Qubit-environment entanglement generation and the spin echo
QQubit-environment entanglement generation and the spin echo
Katarzyna Roszak and (cid:32)Lukasz Cywi´nski Department of Theoretical Physics, Faculty of Fundamental Problems of Technology,Wroc(cid:32)law University of Science and Technology, 50-370 Wroc(cid:32)law, Poland Institute of Physics, Polish Academy of Sciences, 02-668 Warsaw, Poland (Dated: July 7, 2020)We analyze the relationship between qubit-environment entanglement that can be created duringthe pure dephasing of the qubit initialized in a superposition of its pointer states, and the effec-tiveness of the spin echo protocol. Commonly encountered intuitions connecting the amount ofdecoherence with the amount of qubit-environment entanglement - suggesting that large echo signalcorresponds to undoing of a large amount of entanglement - hold only for pure initial states of theenvironment, which is obviously a rarely encountered case, and we focus here on mixed states of theenvironment. We show that while the echo protocol can obviously counteract classical environmen-tal noise (but it does not have to, if the noise is not mostly of low-frequency character), it can alsoundo dephasing associated with qubit-environment entanglement, and there is no obvious differencein its efficiency in these two cases. Additionally, we show that qubit-environment entanglement canbe generated at the end of the echo protocol even when it is absent at the time of application of thelocal operation on the qubit (the π pulse). We prove that this can occur only at isolated points intime, after fine-tuning of the echo protocol duration. Finally, we discuss the conditions under whichthe observation of specific features of the echo signal can serve as a witness of the entangling natureof the joint qubit-environment evolution. I. INTRODUCTION
Environmentally induced dephasing of superpositionsof pointer states of a controlled quantum system is com-monly associated with creation of system-environmententanglement, or at least the presence of the latter isdeemed to be necessary in order to call this process quan-tum decoherence [1–3]. However, as has been pointedout in literature, this association holds only when theinitial states of both the qubit and the environment arepure [1–4]. In the more general, and much more real-istic case of mixed environmental states, dephasing ofthe system does not have to be accompanied by estab-lishment of system-environment entanglement, and intu-itions concerning distinguishing between “quantum deco-herence” and “dephasing due to classical environmentalnoise” (understood here strictly as leading to no system-environment entanglement) that are built in works fo-cusing on pure-state vs “classical” environments becomeunreliable [5–13].We shed light on this general problem by focusing onthe relationship between the effectiveness of qubit coher-ence recovery in a spin echo experiment [14–16], which iswell known to lead to such a recovery when the environ-ment is a source of external noise of mostly low-frequencycharacter [17, 18]. We show that the echo procedure can(but does not have to) lead to coherence recovery whenthe dephasing is not associated with qubit-environmententanglement (QEE), but it can also undo QEE, whileusing only local operations on the qubit. Interestingly,there is no obvious correlation between the efficiency ofcoherence recovery and presence or absence of QEE gen-erated during the evolution of the qubit and its environ-ment.In fact, we show that it is possible for QEE to appear at the end of the echo protocol, with no entanglementpresent at the time of application of the unitary oper-ation to the qubit. This should not be surprising, asthe evolutions that are most interesting in the contextof echo protocol typically have non-Markovian charac-ter, and at the time of application of the local unitaryoperation the state of the qubit and the environment istypically correlated. This effect can however only occurat isolated points in time, and this is the only feature ofthe echo experiment that conforms to the commonly en-countered (but generally incorrect) intuitions that echoprotocol should undo the generation of QEE, as is typi-cally undoes qubit dephasing.While most of our results underline the lack of strongcorrelation between efficacy of coherence recovery in spinecho protocol and the presence of QEE during evolution,we show that there is at least one situation - that of astationary environment and a particular form of qubit-environment coupling - in which the appearance of aphase shift between the initial and the echoed coher-ence of the qubit signifies that the evolution is of QEE-generating character.The paper is organized as follows. In Sec. II we intro-duce the echo protocol for the qubit undergoing pure de-phasing due to an interaction with its environment, andrecapitulate the basic criterion for appearance of QEEduring pure dephasing evolution. In Sec. III we discussthe conditions for the echo to work prefectly, i.e. to leadto the recovery of the initial pure state of the qubit. Asthe perfect echo necessarily leads to removal of any entan-glement (if any was in fact present during the evolution),in Section IV we focus on the imperfect echo and its rela-tion to generation of entanglement during the evolution.There is no simple relation, and we show there that theecho can in fact lead to creation of entanglement in the a r X i v : . [ qu a n t - ph ] J u l final state even if there was none at the time of applica-tion of the local operation to the qubit. However, as weshow in Section V, it can happen only at certain pointsin time, and the π pulse applied to the qubit cannottransform a joint system evolution which is essentiallynonentangling into an entangling one. Finally, in Sec. VIwe describe the conditions for the initial environmentalstate and qubit-environment coupling that allows to usethe echo signal as a witness of the entangling nature ofthe evolution of the qubit and its environment. Sec. VIIconcludes the paper. II. PURE DEPHASING, ENTANGLEMENT,AND ECHOA. Pure dephasing
In the following, we study the spin echo performed ona qubit in an arbitrary pure-dephasing scenario, meaningthat the only constraint on the qubit-environment inter-action is that it does not disturb the occupations of thequbit [9, 19, 20]. The most general form of the Hamilto-nian which describes the pure dephasing case isˆ H = ˆ H Q + ˆ H E + | (cid:105)(cid:104) | ⊗ ˆ V + | (cid:105)(cid:104) | ⊗ ˆ V . (1)The first term of the Hamiltonian describes the qubit andis given by ˆ H Q = (cid:80) i =0 , ε i | i (cid:105)(cid:104) i | , the second describesthe environment, while the remaining terms describe thequbit-environment interaction with the qubit states writ-ten on the left side of each term (the environment oper-ators ˆ V and ˆ V are arbitrary, as is the free Hamiltonianof the environment ˆ H E ). Hence, the only constraint onthe Hamiltonian, which restricts the qubit evolution topure dephasing, is that the interaction term is diagonalwith respect to the qubit eigenstates.The evolution operator corresponding to the Hamilto-nian (1) may in general be written in the formˆ U ( t ) = e − i (cid:126) ε t | (cid:105)(cid:104) | ⊗ ˆ w ( t ) + e − i (cid:126) ε t | (cid:105)(cid:104) | ⊗ ˆ w ( t ) , (2)where ˆ w i ( t ) = exp( − i (cid:126) ˆ H i t ), with ˆ H i = ˆ H E + ˆ V i . Notethat while ˆ H Q commutes with all the other terms in ˆ H ,this is not necessarily the case with ˆ H E . We assume thatthe intial state has no correlations between the qubit andthe environment, ˆ σ (0) = | ψ (cid:105)(cid:104) ψ | ⊗ ˆ R (0) , (3)with the initial qubit state | ψ (cid:105) = a | (cid:105) + b | (cid:105) and ˆ R (0)being the initial state of the environment. The qubit-environment density matrix at later time can be writtenas ˆ σ ( t ) = (cid:18) | a | ˆ w ( t ) ˆ R (0) ˆ w † ( t ) ab ∗ ˆ w ( t ) ˆ R (0) ˆ w † ( t ) a ∗ b ˆ w ( t ) ˆ R (0) ˆ w † ( t ) | b | ˆ w ( t ) ˆ R (0) ˆ w † ( t ) (cid:19) . (4) Here the matrix form only pertains to the qubit subspaceand is written in terms of qubit pointer states. If onlythe state of the qubit is of interest, then the reduceddensity matrix of the qubit is obtained by tracing outthe environment from the matrix (4) and we getˆ ρ ( t ) = Tr E ˆ σ ( t ) = (cid:18) | a | ab ∗ W ( t ) a ∗ bW ∗ ( t ) | b | (cid:19) , (5)with normalized coherence W ( t ) = Tr (cid:104) ˆ R (0) ˆ w † ( t ) ˆ w ( t ) (cid:105) . (6) B. Spin echo during pure dephasing
The procedure which is known as the spin echo [14–16] can be described as follows. After the initializationof the qubit state, the qubit and environment evolve fora certain time τ , after which a π -pulse about x or y axisis applied to the qubit (for concreteness we focus hereon pulses about x axis). Such a pulse interchanges theamplitudes of | (cid:105) and | (cid:105) states. Then the system isallowed to evolve for the same time period τ and an-other π -pulse is applied. In the ideal case, this leads tothe qubit regaining its initial state at time 2 τ (after thesecond π -pulse), but even in non-ideal scenarios the de-coherence which is observed after the echo sequence canbe much smaller compared to the evolution without theecho when the environment is a source of external noiseof mostly low-frequency character [17] (see Section III Bbelow for a concise formal explanation of this fact).The evolution in echo experiment with the final time2 τ is described by the operatorˆ U echo (2 τ ) = ˆ σ x ˆ U ( τ )ˆ σ x ˆ U ( τ ) , (7)where ˆ σ x is the appropriate Pauli matrix which describesthe action of the π -pulse on the qubit and ˆ U ( τ ) is a jointsystem-environment evolution operator, which for puredephasing is given by eq. (2). The second π pulse at time2 τ interchanges the two complex-conjugate coherences inthe final reduced state of the qubit, and it is added forconvenience, to make the final coherence equal to theoriginal one, not to its complex conjugate.We assume that the initial state of the qubit-environment system is given by eq. (3). Then the jointsystem-environment state at time τ before the first π -pulse is given by the desity matrix (4). Modeling thewhole procedure with the evolution operator (7) we getthe qubit-environment state directly after the echo se-quence is performed, which is given byˆ σ (2 τ ) = (cid:18) | a | ˆ w ( τ ) ˆ w ( τ ) ˆ R (0) ˆ w † ( τ ) ˆ w † ( τ ) ab ∗ ˆ w ( τ ) ˆ w ( τ ) ˆ R (0) ˆ w † ( τ ) ˆ w † ( τ ) a ∗ b ˆ w ( τ ) ˆ w ( τ ) ˆ R (0) ˆ w † ( τ ) ˆ w † ( τ ) | b | ˆ w ( τ ) ˆ w ( τ ) ˆ R (0) ˆ w † ( τ ) ˆ w † ( τ ) (cid:19) . (8)The echoed qubit state is obtained, as in the case of sim-ple decoherence (5), by tracing out the environment fromeq. (8), which yields ˆ ρ (2 τ ) = Tr E ˆ σ (2 τ ), which has thesame structure as eq. (5), but with normalized coherence W (2 τ ) = Tr (cid:104) ˆ R (0) ˆ w † ( τ ) ˆ w † ( τ ) ˆ w ( τ ) ˆ w ( τ ) (cid:105) . (9) C. QEE condition for pure dephasing with andwithout echo
For any bipartite density matrix which can be writtenin the form (4), the if and only if condition of qubit-environment separability is[ ˆ w † ( t ) ˆ w ( t ) , ˆ R (0)] = 0 , (10)as has been proven in Ref. [9]. Since the qubit-environment state at time τ before the π -pulse is appliedis given precisely by eq. (4), the condition can be ex-plicitly used to check for QEE present just before theapplication of the pulse (the pre-pulse entanglement).The QEE present in the system after the echo proce-dure is performed is similarly straightforward to study,because the qubit-environment density matrix (8) is ofthe same form as the one that is obtained by a simplepure-dephasing interaction (4). The two can be reducedto one another by the transformationˆ w (cid:48) (2 τ ) = ˆ w ( τ ) ˆ w ( τ ) , (11a)ˆ w (cid:48) (2 τ ) = ˆ w ( τ ) ˆ w ( τ ) . (11b)Then the condition for separability of the echoed state is[ ˆ w (cid:48)† (2 τ ) ˆ w (cid:48) (2 τ ) , ˆ R (0)] = [ ˆ w † ( τ ) ˆ w † ( τ ) ˆ w ( τ ) ˆ w ( τ ) , ˆ R (0)] = 0 . (12) III. CONDITIONS FOR PERFECT ECHOA. General considerations
For the echo to be perfect, meaning that the qubit statewhich is obtained after performing the echo is equal tothe initial qubit state, Tr E ˆ σ (2 τ ) = | ψ (cid:105)(cid:104) ψ | , the followingcondition needs to be met,[ ˆ w † ( τ ) , ˆ w ( τ )] = 0 . (13)The complementary condition [ ˆ w ( τ ) , ˆ w ( τ )] = 0 followsfrom the above equation, since commutation of two op-erators implies that there exists a basis in which bothoperators are diagonal and the Hermitian conjugate of any operator is always diagonal in the same basis as theoperator itself.In the situation when the echo reinstates the initialqubit state, it also severs any entanglement which mayhave been generated between the qubit and the environ-ment during their joint evolution. However, the conditionfor perfect echo is not related in any way to the conditionfor absence of QEE at time τ , which is given by eq. (10).The latter depends on the initial state of the density ma-trix of the environment and can be fulfilled both whenthe conditional evolution operators of the environmentcommute, and when they do not.It is fairly straightforward to find an evolution whichleads to a perfect echo for a given τ , or even for any τ ,but does not lead to any QEE generation, and one thatdoes lead to entanglement generation. For example, if[ ˆ V i , ˆ H E ] = 0 for i = 0, 1, and ˆ R (0) ∝ exp( − β ˆ H E ), i.e. theenvironment is in a thermal equlibrium state achievedin absence of the qubit, then there is no entanglementgenerated at time τ , as eq. (10) is fulfilled. However, theecho is perfect only if additionally [ ˆ V , ˆ V ] = 0.On the other hand, if we assume all the commutationrelations from the previous example to be fulfilled, buttake ˆ R (0) such that [ ˆ R (0) , ˆ V − ˆ V ] (cid:54) = 0, we have perfectecho at time 2 τ , but the qubit-environment state is entan-gled at time τ . These examples already show that the be-havior of “echoed” coherence reflects the general featureof dephasing caused by an environment in a mixed state:there is no direct correspondence between the generationof QEE and the amount of dephasing. The echo proce-dure can undo dephasing (even perfectly) not only in the“classical dephasing” case (using the terminology fromRef. [1]), in which no entanglement is established, butalso in the “true quantum decoherence” case, in whichthe entanglement is created during the evolution. B. Small decoherence limit
If the echoed coherence W (2 τ ) is close to unity, as it ofcourse happens when 2 τ is close to the time at which theecho is perfect, one can approximate it by an expressionvalid to second-order in qubit-environment coupling. Forsimplicity, let us focus now on a slightly less general formof the ˆ V i operators, and take them asˆ V = 12 λ ( η + 1) ˆ V , ˆ V = 12 λ ( η −
1) ˆ
V , (14)so that the qubit-environment coupling takes the form of λ ( η ˆ − ˆ σ z ) ⊗ ˆ V . In the formulas above, λ is a dimension-less parameter controlling the strength of the coupling,while η controls the “bias” of the coupling. A commonlyused “unbiased” coupling, ∝ ˆ σ z ⊗ ˆ V , which occurs for ex-ample for qubits based on spin-1 / η = 0, while the “biased” case of η = − m = 0 and m = ± λ ordergives [29, 30] W (2 τ ) ≈ − λ χ (2 τ ) − iηλ Φ(2 τ ) , (15)where the attenuation function χ ( t ) and the phase shiftΦ( t ) are real functions given by χ (2 τ ) = 12 (cid:90) τ d t (cid:90) t d t f ( t ) f ( t ) C ( t , t ) , (16)Φ(2 τ ) = 12 (cid:90) τ d t (cid:90) t d t f ( t ) K ( t , t ) , (17)where C ( t , t ) = Tr E (cid:16) ˆ R (0) { ˆ V ( t ) , ˆ V (0) } (cid:17) , (18)is the autocorrelation function of the operator ˆ V ( t ) =exp( i ˆ H E t ) ˆ V exp( − i ˆ H E t ), K ( t , t ) = − iθ ( t − t )Tr E (cid:16) ˆ R (0)[ ˆ V ( t ) , ˆ V (0)] (cid:17) , (19)is the linear response function [31, 32] associated withthis operator, and the temporal filter function [17, 33]for the echo experiment is given by f ( t ) = Θ( t )Θ( τ − t ) − Θ( t − τ )Θ(2 τ − t ), i.e. | f ( t ) | = 1 for t ∈ [0 , τ ] and iszero otherwise, and it changes sign at t = τ . For deriva-tion of the expression for χ (2 τ ) see Ref. [18], while thederivations of the formula for phase Φ(2 τ ) can be foundin Refs [29] and [30].We assume now that the environment is in a stationarystate, [ ˆ R (0) , ˆ H E ] = 0, which implies that C ( t , t ) is infact a function of a single variable, ∆ t = t − t . We canthen introduce the power spectral density (PSD) of thenoise, defined by S ( ω ) = (cid:90) ∞−∞ e iω ∆ t C (∆ t )d∆ t , (20)and express the attenuation function and the phase shiftas χ (2 τ ) = (cid:90) ∞−∞ ωτ ω S ( ω ) d ω π , (21)Φ(2 τ ) = (cid:90) ∞−∞ ωτ ω cotan ωτ βω S ( ω ) d ω π , (22)where in order to derive the second of these expressionswe have assumed that the environment is actually in athermal state, i.e. ˆ R (0) = e − β ˆ H E / Tr e − β ˆ H E . Vanishing χ (2 τ ) is necessary for occurence of perfectecho, and from the above formulas we see that, takinginto account that S ( ω ) is positive-definite, this can hap-pen at τ (cid:54) = 0 only when PSD consists of a series of deltapeaks at frequencies ω k = 2 πk/τ for integer k . The mostcommonly encountered case is that of PSD concentratedonly at very low frequencies (only k = 0 peak is present),i.e. S ( ω ) ∝ δ ( ω ). This corresponds to time-independentsymmetric correlator of ˆ V ( t ), i.e. C (∆ t ), which requires[ ˆ H E , ˆ V ] = 0. This situation is thus equivalent to the pre-viously discussed case of perfect echo, which might ormight not be accompanied by generation of QEE duringthe evolution of the system, depending on [ ˆ R (0) , ˆ V ] be-ing finite or zero. The situation of S ( ω ) with periodicallypositioned narrow peaks in frequency is more interesting,as it corresponds to ˆ V ( t ) that has nontrivial dynamics.It is also not particularly artificial: it corresponds to sit-uation in which the second-order correlation function ofenvironmental operator ˆ V has a well-defined periodicity.A perfect echo can occur at isolated points in time in thiscase.Let us note that while the response function K (∆ t )vanishes when the environment is completely mixed, thesymmetric correlation function C (∆ t ) has no reason tovanish in this situation. The presence of finite attenu-ation function χ , and thus of finite decay of qubit’s co-herence, obviously does not require the presence of QEE:note that the condition (10) for Q-E separability is ful-filled for a completely mixed initial environmental state. IV. IMPERFECT ECHO AND QEEA. Echo-induced entanglement
Let us consider the situation when at time τ , at whichwe apply a local operation to one part (the qubit) of ourbipartite system, the condition of qubit-environment sep-arability is fulfilled (10), but the perfect-echo condition(13) is not. Based on widespread notion that “local oper-ations cannot increase entanglement” it might seem obvi-ous that, if the evolution does not entangle the qubit withis environment at the time the first π -pulse is applied,it should not lead to QEE after the whole echo proce-dure is performed. Of course, a careful reconsideration ofprecise formulation of the “local operations and classicalcommunications (LOCC) not increasing entanglement”statement shows that this expectation is not necessarilytrue in the situation at hand. When the initial state, withrespect to which we want to look at subsequent changesin entanglement, is a correlated bipartite state, entangle-ment can increase during the evolution, and there is noreason for which a local operation could not aid in theoccurence of this increase [34] (see also discussion in [35]for a different, but in this context analogous situation oftwo-qubit echo caused by local operations on both qubitsleading to revival of two-qubit entanglement).However, there is another intuition that could be usedto support such an expectation: since the perfect echokills any QEE that was generated during the evolution,one could expect that non-perfect echo, albeit still lead-ing to partial recovery of coherence, should diminish itsamount compared to values attained during the evolu-tion, for example at the time of application of the pulse.In the following, we will show that this is in fact not nec-essarily the case. This is nothing else, but another resultof the general fact that the magnitude of system dephas-ing is rather weakly affected by presence or absence ofsystem-environment entanglement when the environmen-tal state is far from being pure .The condition for nonentangling evolution (10) isequivalent to the statement that the operator ˆ w † ( τ ) ˆ w ( τ )has block form in the basis which diagonalizes the initialdensity matrix of the environment and the blocks cor-respond to blocks in which the density matrix ˆ R (0) isproportional to unity. If we write ˆ R (0) = (cid:80) n c n | n (cid:105)(cid:104) n | (where {| n (cid:105)} is the set of eigenstates of ˆ R (0)), wecan rewrite this condition as that either c n = c m or (cid:104) n | ˆ w † ( τ ) ˆ w ( τ ) | m (cid:105) = (cid:104) m | ˆ w † ( τ ) ˆ w ( τ ) | n (cid:105) = 0 for all m and n [9]. Obviously the same condition is valid in caseof the conjugate (cid:16) ˆ w † ( τ ) ˆ w ( τ ) (cid:17) † = ˆ w † ( τ ) ˆ w ( τ ).It is now important to note that, if the condition forthe lack of QEE at time τ (10) is fulfilled, this meansthat there exists a basis in which both the operatorˆ w † ( τ ) ˆ w ( τ ) and the initial density matrix of the envi-ronment ˆ R (0) are diagonal. This is true, because theparts of the density matrix which correspond to non-diagonal blocks in ˆ w † ( τ ) ˆ w ( τ ) are proportional to unity,so the transformation that diagonalizes each block inˆ w † ( τ ) ˆ w ( τ ) cannot change the corresponding part of thedensity matrix which is still proportional to unity. Hence,we can work in the eigenbasis in which both operators arediagonal and we will denote it in the following as {| n (cid:48) (cid:105)} , which yields ˆ R (0) = (cid:88) n (cid:48) c n | n (cid:48) (cid:105)(cid:104) n (cid:48) | , (23)ˆ w † ( τ ) ˆ w ( τ ) = (cid:88) n (cid:48) (cid:16) ˆ w † ( τ ) ˆ w ( τ ) (cid:17) n (cid:48) | n (cid:48) (cid:105)(cid:104) n (cid:48) | , (24)with c n = c n (cid:48) because during the process of diagonaliza-tion of ˆ w † ( τ ) ˆ w ( τ ), transformations in the density matrixremain within subspaces of equal occupations c n .Although diagonality in this basis is obviously pre-served for the conjugate of ˆ w † ( τ ) ˆ w ( τ ) there is no reasonwhy the operators ˆ w † ( τ ) and ˆ w ( τ ) should be diagonalin this basis. The only condition is (cid:88) p (cid:48) (cid:16) ˆ w † ( τ ) (cid:17) n (cid:48) p (cid:48) ( ˆ w ( τ )) p (cid:48) m (cid:48) = (cid:16) ˆ w † ( τ ) ˆ w ( τ ) (cid:17) n (cid:48) δ n (cid:48) m (cid:48) , (25)sinceˆ w † ( τ ) ˆ w ( τ ) = (cid:88) n (cid:48) m (cid:48) (cid:88) p (cid:48) (cid:16) ˆ w † ( τ ) (cid:17) n (cid:48) p (cid:48) ( ˆ w ( τ )) p (cid:48) m (cid:48) | n (cid:48) (cid:105)(cid:104) m (cid:48) | = (cid:88) n (cid:48) (cid:16) ˆ w † ( τ ) ˆ w ( τ ) (cid:17) n (cid:48) | n (cid:48) (cid:105)(cid:104) n (cid:48) | . In other words, for any two evolution operators ˆ w † ( τ )and ˆ w ( τ ) which do not commute at a given time τ (which means that ˆ w † ( τ ) is diagonal in a different ba-sis than ˆ w ( τ )), there exists a set of initial environmen-tal states for which [ ˆ w † ( τ ) ˆ w ( τ ) , ˆ R (0)] = 0. If the initialstate of the environment is described by one of these den-sity matrices then at time τ (both before and after thefirst π -pulse), the qubit-environment density matrix ob-tained by using the evolution operator (2) is separable,but is no longer a product state. The state (after the π -pulse) can be written as σ ( τ ) = (cid:18) | b | ˆ R ( τ ) a ∗ be i ∆ εt ˆ w ( τ ) ˆ w † ( τ ) ˆ R ( τ ) ab ∗ e − i ∆ εt ˆ R ( τ ) ˆ w ( τ ) ˆ w † ( τ ) | a | ˆ R ( τ ) (cid:19) , (26)where ∆ ε = ε − ε , ˆ R ( τ ) = ˆ w ( τ ) ˆ R (0) ˆ w † ( τ ), and thefact that ˆ w ( τ ) ˆ R (0) ˆ w † ( τ ) = ˆ w ( τ ) ˆ R (0) ˆ w † ( τ ) (27)is a straightforward consequence of the separability crite- rion (10) being fulfilled at time τ (the two are equivalent).Applying the other half of the echo procedure (unitaryevolution U ( τ ) followed by the σ x operator) yields σ (2 τ ) = (cid:18) | a | ˆ w ( τ ) ˆ R ( τ ) ˆ w † ( τ ) ab ∗ ˆ w ( τ ) ˆ R ( τ ) ˆ w ( τ ) ˆ w † ( τ ) ˆ w † ( τ ) a ∗ b ˆ w ( τ ) ˆ w ( τ ) ˆ w † ( τ ) ˆ R ( τ ) ˆ w † ( τ ) | b | ˆ w ( τ ) ˆ R ( τ ) ˆ w † ( τ ) (cid:19) . (28)This qubit-environment density matrix is separable, if and only if the condition (cid:104) ˆ w † ( τ ) ˆ w ( τ ) , ˆ R ( τ ) (cid:105) = 0 (29)is fulfilled. The condition is equivalent to the separa-bility criterion for a product initial state of the qubitand the environment initially in state ˆ R ( τ ), when theevolution is governed by the operators ˆ w ( τ ) and ˆ w ( τ ),eq. (10). Interestingly, the resulting state (28) is dif-ferent than the state which would be obtained at time τ from an initial environmental state ˆ R (0) = ˆ R ( τ ).This becomes obvious when the elements of the densitymatrix proportional to ab ∗ are compared in both cases,since ˆ w ( τ ) ˆ w † ( τ ) ˆ w † ( τ ) (cid:54) = ˆ w † ( τ ) (because we assumedthat ˆ w ( τ ) and ˆ w † ( τ ) do not commute with ˆ w † ( τ )). B. Example of qubit-environment entanglementgenerated via the spin echo at time τ for separablestate at time τ As an example let us study a qubit interacting with anenvironment of dimension N = 2. We will study a pair ofinteraction operators ˆ w ( τ ) and ˆ w ( τ ) that do not leadto entanglement in the density matrix (26), but lead toentanglement in the echoed density matrix (28) for a setof initial environmental states.Our exemplary operators ˆ w ( τ ) and ˆ w ( τ ) written inthe eigenbasis of the initial environment density matrixˆ R (0) = c | (cid:105)(cid:104) | + c | (cid:105)(cid:104) | areˆ w † ( τ ) = 1 √ (cid:18) − (cid:19) , (30a)ˆ w ( τ ) = ˆ w † ( τ ) = 1 √ (cid:18) − (cid:19) . (30b)The operators do not commute and we find thatˆ w † ( τ ) ˆ w ( τ ) = ˆ w † ( τ ) ˆ w ( τ ) = (cid:18) − (cid:19) (31)are diagonal in the eigenbasis of ˆ R (0) meaning that theevolution (without the echo) does not yield entanglementat time τ for any c , since [ ˆ w † ( τ ) ˆ w ( τ ) , ˆ R (0)] = 0. Onthe other hand, this does not mean that there is no qubitdecoherence, since the off-diagonal elements of the qubitdensity matrix are proportional toTr (cid:104) ˆ w † ( τ ) ˆ w ( τ ) ˆ R (0) (cid:105) = c − c . (32)Hence, the qubit state remains pure only for an initialpure state of the environment, c = 1 or c = 0, with thepurity reaching its minimal possible value in the type ofevolutions described for a completely mixed environment, c = c = 1 / w † ( τ ) ˆ w † ( τ ) ˆ w ( τ ) ˆ w ( τ ) = (cid:18) − (cid:19) . (33) This operator is obviously not diagonal in the eigenbasisof the initial environment density matrix. Furthermore, (cid:104) ˆ w † ( τ ) ˆ w † ( τ ) ˆ w ( τ ) ˆ w ( τ ) , ˆ R (0) (cid:105) = ( c − c ) (cid:18) (cid:19) (34)and the condition for separability (12) is fulfilled only for c = c = , another words, only when the initial den-sity matrix of the environment is proportional to unity,ˆ R (0) ∼ I .When it comes to qubit decoherence, we always haveTr (cid:104) ˆ w † ( τ ) ˆ w † ( τ ) ˆ w ( τ ) ˆ w ( τ ) ˆ R (0) (cid:105) = 0 , (35)which means that the qubit at time 2 τ is always fullydecohered, regardless of the initial state of the environ-ment. In this extreme case, the spin echo can do nodamage in the best scenario, while for most states of theenvironments, the procedure strongly enhances decoher-ence. This should not be surprising in light of discussionfrom Sec. III B, as for such a small (two-dimensional) en-vironment the correlation function of any environmentaloperator has to be periodic.This example shows that the echo may lead to the in-crease of entanglement with respect to the entanglementpresent in the system at the end of the free-evolutionperiod in the echo procedure (since it can create suchentanglement). This is contrary to intuition, since it isnatural to try to extend the notion, that since a per-fect echo procedure diminishes all QEE (while diminish-ing all decoherence), an imperfect echo should lead tolesser entanglement while it leads to lesser decoherencein the echoed state. As we see here, there exist situ-ations when the echo not only increases entanglement,but also increases decoherence, and can be counterpro-ductive. Using the physical picture discussed for weakdephasing in Sec. III B (and taking it strictly speakingoutside of domain of its quantitative applicability, un-less we assume a Gaussian environment [18] for which | W (2 τ ) | = exp[ − χ (2 τ )]), we see that this can occur whenthe PSD of the environmental noise is periodic, but τ is such that it is the maximum of the filter | ˜ f ( ω ) | ineq. (21) that overlaps with the peaks of S ( ω ). C. Entangling evolution - pure environmentalstates
Let us study the special case of a pure initial state ofthe environment (we expect from the results of the pre-vious subsection that this situation will enhance the dif-ferences between the pre-pulse entanglement and echoedentanglement). Then the joint state of the system andthe environment is pure at any time, so it is pure attime τ (pre-pulse) and at echo time 2 τ . In this situation,entanglement at any time can be evaluated in a straight-forward manner using the von Neumann entropy of oneof the entangled subsystems, which is a good entangle-ment measure for pure states. The measure is definedas E ( | ψ ( t ) (cid:105) ) = − ρ ( t ) ln ρ ( t )) , (36)where | ψ ( t ) (cid:105) is the pure system-environment state so σ ( t ) = | ψ ( t ) (cid:105)(cid:104) ψ ( t ) | , ρ ( t ) = Tr E | ψ ( t ) (cid:105)(cid:104) ψ ( t ) | is the den-sity matrix of the qubit at time t (obtained by tracingout the environment), and the entanglement measure isnormalized to yield unity for maximally entangled states.The same result would be obtained when tracing out thequbit degrees of freedom instead of the environmentaldegrees of freedom, but the small dimensionality of thequbit makes this way much more convenient.Let us denote the pure initial state of the environmentas | R (cid:105) . Then qubit-environment state at time τ (pre-pulse) is given by | ψ ( τ ) (cid:105) = a | (cid:105) ⊗ ˆ w ( τ ) | R (cid:105) + b | (cid:105) ⊗ ˆ w ( τ ) | R (cid:105) (37)and the corresponding echoed state (at time 2 τ ) is | ψ (2 τ ) (cid:105) = a | (cid:105)⊗ ˆ w ( τ ) ˆ w ( τ ) | R (cid:105) + b | (cid:105)⊗ ˆ w ( τ ) ˆ w ( τ ) | R (cid:105) . (38)The qubit density matrices are then of the general form(5) with W ( τ ) = (cid:104) R | ˆ w † ( τ ) ˆ w ( τ ) | R (cid:105) pre-pulse, and W (2 τ ) = (cid:104) R | ˆ w † ( τ ) ˆ w † ( τ ) ˆ w ( τ ) ˆ w ( τ ) | R (cid:105) for the echoedstate. Hence, the absolute values of functions W ( τ ) and W (2 τ ) constitute the degrees of coherence retained in thequbit system at the time of application of the pulse andat the echo time, respectively.The entanglement measure of eq. (36) can be calcu-lated using eq. (5) which yields E ( | ψ ( t ) (cid:105) ) = − (cid:34) (cid:112) ∆( t )2 ln 1 + (cid:112) ∆( t )2 (39)+ 1 − (cid:112) ∆( t )2 ln 1 − (cid:112) ∆( t )2 (cid:35) , with ∆( t ) = 1 − | a | | b | + | a | | b | | W ( t ) | . Note that∆( t ) is an increasing function of the degree of coher-ence | W ( t ) | , while entanglement measured by E ( | ψ ( t ) (cid:105) )is a decreasing function of ∆( t ), so entanglement is a de-creasing function of coherence | W ( t ) | , which means (asexpected) that the higher the qubit coherence, the lowerthe QEE. Consequently, the situation described at thebeginning of Sec. IV A, when the pre-pulse state σ ( τ ) hasno QEE, but the echoed state σ (2 τ ) is entangled, for apure initial state of the environment translates to the pre-pulse qubit state being more coherent than the echoedqubit state, meaning that the echo can have an oppositeeffect on the qubit coherence than intended. This shouldbe kept in mind when dealing with rather small environ-ments that have a discrete spectrum, and which are closeto being in pure state (e.g. their temperature is very low,or, in case of spin environments, a large nonequilibriumpolarization of the environmental spins was previouslyestablished, see Ref. [12] for discussion of QEE in thiscase). τ τ τ τ τ τ E τ FIG. 1. Exemplary QEE evolution for a single qubit envi-ronment initially in a pure state pre-pulse (at time τ , blackline) and the corresponding echoed entanglement (at time 2 τ ,red line). Fig. (1) shows an exemplary evolution of the QEE,measured by the normalized von Neumann entropy ofeq. (36), for an environment restricted to a single qubitwhich is initially in a pure state. The evolution operators(in the subspace of the environment) are given byˆ w i ( t ) = e iω i t | ψ i (cid:105)(cid:104) ψ i | + e iω (cid:48) i t | ψ (cid:48) i (cid:105)(cid:104) ψ (cid:48) i | , (40)with i = 0 , ω = π/ (4 τ ), ω (cid:48) = − π/ (4 τ ), ω = π/τ , ω (cid:48) = 2 π/τ , and | ψ (cid:105) = 1 √ | R (cid:105) − i √ | R (cid:105) , (41) | ψ (cid:48) (cid:105) = 1 √ | R (cid:105) + i √ | R (cid:105) , (42) | ψ (cid:105) = (cid:112) − √ | R (cid:105) − (cid:112) √ | R (cid:105) , (43) | ψ (cid:48) (cid:105) = (cid:112) √ | R (cid:105) + (cid:112) − √ | R (cid:105) , (44)where | R (cid:105) is the state perpendicular to the initial en-vironmental state | R (cid:105) . Obviously, the evolution is pe-riodic and repeats itself every 4 τ , while at t = τ theevolution operators are equal to the operators introducedin Sec. IV B, for which a non-entangled state before thepulse leads to an entangled echoed qubit-environmentstate.The black line in Fig. (1) (denoted as τ ) shows theamount of entanglement between the qubit and the en-vironment as a function of time τ , when no echo is per-formed. The red line (denoted as 2 τ ), on the other hand,shows qubit-environment entanglement at time 2 τ in thesituation when a π pulse was applied to the qubit attime τ , again as a function of τ . Hence, the two curvesin Fig. (1) show pre-pulse entanglement and the corre-sponding echoed entanglement as a function of the sameparameter τ . The evolution of echoed entanglement ismuch more involved, and the interplay of the two curvesshows that apart from the previously predicted τ = τ case (when no pre-pulse entanglement is observed, butthere is echoed entanglement), there are many situationswhen applying the pulse enhances qubit-environment en-tanglement at a later time. Note, that for a pure initialstate of the environment, there is a strict correspondencebetween QEE and qubit coherence, meaning that everytime entanglement is enhanced by the echo, the coher-ence of the qubit is damped, and the effect of the echo iscontrary to its purpose. V. ECHO INDUCED ENTANGLEMENT IS NOTPOSSIBLE FOR PRINCIPALLYNONENTANGLING EVOLUTIONS
Although the examples discussed above show that thespin echo procedure can lead to the appearance of QEEat echo time when the qubit-environment state was sepa-rable before the application of the pulse to the qubit, thisoccurs in rather special situations. Let us show now thatit is only possible at isolated points of time, and thereare no finite time intervals t ∈ [ τ , τ ] for which the pre-pulse state ρ ( t ) is separable, while the echoed state ρ (2 t )is entangled. Since this is the case, we can extend thetime interval to encompass the whole pre-pulse evolution t ∈ [0 , ∞ ], which yields the result that the echo procedurecannot be used to modify a non-entangling evolution intoan entangling one.The argument is as follows. Separable evolutions,which obviously must fulfill the criterion (27), can bedivided into two categories: One encompasses all typesof evolutions for which the environment does not evolve,Tr Q σ ( t ) = R ( t ) = R ( t ) = R (0) . (45)Here the trace is taken over the qubit degrees of freedom,so what is left is the evolution only in the subspace of theenvironment. Note that such evolutions also lead to puredephasing of the qubit, it is only that this process cannotbe witnessed by any measurements on the environment.The other encompasses all types of evolutions which doinvolve evolution of the environment,Tr Q σ ( t ) = R ( t ) = R ( t ) = R ( t ) (cid:54) = R (0) . (46)The density matrix of the environment conditional onthe qubit being in state | (cid:105) is defined as R ( t ) =ˆ w ( t ) R (0) ˆ w † in analogy to R ( t ).An evolution of the first category can never leadto echoed entanglement, since if ˆ w ( t ) R (0) ˆ w † =ˆ w ( t ) R (0) ˆ w † = R (0), we have R (0) = ˆ w ( t ) R (0) ˆ w † = ˆ w ( t ) ˆ w ( t ) R (0) ˆ w † ˆ w † ,R (0) = ˆ w ( t ) R (0) ˆ w † = ˆ w ( t ) ˆ w ( t ) R (0) ˆ w † ˆ w † , so the separability criterion for the echoed state (29) isobviously fulfilled at all times without any additional as-sumption. Even isolated instances of time, which would lead to entanglement in the echoed state for a separablepre-pulse state are impossible.In the other situation, we know that such instances oftime exist, due to the examples above. To check if thereexist time intervals in the pre-pulse evolution for whichthe echo generates entanglement, let us study a time in-terval t ∈ [ τ , τ ] such that for any time t within thisinterval we have R ( t ) = R ( t ) (which guarantees pre-pulse separability). For there to be entanglement in theechoed state we need ˆ w ( t ) R ( t ) ˆ w † (cid:54) = ˆ w ( t ) R ( t ) ˆ w † ,but because of the pre-pulse separability we can ex-change the conditional environmental states and getˆ w ( t ) R ( t ) ˆ w † (cid:54) = ˆ w ( t ) R ( t ) ˆ w † , or equivalently R (2 t ) (cid:54) = R (2 t ) . (47)Hence, for there to exist time-intervals for which the echoprotocol leads to entanglement generation, the qubit-environment evolution without the echo procedure wouldhave to fulfill a very specific requirement. Namely therewould have to exist time intervals in which the evolutionis separable, followed by time intervals in which QEE isgenerated. In other words, sudden birth of entanglement[36, 37] would have to be possible in the system.The results of Ref. [11] show that for pure dephasingevolutions such as studied here, separability is equiva-lent to the lack of quantum discord [38–40] with respectto the environment. This means that the set of separa-ble states has zero volume, and therefore sudden deathof entanglement (which is a consequence of the geometryof separable states [41]) will not occur. Hence, also thetransformation of separable evolutions to entangling onesvia the quantum echo, when the evolution remains sep-arable for finite or infinite time-intervals is not possible,and such occurrences are limited to isolated instances intime. VI. ECHO SIGNAL ASQUBIT-ENTANGLEMENT ENVIRONMENTWITNESS
In the previous sections we have given examples show-ing that in general there is no correlation between theeffectivenes of the echo protocol (measured by its capa-bility to lead to coherence revival at time 2 τ ) and thegeneration of QEE. While this conclusion stands, as it issimply a manifestation of the fact that for an environ-ment in a mixed state the correlation between amount ofQEE and the strength of dephasing is rather weak, letus finish here with a more “positive” result for a specificcase.Let us use the separability condition for the pre-pulseevolution of the qubit-environment system lasting fortime τ in the form given by eq. (27). Let us then focus ona qubit that couples to the environment in “biased” way[29, 30], so that ˆ V = 0 and only ˆ V = λ ˆ V is nontrivial.This means that ˆ R ( τ ) = ˆ R (0), and QEE is generatedif and only if ˆ R ( τ ) (cid:54) = ˆ R (0). A necessary condition forthe latter is [ ˆ H , ˆ R (0)] (cid:54) = 0. It is also a sufficient condi-tion for QEE to appear at all τ but a subset of isolatedpoints. This follows from an argument about impossibil-ity of sudden death or birth of QEE from the previousSection: for [ ˆ H , ˆ R (0)] (cid:54) = 0, QEE appears at the begin-ning of the evolution, and it cannot then vanish and stayzero for a finite stretch of time.We focus now on system in which the state of the en-vironment is stationary, [ ˆ R (0) , ˆ H E ] = 0. The “if and onlyif” (with exception of isolated points in time) conditionfor nonzero QEE is then [ ˆ V , ˆ R (0)] (cid:54) = 0. A simple cal-culation of the commutator in expression for imaginarycontribution to dephasing, eq. (17), shows that the func-tion Φ(2 τ ) vanishes if the commutator of ˆ V and ˆ R (0) iszero. This leads to the following statement: if the envi-ronment is in a stationary state, and the qubit’s couplingis biased, the appearance of nonzero Φ( t ) contribution toecho signal means that qubit and environment were en-tangled during the evolution (with possible exception ofisolated points in time). This means that if the qubit isinitialized with its Bloch vector in some direction (say x ), then at echo time 2 τ the length of this vector is notonly going to be diminished due to nonzero χ (2 τ ), butdue to nonzero Φ( t ) the direction of the final vector isgoing to be rotated with respect to the original one. Un-der all the listed conditions, the appearance of such anenvironment-induced rotation of the echoed state of thequbit is equivalent to entangling nature of the evolutionof the composite qubit-environment system. VII. CONCLUSION
We have studied the spin echo performed on a qubitthat interacts with an environment due to a type ofHamiltonian which leads to pure dephasing of the qubit.Our intent was to quantify the relation between the per-formance of the echo procedure to reduce decoherence,and the entanglement which can be generated between the qubit and its environment. Quite surprisingly, wehave found that the effectiveness of the echo and entange-ment generation are two distinct issues. The perfect echofor which full coherence is restored can occur both in caseof entangling and separable evolutions.We have further analyzed the situation when the echois not perfect, and found that it is possible for a qubit-environment state to be separable prior to the applicationof the local operation on the qubit (the π pulse) while thefinal echoed state is entangled. It turns out that althoughsuch a possibility does exist, it is limited to isolated in-stances of time. The important consequence here is thatalthough the spin echo can result in the generation ofentanglement from a point of time when there is no pre-pulse entanglement, this is a special case in an evolutionwhich leads to entanglement generation on average. Itcannot result in the change of the nature of evolutionfrom nonentangling to entangling, so it cannot lead to arobust creation of quantum correlations.Finally, we have shown that there is at least one casein which one can use the echo signal as a witness of theentangling charater of the evolution of a qubit and itsenvironment. When the environment is in a stationarystate, and only one of two levels of the qubit is coupled tothe environment (as it happens for qubits for which onlyone of their levels has a finite dipole moment, e.g. exci-tonic qubits [23–26] or spin qubits based on m = 0 and m = 1 levels of spin S = 1 system, such as nitrogen-vacancy center [27, 28]). The appearance of phase shiftof coherence [29, 30] proves then the entangling natureof the evolution. VIII. ACKNOWLEDGMENTS (cid:32)L. C. would like to thank Piotrek Sza´nkowski for stim-ulating discussions. This work was funded from theReseach Projects No. UMO-2012/07/B/ST3/03616 andUMO-2015/19/B/ST3/03152 financed by the NationalScience Centre of Poland (NCN). [1] M. Schlosshauer,
Decoherence and the Quantum-to-Classical Transition (Springer, Berlin/Heidelberg, 2007).[2] K. Hornberger, “Introduction to decoherence theory,”in
Entanglement and Decoherence , Lecture Notes inPhysics, Vol. 768, edited by Andreas Buchleitner, CarlosViviescas, and Markus Tiersch (Springer Berlin Heidel-berg, 2009) pp. 221–276.[3] Wojciech Hubert ˙Zurek, “Decoherence, einselection, andthe quantum origins of the classical,” Rev. Mod. Phys. , 715 (2003).[4] O. K¨ubler and H. D. Zeh, “Dynamics of quantum corre-lations,” Ann. Phys. , 405 (1973).[5] Jens Eisert and Martin B. Plenio, “Quantum and clas-sical correlations in quantum brownian motion,” Phys.Rev. Lett. , 137902 (2002). [6] Stefanie Hilt and Eric Lutz, “System-bath entanglementin quantum thermodynamics,” Phys. Rev. A , 010101(2009).[7] J. Maziero, T. Werlang, F. F. Fanchini, L. C. C´eleri, andR. M. Serra, “System-reservoir dynamics of quantum andclassical correlations,” Phys. Rev. A , 022116 (2010).[8] A. Pernice and Walter T. Strunz, “Decoherence and thenature of system-environment correlations,” Phys. Rev.A , 062121 (2011).[9] Katarzyna Roszak and (cid:32)Lukasz Cywi´nski, “Charac-terization and measurement of qubit-environment-entanglement generation during pure dephasing,” Phys.Rev. A , 032310 (2015).[10] Katarzyna Roszak, “Criteria for system-environment en-tanglement generation for systems of any size in pure- dephasing evolutions,” Phys. Rev. A , 052344 (2018).[11] Katarzyna Roszak and (cid:32)Lukasz Cywi´nski, “Equivalence ofqubit-environment entanglement and discord generationvia pure dephasing interactions and the resulting conse-quences,” Phys. Rev. A , 012306 (2018).[12] Katarzyna Roszak, Damian Kwiatkowski, and (cid:32)LukaszCywi´nski, “How to detect qubit-environment entangle-ment generated during qubit dephasing,” Phys. Rev. A , 022318 (2019).[13] Piotr Sza´nkowski and (cid:32)Lukasz Cywi´nski, “Noise repre-sentations of open system dynamics,” arXiv:2003.09688(2020).[14] E. L. Hahn, Phys. Rev. , 580 (1950).[15] A. Abragam, The Principles of Nuclear Magnetism (Ox-ford University Press, New York, 1983).[16] L. M. K. Vandersypen and I. L. Chuang, “Nmr techniquesfor quantum control and computation,” Rev. Mod. Phys. , 1037–1069 (2005).[17] Rogerio de Sousa, “Electron spin as a spectrometer ofnuclear-spin noise and other fluctuations,” Top. Appl.Phys. , 183 (2009).[18] P. Sza´nkowski, G. Ramon, J. Krzywda, D. Kwiatkowski,and (cid:32)L. Cywi´nski, “Environmental noise spectroscopywith qubits subjected to dynamical decoupling,” J.Phys.:Condens. Matter , 333001 (2017).[19] Hong-Bin Chen, Clemens Gneiting, Ping-Yuan Lo, Yueh-Nan Chen, and Franco Nori, “Simulating open quantumsystems with hamiltonian ensembles and the nonclassi-cality of the dynamics,” Phys. Rev. Lett. , 030403(2018).[20] Hong-Bin Chen, Ping-Yuan Lo, Clemens Gneiting, Joon-woo Bae, Yueh-Nan Chen, and Franco Nori, “Quan-tifying the nonclassicality of pure dephasing,” NatureComm. , 3794 (2019).[21] (cid:32)Lukasz Cywi´nski, “Dephasing of electron spin qubits dueto their interaction with nuclei in quantum dots,” ActaPhys. Pol. A , 576 (2011).[22] E. A. Chekhovich, M. N. Makhonin, A. I. Tartakovskii,A. Yacoby, H. Bluhm, K. C. Nowack, and L. M. K. Van-dersypen, “Nuclear spin effects in semiconductor quan-tum dots,” Nature Materials , 494 (2013).[23] P. Borri, W. Langbein, S. Schneider, U. Woggon, R. L.Sellin, D. Ouyang, and D. Bimberg, “Ultralong dephas-ing time in InGaAs quantum dots,” Phys. Rev. Lett. ,157401–1–4 (2001).[24] A. Vagov, V. M. Axt, and T. Kuhn, “Impact of pure de-phasing on the nonlinear optical response of single quan-tum dots and dot ensembles,” Phys. Rev. B , 115338(2003).[25] A. Vagov, V. M. Axt, T. Kuhn, W. Langbein, P. Borri,and U. Woggon, “Nonmonotonous temperature depen-dence of the initial decoherence in quantum dots,” Phys.Rev. B , 201305(R)–1–4 (2004).[26] Katarzyna Roszak and Pawe(cid:32)l Machnikowski, “Completedisentanglement by partial pure dephasing,” Phys. Rev. A , 022313 (2006).[27] Nan Zhao, Sai-Wah Ho, and Ren-Bao Liu, “Decoherenceand dynamical decoupling control of nitrogen vacancycenter electron spins in nuclear spin baths,” Phys. Rev.B , 115303 (2012).[28] Damian Kwiatkowski and (cid:32)Lukasz Cywi´nski, “Decoher-ence of two entangled spin qubits coupled to an inter-acting sparse nuclear spin bath: Application to nitrogenvacancy centers,” Phys. Rev. B , 155202 (2018).[29] Gerardo A. Paz-Silva, Leigh M. Norris, and LorenzaViola, “Multiqubit spectroscopy of gaussian quantumnoise,” Phys. Rev. A , 022121 (2017).[30] D. Kwiatkowski, P. Sza´nkowski, and (cid:32)L. Cywi´nski, “Influ-ence of nuclear spin polarization on the spin-echo signalof an nv-center qubit,” Phys. Rev. B , 155412 (2020).[31] John W. Negele and Henri Orland, Quantum Many-Particle Systems (Addison-Wesley, Redwood City, CA,1988).[32] H. Bruus and K. Flensberg,