Quantumness of the quantum channels
Quantum coherence is a fundamental aspect of quantum physics and plays a central role in quantum information science. This essential property of the quantum states could be fragile under the influence of the quantum operations. The extent to which quantum coherence is diminished depends both on the channel and the incoherent basis. Motivated by this, we propose a measure of nonclassicality of a quantum channel as the average quantum coherence of the state space after the channel acts on, minimized over all orthonormal basis sets of the state space. Utilizing the squared -norm of coherence for the qubit channels, the minimization can be treated analytically and the proposed measure takes a closed form of expression. If we allow the channels to act locally on a maximally entangled state, the quantum correlation is diminished making the states more classical. We show that the extent to which quantum correlation is preserved under local action of the channel cannot exceed the quantumness of the underlying channel. We further apply our measure to the quantum teleportation protocol and show that a nonzero quantumness for the underlying channel provides a necessary condition to overcome the best classical protocols.
Quantum coherence arising from superposition is the most fundamental aspect of quantum mechanics and is responsible for nonclassical properties of quantum systems. The role of quantum coherence in the wide range of areas, e.g. thermodynamics, quantum algorithms, quantum correlations, biology and etc. streltsovrev2017 (); streltsovprl2015 (); streltsov distil (); napoli2016 (); anand (); hillery2016 (); shi2017 (); piani2016 (); bu2016 (); karpat2014 (); marvian2016 (); marvian2016 2 () shows the importance of its investigation. Although it is not a new concept and is as long-standing as quantum mechanics itself, the study of quantum coherence as a quantum resource theory (QRT) pleniocoherence2014 () has attracted a lot of interest recently.
On the other hand it is understood that quantum channels, the completely positive and trace preserving maps, could reduce quantum coherence of the states. Quantum channels provide a framework to investigate coherence behavior under different dynamics and noises. Cohering power and decohering power for quantum operations manikarimipour () and also classification of quantum channels based on coherence-breaking property coherencebreaking () in comparison with entanglement-breaking channels entanglementbreaking () arose from these investigations. Another important class of channels is the so-called semiclassical channels, those that map all input quantum states into ones which are diagonal in the same basis; streltsovprl2011 (). Clearly, at the result of applying a semiclassical channel the space of quantum states changes into classical one meaning that all states are distinguishable by a classical observer measuring states with respect to the channel’s diagonal basis meznaric 2013 ().
The QRT provides preliminaries to recognize the interrelations between coherence and different physical theories streltsovrev2017 (); streltsovprl2015 (); streltsov distil (); chitambar coh vs ent (); zhu 2017 (); streltsovprx2017 (); streltsov merging (); coh vs cor (); frozencoh (). Particularly, it has been shown that coherence is the resource in the process of generating entanglement in bipartite systems streltsovrev2017 (); streltsovprl2015 (). Furthermore, one of the most significant results is that quantum discord type correlations are nothing but the minimum amount of quantum coherence of the bipartite system with respect to the local product basis frozencoh (). In view of this, inability of semiclassical channels in local creation of quantum correlations is predictable as they remove all coherence of the state they applied on. Actually it has been shown that for creation of quantum correlation via local channels in finite-dimensional systems, a necessary and sufficient condition for channels is that they should not be commutativity preserving hu 2012 (). A quantum channel is said to be a commutativity preserving channel if it preserves commutativity of any two compatible states, i.e. , then hu 2012 (). Clearly semiclassical channels are commutativity preserving, so nonsemiclassicality is a necessary condition to create quantum correlations locally. Particularly in qubit systems, necessary and sufficient conditions for channels are nonunitality and nonsemiclassicality streltsovprl2011 (). These results lead us to conclude that there should be connections between the amount of nonsemiclassicality of one-party channels and quantum correlations of the bipartite states.
Furthermore, one can claim that nonunitary quantum channels generally induce decay in quantum properties of the states with respect to some orthonormal basis that the coherence of most outputs is reduced. This is at the heart of channels contraction effect on the space of states, the case of semiclassical channels is the special one making the states completely classical. In the other words, a semiclassical channel washes out quantum coherence of the input states making the states distinguishable by a classical observer. We adapt this as our notion of classicality of channels and pursue the question that how much a channel is far from this very notion of classicality. Our notion of classicality is different from the one given in meznaric 2013 () based on the concept of einselection, characterizing a classical operation as a one that commute with the completely dephasing process. Here we introduce a measure to quantify the amount of quantumness preserved by the channel, along with a natural coherence-based criterion to decide whether a given channel is semiclassical or not. We further investigate the local action of the channels and show that the quantumness of the channel provides an upper bound for the quantum correlations of the associated Choi states. Moreover, by applying the measure to quantum teleportation protocol, we show that the quantumness of the underlying channel provides a necessary, but not sufficient, condition to overcome the best classical protocols.
The paper is organized as follows. In Sec. II, we briefly review quantum coherence measures and introduce the squared -norm of coherence. In Sec. III, we introduce our method to quantify the quantumness of the channels. A closed form for the quantumness of the qubit channels and its properties are also presented in Sec. III. Section IV is devoted to study local action of the qubit channels. In particular, we identify a relation between the channel’s quantumness and the quantum correlation of its associated Choi state. Application of the presented measure to quantum teleportation protocol is presented in Sec. V. The paper is concluded in Sec. VI.
Ii Preliminary: MEASURE FOR QUANTUM COHERENCE
Resource theory of quantum coherence is based on three ingredients; incoherent states denoted by , incoherent operations, and coherence monotones. This framework for quantum coherence provides a way to quantify coherence napoli2016 (); streltsovrev2017 (); streltsovprl2015 (); winter2016 (); zhu 2017 (); yu 2016 (); shao 2015 (); pleniocoherence2014 (); Aberg2006 (). A coherence measure is a map from the set of states to the set of non-negative real numbers, with the following conditions imposed by QRT pleniocoherence2014 (); Aberg2006 (); streltsovrev2017 (). (a) Non-negativity: and if and only if . (b) Monotonicity: for any non-selective incoherent operation, meaning that one cannot create coherence via incoherent operations. (c) Strong monotonicity: where and is an incoherent Kraus operator. This property assures that coherence cannot be increased on average using incoherent operations in case individual outputs are accessible. (d) Convexity: , means that mixing cannot increase coherence. Also it has been proved that convexity and strong monotonicity together imply monotonicity pleniocoherence2014 ().
The -norm of coherence defined by pleniocoherence2014 ()
is a measure satisfying the desired properties. In contrast, it is shown that the sum of the squared absolute values of all off-diagonal elements violates monotonicity pleniocoherence2014 (). However, for further use let us mention here that for qubit systems it coincides with the squared -norm
and satisfies the properties of a bona fide measure of coherence (see Appendix A for the proof).
Iii Quantumness of the quantum channels
Quantum coherence is an evidence of existence of the quantum properties in a given state. In order to investigate the amount of quantumness of the state space, a good candidate is to average on coherence of all states. It turns out that minimization over all orthonormal basis sets results in finding the one that most of the states behave more classically with respect to it. Obviously, without any evolution in states, the state space is so symmetric that the minimization is completely pointless. The importance of such optimization appears in case of nonunitary evolution of states and the result addresses the question that how much quantumness does a quantum channel preserve? Clearly, this is a restatement of asking “how much does a quantum channel deviate from a semiclassical one?”, as this class of operations erases quantum coherence of all states completely. Motivated by this, we define the quantumness (or equivalently the nonsemiclassicality) of a quantum channel as
where subindex denotes that the defined quantity depends on the chosen measure of coherence, and the factor is introduced in order to normalize the desired quantity at the end. Above, the integration is to average on coherence of all states. It could be straightforwardly proved, because of non-negativity condition of any bona fide measure of coherence, that and it is zero if and only if the quantum channel is semiclassical. Furthermore, at the result of integration over all input states, the equality holds and minimization over all incoherent bases leads to , where is an arbitrary unitary map, for unitary operator . So as an immediate corollary, the nonsemiclassicality of any unitary map is the same as the one of identity channel with no action on its input states, .
In the following subsection we focus on qubit channels and show that the minimization specified in Eq. (3) can be calculated analytically for the squared -norm of coherence.
iii.1 Qubit channels
The effect of the qubit channel is an affine transformation on the Bloch sphere; , where is the Bloch vector of the input state. Here the matrix and the vector are features of the channel . Having this in mind, to be more computable, we choose the squared -norm as the measure of coherence and derive an explicit expression for the quantumness of qubit channels. To apply Eq. (3) in this case, one has to measure quantum coherence of any state with respect to an arbitrary orthonormal basis , where and . A straightforward calculation shows that
which can be used to obtain the quantumness of the qubit channels
Above we set and the minimum is taken over all unit vectors . Also the last equality follows from , arisen from the fact that for any in the integration there is a corresponding . To continue, we use for an arbitrary vector and any unit vector . Moreover, by writing and and using , we get
where . The minimum is achieved when is an eigenvector of corresponding to the largest eigenvalue, so that
where are eigenvalues of in nonincreasing order. Equation (7) provides an explicit expression for the quantumness of qubit channels.
holds for any unit vector . Recalling that the trace of a matrix is independent of the basis chosen, one can define as a set of orthonormal basis, so . This holds for any orthonormal basis, hance for any basis with we find
Here, the last inequality follows from the fact that singular values of cannot be greater than 1, so expectation value of on any normalized vector is bounded above by .
Interestingly, the unitary channels are the only channels that reach the maximum quantumness . Recalling that quantumness of the unitary channels is equal to the one of identity channel, we have to proof the above assertion for the identity channel. Obviously, for identity channel for which and , the bound is saturated. On the other hand, if then , implies that the channel should be unital, i.e. , as the singular values of cannot reach its maximum value unless . Putting , then there will be no condition on and , as such and the channel is identity.
In the rest of this paper, we drop the subscript from the quantumness and denote the quantumness, Eq. (6), by for the sake of brevity. In what follows we go into more details about qubit channels and their local action on two-qubit system.
Iv Local action of the quantum channels
Equation (7) provides an analytical expression for the quantumness of qubit channels and should facilitate the investigation of questions concerning the quantum channels. In particular, it may be helpful to investigate questions concerning local actions of quantum channels such as; wether a channel with high nonclassicality will produce more quantum effects when the channel acts locally on composite systems? Is there any relation between the channel quantumness and the quantum correlation preserved under local action of the channel? To address these questions, we need to recall some important classes of quantum channels with special focus on their local action. A quantum channel is called entanglement-breaking channel, channels that wash out entanglement when applied locally entanglementbreaking (), if and only if it can be written in the so-called Holevo form holevo ()
where is a density matrix and forms a POVM. An important subclass of the entanglement-breaking channels is the so-called quantum-classical (QC) channels holevo (), defined by Eq. (10) when where is an orthonormal basis. As it is clear from definition, any QC channel is semiclassical. On the other hand, any semiclassical channel is at least coherence-breaking, with respect to some basis, but the latter is a subset of QC channels coherencebreaking (). Accordingly, a quantum channel is quantum-classical if and only if it is semiclassical.
Considering the effect of semiclassical channels in removing quantum aspects of input states, it has been mentioned that local semiclassical channels cannot create discord type correlations. It follows from the equivalency between quantum-classical and semiclassical channels that there is a much more stronger fact; semiclassical channels not only cannot create quantum correlations locally but also destroy quantum correlations of the party they applied on correlationbreaking (). Even more, only these channels have such effect. In summary, the following equivalent statements will clarify semiclassical channels entanglementbreaking (); correlationbreaking (); (i) is a semiclassical or QC channel. (ii) is a discord-breaking channel. (iii) is a zero-discord state on part where is a maximally entangled state.
The above discussion suggests that for preserving quantum correlations in a composite system, the assumed quantum channel should retain some quantumness in the whole space of the one-party system. In the following, we will show that there is actually an interesting relation between the channel quantumness and the quantum correlation preserved under local action of the channel. Not surprisingly, such relation is based on the Choi-Jamiołkowski isomorphism, stating that there is a one-to-one correspondence between quantum channel and quantum state mathematical ().
The following relation exists between quantumness of , measured by Eq. (7), and quantum correlation of , measured by the (normalized) geometric discord
To proof this we first need to assert the following lemma.
Let be a qubit channel described by the affine parameters and . Then acts locally on the Bob’s side of a two-qubit state as
where . Above , are the Bloch vectors of Alice and Bob, respectively, and is the correlation matrix of the initial state.
See appendix B for a proof of the above lemma.
where equals to except for the sign of its second column which is opposite to the one of . (ii) The normalized geometric discord vedral () of is equal to the sum of two lower eigenvalues of . Recalling we get . Evidently HornBook1985 () which completes the proof of observation.
Now, several properties of the inequality (11) are in order. (i) Looking at two matrices and shows that if and only if (unital channels), or be an eigenvector of corresponding to its largest eigenvalue. (ii) if and only if , follows from the fact that both quantities vanish if and only if , that is to say, if and only if there exists a unit vector such that and . This was an expected result since QC channels are the only class of channels removing quantum discord and transforming any state into a quantum-classical state. (iii) Both quantities attain their maximum value if and only if is an orthogonal matrix (consequently ), i.e. for the unitary channels which transform a maximally entangled state into a maximally entangled state. (iv) Both quantities are invariant under (local) unitary maps.
Paying attention to and its common features with geometric discord we argue that, for the states defined by Eq. (IV), this quantity has the properties necessary for a good measure of quantum correlation. The duality between and , described by the Choi-Jamiołkowski isomorphism, reveals that the quantumness of is a necessary resource for quantum correlation of . In the other words, the preserved quantum correlation, measured by geometric discord, cannot exceed the quantumness of the channel. Moreover, the fact if and only if , states that a channel with nonzero quantumness will preserve some quantum correlation. This sheds light on the claim that more nonsemiclassical a quantum channel is more quantum correlation is preserved when the channel acts locally on a composite system. A semiclassical channel, on the other hand, destroys all one-party quantumness, and as such it cannot create or preserve any discord-like quantum correlation.
To demonstrate this, let us consider the amplitude damping channel characterized by the following Kraus operators nielsen ()
The affine parameters associated with this channel are and . Straightforward calculations show that for this channel the quantumness and geometric discord are given by
respectively. Figure 1 shows the quantumness and geometric discord for this channel in terms of the channel’s parameter . As can be seen from this figure both quantumness and geometric discord are monotonically decreasing from to as channel’s parameter goes from to . Indeed, when the damping rate passes from , quantum correlation reaches the channel’s quantumness.
For entanglement-breaking channels, despite they may preserve a fraction of one-party quantumness, significant amount of quantumness should be removed. More precisely, for such channels the quantumness is bounded above as
holds for any unit vector which lies in the direction of . This follows easily from Eq. (9) and noting that for entanglement-breaking channels we have entanglementbreakingqubit () , leads to . Clearly, both of inequalities in Eq. (17) cannot be saturated simultaneously, so does not definitely reach for an entanglement-breaking channel. On the other hand, for the unital channels whose necessary and sufficient condition to be entanglement-breaking is entanglementbreakingqubit (), the quantumness is bounded above by , saturated for example by a channel described by the Kraus operators , and .
Before we conclude this section, let us mention here that the quantity defined above measures the amount of quantumness preserved in the state during the action of the channel. Clearly, a nonzero value for the channel’s quantumness does not offer an ability to create quantum correlation. For example, although unital qubit channels mostly preserve quantumness of the input states, they cannot create quantum correlation locally streltsovprl2011 (). On the other hand, a nonunital channel described by the Kraus operators and creates the maximum amount of quantum discord locally correlating power (), although its quantumness is only . In summary, for creation of quantum correlations by a local channel, other additional properties rather than preserving quantumness are required.
V Application to quantum teleportation
Consider the generalized depolarizing channel described by the Kraus operators for where , and , and are the Pauli matrices. For this channel, the affine parameters are given by and , where , and . If then and we get for quantumness of this channel .
The generalized depolarizing channel is of particular importance because of its usefulness in studying quantum teleportation. The teleportation protocol BennettPRL1993 () consists of a two-qubit states shared between two separated parties, say Alice and Bob, and an unknown qubit state where with and . The protocol is described by the generalized depolarizing channel
where and . Here where are the four maximally entangled Bell states associated with the Pauli matrices , i.e. for . Furthermore, for optimal utilization of a given entangled state as resource, one has to choose the local basis in such a way that .
To characterize the quality of the teleported state , it is useful to look at the fidelity between and . We find where is the Bloch vector of the input qubit. Moreover, the average fidelity is defined by averaging the fidelity over all possible input states , turns out that . It is clear that Alice and Bob could gain a fidelity better than (the best possible fidelity when they communicate only through classical channel), if and only if . A simple investigation shows that if then . Equivalently if then and we do not benefit the quantum advantages. However, the inverse is not correct meaning that it is possible for the channel to possesses nonzero quantumness but the average fidelity of the teleportation be less than . This implies that in order to have a teleportation protocol with fidelity better than the classical one, a nonzero quantumness for the associated channel is necessary, although it is not sufficient. For example, for the Werner state , we have and . This state is separable (disentangled) if and only if and, not surprisingly, only for such values of parameter the fidelity of the protocol is less than . However, as it is clear from Fig. 2, even for , the channel possesses a nonzero quantumness.
Regarding that quantum coherence is the most fundamental aspect of quantum physics, it can be used to characterize and quantify other nonclassical features identified in the emerging quantum information theory. Using this very signature of quantumness, we propose the average coherence of the channel’s outputs, minimized over all incoherent bases, as the degree of nonclassicality of the channels, . This, intuitively, quantifies the amount of quantumness preserved by a quantum channel and measures the deviation of the assumed channel from the so-called semiclassical channels. Applying the squared of as a qualified measure of coherence for qubit systems, an analytical expression is gained for any qubit channel. It is shown that attains its maximum value for any unitary map, however, in other cases it cannot reach its maximum value due to the contraction of the Bloch sphere to its associated channel’s ellipsoid, hence decreasing happens in quantum properties. By decrease in quantum properties we mean that the error of description of states using a family of classical probability distributions is getting more and more close to be vanished when the measurement is in the minimized basis. Obviously, doing maximization rather than minimization in our definition of quantumness introduces the basis that most states are noncompatible with, meaning that most states show quantumness if we perform measurement in such a basis. In the case of maximization, it could be straightforwardly proved that the contraction into totally mixed state has the lowest amount and contraction into a pure state, which is a semiclassical map, has the largest amount.
Moreover, utilizing the Choi-Jamiołkowski isomorphism between the quantum channel and Choi state , we show that the amount of quantum correlation preserved in the Choi state is bounded above by the amount of quantumness of the quantum operation. The bound is saturated both for the zero-discord states and the maximally entangled ones. Even more, the saturation occurs faithfully in a sense that one quantity attains its respective minimum (maximum) if and only if the other one attains its minimum (maximum). This, in turn, proposes to consider the quantumness of as a necessary resource for the quantum correlation of and suggests that such notion of quantumness can be considered as a quantum correlation measure for the class of Choi states. For an entanglement-breaking channel whose local action washes out all quantum entanglement of the input states, the preserved discord type quantum correlation of the Choi states cannot be larger than the washed out one, meaning that the channel should clear more than half the discord type correlations in order to remove all quantum entanglement. Due to the widespread application of the quantum channels in quantum information theory, presenting a measure to quantify nonclassicality of the quantum channels could be helpful in identifying the resources of the quantum advantages. We further introduce such application by applying our measure to quantum teleportation protocol and showing that the quantumness of the underlying channel is a necessary resource to gain fidelity better than the one offered by the best classical protocol.
This work was supported by Ferdowsi University of Mashhad under Grant No. 3/45469 (1396/10/09).
Appendix A Proof of the monotonicity of
Here we are going to prove properties of the squared -norm of coherence qualifying it as a coherence measure in qubit systems. To do this note that this quantity satisfies positivity and convexity as the squared of any convex function is convex too. It remains only to prove that the aforementioned quantity is strong monotone under incoherent kraus operators. In qubit case, it has been shown that a general incoherent operation admits a decomposition with at most Kraus operators streltsov2017prl ()
where , , , and . For the input state , the squared -norm coherence equals . On the other hand
and we have defined and (for ) with conditions and . In Eq. (A), the first line follows from the fact that the three last Kraus operators remove coherence of the input states. To complete the proof of monotonicity of the squared -norm, we need to show that cannot exceed 1. A straightforward calculation can be applied to see that for the parameters , the factor reaches its maximum value . This completes the proof.
Appendix B Proof of the Lemma 1
To prove Eq. (12), we start with the Hilbert-Schmidt representation of a general two-qubit state
Using this and linear property of the quantum channels we get
In affine representation, one can easily shows that and , where can be used in Eq. (23) to get
After rearranging the terms, we arrive at Eq. (12).
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