properties like skew information, it is naturally to extend our previous discussion by utilizing this unified entropy.
for operator and density operator and , and when a TPCP map written as , we define
for . If , is very similar to in Theorem 4 but there are several subtleties due to the linear term. As a consequence, we only need to require that , or equivalently, for all . We must show that share the same properties as when . We first show that is independent of the Kraus operator representations and thus is indeed a quantity about and channel . In fact, if for all density operator
, then there exists a unitary matrixsuch that . By some direct calculation one can see that , which is independent of the Kraus operators of . The non-negativity and convexity of can be found in jenvcova2010unified . For , the function is operator monotone as well as operator concave bhatia2013matrix . So the monotonicity and strong monotonicity of under TPCP map follows from the same line in Theorem 4. However, there does not exist any discussion about the monotonicity when . Since in this case, the function is a operator convex function, the method in Lemma 2 is not available anymore. We leave it as an open problem. For now we can conclude the following theorem.
Theorem 5: For a TPCP map with Kraus operators , the other TPCP map satisfies and for all and , the quantity
is nonnegative, convex on and monotonicity under the action of . That is, with equality if and only if , is convex on and more crucially,
Furthermore, it is of strong monotonicity:
where and .
Iii Applications in Resource framework
Given a Hilbert space , we say that a TPCP map is a resource destroying map liu2017resource if it satisfies the following two properties:
It maps any free state to itself; i.e., .
It maps any (possibly not free) density operator to a free state; i.e., .
From its definition, it is not clear at all that a resource destroying map exists for a given QRT. However, the full necessary and sufficient conditions for the existence of a resource destroying map were derived in gour2017quantum .
In this section we show that the skew information and its generalization can be acted as a proper resource measure with respected to resource destroying map . The typical example is in quantum asymmetry theory. The resource theory of asymmetry with respect to a given representation of a symmetry group has been used extensively to distinguish and quantify the symmetry-breaking properties of both the states and the operations marvian2012symmetry marvian2013theory . The asymmetry theory can be described by a resource destroying map liu2017resource , namely,
where is the unitary representation of and is the Haar measure with respect to . For simplicity we confine our focus on finite group but note that the conclusions hold also for compact Lie group. The symmetric state of the free state are those invariant under , i.e.,
An equivalently characterization is that marvian2012symmetry
The free operation we consider in this paper is the TPCP map such that for all . Since are unitary thus normal the second type of condition is satisfied automatically. By Lemma 3, it holds that for all and all . Therefore, the generalized skew information between state and channel can be served as a proper resource measure of asymmetry due to Theorem 5.
Another typical example is the resource theory of coherence winter2016operational streltsov2017colloquium marvian2016quantify yu2017quantum . In this case the resource destroying map is the dephasing map
for a fixed basis . The Kraus operators are and for , reproduces the result in yu2017quantum up to a constant.
In this paper we formulated a family of resource measure extended by skew information when the resource theory can be characterized by a resource destroying map and for a class of particular free operations. The new measure acted well in asymmetry theory and coherence theory. There are also some unsolved problems. The first one is how to generalize our measure to a more wide free operations. After all, the requirement for commute relation between Kraus operator seems too severe. The second problem is to consider the monotonicity of under the TPCP maps when . Recently, the author in vershynina2018quantum showed the monotonicity of quantum quasi-entropy under partial trace, but it is different from our open problem. It is expected that a similar operator algebra approach proof should be applied. And the last one is to figure out the relationship between our measure and the measure introduced in zhao2018coherence .
The author is very grateful to professor Shunlong Luo, professor Shaoming Fei and Wei Xie for insightful discussions.
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