The ultimate precision limits for noisy frequency estimation
Abstract
Quantum metrology protocols allow to surpass precision limits typical to classical statistics. However, in recent years, no-go theorems have been formulated, which state that typical forms of uncorrelated noise can constrain the quantum enhancement to a constant factor, and thus bound the error to the standard asymptotic scaling. In particular, that is the case of time-homogeneous (Lindbladian) dephasing and, more generally, all semigroup dynamics that include phase covariant terms, which commute with the system Hamiltonian. We show that the standard scaling can be surpassed when the dynamics is no longer ruled by a semigroup and becomes time-inhomogeneous. In this case, the ultimate precision is determined by the system short-time behaviour, which when exhibiting the natural Zeno regime leads to a non-standard asymptotic resolution. In particular, we demonstrate that the relevant noise feature dictating the precision is the violation of the semigroup property at short timescales, while non-Markovianity does not play any specific role.
Introduction.—Parameter estimation, ranging from the precise determination of atomic transition frequencies to external magnetic field strengths, is a central task in modern physics Caves (1981); Wineland et al. (1992); Giovannetti et al. (2004); Leibfried et al. (2004); Roos et al. (2006); Pezzé and Smerzi (2009); Jones et al. (2009). Quantum probes made up of entangled particles can attain the so-called Heisenberg limit (HL), where the estimation mean squared error (MSE) scales as , as compared with the standard quantum limit (SQL) of classical statistics Demkowicz-Dobrzański et al. (2015); *Toth2014.
Heisenberg resolution relies on the unitarity of the time evolution. In realistic situations, however, quantum probes decohere as a result of the unavoidable interaction with the surrounding environment Breuer and Petruccione (2002); *Rivas2012. Such interactions can have a dramatic effect on estimation precision—even infinitesimally small uncorrelated dephasing noise, modelled as a semigroup (time-homogeneous-Lindbladian) evolution Gorini et al. (1976); *Lindblad1976, forces the MSE to eventually follow the SQL Huelga et al. (1997). This result was proven to be an instance of the quantum Cramér-Rao bound (QCRB) Braunstein and Caves (1994) for generic Lindbladian dephasing and thus holds even when using optimized entangled states and measurements Escher et al. (2011); Demkowicz-Dobrzański et al. (2012); Kołodyński and Demkowicz-Dobrzański (2013); *Kolodynski2014; Knysh et al. (2014). The question then arises of what is the ultimate precision limit when the noisy time evolution is not governed by a dephasing dynamical semigroup Escher et al. (2011); Demkowicz-Dobrzański et al. (2012); Kołodyński and Demkowicz-Dobrzański (2013); *Kolodynski2014; Knysh et al. (2014); Ulam-Orgikh and Kitagawa (2001); Shaji and Caves (2007); Fujiwara and Imai (2008); Huver et al. (2008); Dorner et al. (2009); *Knysh2011; *Kacprowicz2010; *Kolodynski2010; Brivio et al. (2010); *Genoni2011; Chwedeńczuk et al. (2012); Matsuzaki et al. (2011); *Chin2012; Chaves et al. (2013); *Brask2015; Berrada (2013). The SQL has been shown to be surpassable in the presence of time-inhomogeneous (non-semigroup) dephasing noise Matsuzaki et al. (2011); *Chin2012, noise with a particular geometry Chaves et al. (2013); *Brask2015 and correlated time-homogeneous dephasing Dorner (2012); *Jeske2014, or when the noise geometry allows for error correction techniques Kessler et al. (2014); *Duer2014; *Arrad2014; *Lu2015; *Plenio2015.
Here, we derive the ultimate lower bounds on the MSE for the noisy frequency estimation scenario depicted in Fig. 1 where probe systems are independently affected by the decoherence. In particular, we focus on uncorrelated phase-covariant noise, that is, noise-types commuting with the parameter-encoding Hamiltonian, as these underpin the asymptotic SQL-like precision in the semigroup case (Chaves et al., 2013; Knysh et al., 2014). Yet, most importantly, we allow for any form of time-inhomogeneity and non-Markovian features in the noise. Our results show that, when moving away from the semigroup regime, entanglement generally improves the precision beyond the constant-factor enhancement, so that the SQL is truly overcome. As a special case, we confirm the conjecture made in Matsuzaki et al. (2011); *Chin2012, where by considering a Ramsey interferometry scheme and non-semigroup dephasing dynamics, a error scaling was shown to be achievable. This was argued to be a consequence of the Zeno regime at short time scales. The generality of this scaling has been recently verified for pure dephasing noise Macieszczak (2015). We formally prove the emergence of non-SQL scaling for any non-semigroup phase covariant noise. We demonstrate that it is solely the short-time expansion of the effective noise parameters that determines the ultimate attainable precision. In particular, any memory (non-Markovian) effects, which may be displayed by the system at later times, are irrelevant for the asymptotic limit.
Noisy frequency estimation.—In a typical frequency estimation setting, a parameter is unitarily encoded on sensing particles (probes), specifically qubits, over the interrogation time during which the probes are also independently disturbed by the decoherence Huelga et al. (1997); Escher et al. (2011). As depicted in Fig. 1, we generalise such a setup to allow for an arbitrary number of ancillary particles, that can be initially entangled with the probes and measured at the end of the protocol. Hence, the combined final state of the system reads:
(1) |
with being the initial state, and a completely positive and trace preserving (CPTP) linear map Bengtsson and Życzkowski (2006) representing the identical, but independent, evolution of each probe (see App. A). We assume full control and noise-free evolution for the ancillae, so that to allow for single-step error-correction protocols Kessler et al. (2014); *Duer2014; *Arrad2014; *Lu2015. The dependent parameter estimate, , relies on sufficiently large statistical data after performing repetitions, provided the total experimental time .
We quantify the performance of the estimation protocol by the MSE, —describing the average deviation of the estimate from the true value. Crucially, requiring unbiasedness and consistency for the estimate, the QCRB directly provides us with the ultimate lower bound on the MSE that is optimised over all potential measurement strategies Braunstein and Caves (1994). Hence, possessing also the freedom to adjust the single-shot duration time , the ultimate attainable precision can be written as
(2) |
where is the quantum Fisher Information (QFI) evaluated with respect to (w.r.t.) the estimated parameter encoded in the final state (1). Importantly, the that minimises the right-hand side in Eq. (2), i.e., the optimal single-shot duration, generally depends on the system size and we thus denote it as .
Phase covariant dynamics.—The frequency parameter is unitarily encoded within the phase, , accumulated during the free evolution of qubit probe, which in the Bloch ball picture corresponds to a rotation around a known direction— in Fig. 2. We consider systems exhibiting uncorrelated forms of noise that commute with such rotations, which formally correspond to the so-called phase covariant qubit maps Holevo (1993); *Holevo1996. Such noise-types are known to most severely limit the attainable precision in case of semigroup dynamics, for which they constrain the quantum enhancement to a constant factor above the SQL Escher et al. (2011); Demkowicz-Dobrzański et al. (2012); Kołodyński and Demkowicz-Dobrzański (2013); Kołodyński (2014); Knysh et al. (2014). Although such negative conclusion cannot be drawn for other less severe but still semigroup noises: purely transversal Chaves et al. (2013) and correlated Dorner (2012); *Jeske2014; the phase covariant noise if present, no matter how weak, must always asymptotically dominate and limit the ultimate quantum improvement to a constant factor Chaves et al. (2013); Knysh et al. (2014). Thus, in what follows, we focus on the frequency estimation scenario of Fig. 1 in the presence of general independent, identical and phase covariant (IIC) noise, where each single probe at any instance of time may be described by the action of a map
(3) |
with and being its unitary encoding and -independent dissipative parts respectively. We fix
(4) |
that acts on a four-component Bloch vector. As depicted in Fig. 2, the qubit evolution amounts then to: a rotation around the axis by an angle containing the parameter encoding ( plane by a factor direction by a factor case corresponds to an additional reflection with respect to the plane), and a displacement in the z direction by 4) fulfils the CPTP condition as long as B). (see App. and . The map ( ( , a contraction in the ), a contraction in the
It is important to stress that any single qubit phase covariant dynamics can always be put on physical grounds by considering a corresponding time-local master equation of the form
(5) | |||||
The proof as well as explicit relations between , , , and , , , are given in App. B. Phase covariant dynamics therefore describes any physical evolution that may arise from combinations of time-varying absorption, emission and dephasing processes, as well as Lamb shift corrections to free Hamiltonian Breuer and Petruccione (2002); *Rivas2012. Moreover, Eq. (5) generally allows for the quantitative characterisation of non-Markovian effects Breuer et al. (2009); *Breuer2015; Rivas et al. (2010); *Rivas2014. In the special case of positive constant rates (time homogeneity), Eq. (5) provides the generator of any phase covariant quantum dynamical semigroup Vacchini (2010).
Bounding the ultimate precision.—Having fully characterized the class of qubit IIC dynamics, we can now state the main result of the paper. Given qubit probes and ancillae evolving according to Eq. (1), with the single qubit dynamics given by a phase covariant map as in Eq. (4), and provided that at all times () , the MSE in estimating the frequency is asymptotically determined by the short-time expansion of the noise parameters:
(6) |
and it satisfies the following inequality
(7) |
where and
(8) |
Crucially, as (see below) the bound in Eq. (7) is always attainable up to a constant factor, the asymptotic precision is fully determined by the short-time expansion of the radius in the plane perpendicular to the rotation axis, which fixes the asymptotic scaling to . For semigroup dynamics () one accordingly recovers the SQL-like limit, while with increasing one finds a progressively more favourable scaling that tends to HL for unrealistic . Besides the assumption of IIC noise (3), the only condition assuring the bound (7) to be valid is . In fact, if at some finite then by CPTP-property also and . In other words, a “full revival” of the Bloch vector length occurs and the only effect of the interaction with the environment is a rotation about the -axis by some angle . Not surprisingly, the best estimation strategy is then to measure the frequency at such pseudo-noiseless moment, at which the HL is attainable. However, note that such a behaviour is quite unlikely when dealing with open systems subject to realistic sources of noise Breuer and Petruccione (2002).
The sketch of the proof is provided below, while a complete version is given in App. C. Firstly, we fix the evolution time (and omit it for simplicity), to use the finite- channel extension (CE) method Fujiwara and Imai (2008); Kołodyński and Demkowicz-Dobrzański (2013); *Kolodynski2014, which provides an upper-bound on the QFI that is already optimised over all initial states:
(9) |
The minimisation above is performed over Kraus representations of the channel , describing the dynamics of a single probe. denotes the operator norm, whereas and with 2013); *Kolodynski2014 automatically provide the correct ansatz, with which one may then proceed analytically. . Identifying the optimal Kraus representation is usually non-trivial, however, the numerical semidefinite programming (SDP) methods introduced in Kołodyński and Demkowicz-Dobrzański (
In App. C, we explicitly deal with the general case of phase covariant qubit map, where we additionally prove the convexity of the bound (9) w.r.t. mixing of quantum channels. This allows us to analytically apply Eq. (9) to any map of the form (4), after adequately decomposing it into an optimal mixture of unital () and amplitude damping channels (
(10) |
Thus, substituting into Eq. (2) we obtain the precision bound
(11) |
which in the case of semigroup dynamics coincides with the asymptotically tight limit derived in Knysh et al. (2014).
First, beating the SQL-like scaling necessarily requires . Assume on the contrary that the optimal evolution time attains some as . Then, inspecting Eq. (10), one sees that the “no full-revival” assumption , along with the CPTP constraints, implies and hence Eq. (11) directly restricts the precision to asymptotically follow . As a consequence, we can focus on the short-time regime and expand , as in Eq. (6) to get
(12) |
Plugging the above expansion into Eq. (11), we find that its minimum is reached for
(13) |
which yields the bound (7) with , so that correctly Eqs. (8) and (12) coincide for .
Attaining the ultimate precision.—As Eq. (9) provides us only with an upper limit on the QFI, we still must investigate the tightness of bound (7). Yet, note that also the QCRB (2) itself is guaranteed to be saturable only in the limit of infinite independent experimental repetitions . This issue is particularly important in the noiseless case, when the minimisation of the MSE (2) over yields , indicating that a single experimental shot consuming all time-resources should be performed Escher et al. (2011). The QCRB is then not saturable, what can be demonstrated by means of rigorous Bayesian approach Berry and Wiseman (2000); *Jarzyna2015. Fortunately, in the presence of IIC noise the optimal single-shot duration, , is independent of and decays as with , see Eq. (13), so that always diverges as . Thus, only due to noise we may assure that for any there exists large enough for which the QCRB is saturable.
We now show that the scaling exponent in Eq. (7) is always correct and it is only the constant that in some cases may be underestimated. Consider a GHZ state may be analytically derived: . Thanks to its simple structure, the expression for its QFI w.r.t. the estimated
(14) |
with . Focusing again for simplicity on unital maps with (see App. E for the general case), expanding the above formula for short times and using optimal that minimises asymptotically the QCRB (2) for the GHZ-based scenario, we arrive at
(15) |
For the semigroup case () the asymptotic coefficient (15) differs by a factor from of Eq. (7)—a known fact for the pure dephasing model Huelga et al. (1997); Escher et al. (2011) which may be remedied by replacing GHZ with spin-squeezed states Ulam-Orgikh and Kitagawa (2001)—yet the discrepancy decreases with increasing . Crucially, Eq. (15) proves that the scaling of the MSE predicted by Eq. (7) is indeed always achievable when , however, such claim applies to all phase covariant maps, see App. E.
Role of non-Markovianity and Zeno regime.—We have shown that by going beyond the semigroup regime one can overcome the SQL for a relevant class of open system dynamics. A natural question is whether non-Markovian features are of some relevance. Since non-Markovianity is typically associated with backflow of information to the system of interest Breuer et al. (2009); *Breuer2015; Rivas et al. (2010); *Rivas2014, one may think that such recovered information (also about the estimated parameter) could be advantageous for metrological purposes, possibly leading to improved scalings of precision. Our results clearly indicate that this is not the case. As any measurement strategy outside the short-time regime will be asymptotically bounded by a scaling, to beat the SQL one must perform measurements on shorter and shorter timescales as , whatever the subsequent memory effects are. The attainable asymptotic precision is then fully dictated by the time-inhomogeneous, i.e., non-semigroup, nature of the dynamics.
The characterisation of non-semigroup dynamics is a complex task, which calls for a detailed knowledge of the environmental properties, as well as the interaction mechanism Breuer and Petruccione (2002). Yet, a general property of any evolution derived exactly from the global (system+environment) unitary dynamics is the quadratic decay of the survival probability at short timescales—the emergence of the so-called quantum Zeno regime Facchi and Pascazio (2008); *Pascazio2014. In App. D, we explicitly show that for any phase covariant such quadratic decay implies that and Eq. (7) then reduces to . Thus, we can conclude that the ultimate precision scaling—provably attainable—is a general feature of any reduced dynamics exhibiting the Zeno regime and phase covariance. In particular, if we restrict to the specific case of pure dephasing, we provide further confirmation of the conjecture made in Chin et al. (2012) and also recently proved in Macieszczak (2015).
Shabani-Lidar post-Markovian noise model.—To demonstrate the applicability of our methods for general phase covariant dynamics, we consider the post-Markovian model of Shabani and Lidar Shabani and Lidar (2005) (SL), that has been widely used to study non-semigroup evolutions and their non-Markovian properties Maniscalco and Petruccione (2006); *Maniscalco2007; *Mazzola2010. The SL master equation (5) contains all the emission, excitation and dephasing contributions, yet it is fully described by three parameters: –dissipation constant, –the effective memory rate, and –the mean number of excitations in the reservoir. In App. F, we show the short-time expansions (6) of its noise parameters (, and ) to be quadratic in (
Conclusions.—We have derived a novel limit on the attainable precision in frequency estimation, which holds for all forms of phase covariant uncorrelated noise. Our results show that, despite the noiseless HL not being within reach, by exploiting the non-semigroup, time-inhomogeneous system dynamics arising at short-times in the Zeno regime, the asymptotic SQL-like scaling of precision can be beaten. Any measurement strategies performed on longer timescales are always ultimately limited by the SQL, irrespectively of the possible non-Markovian effects exhibited by the evolution. We leave it as an open question whether the asymptotic precision can be further improved by means of general active-ancilla assisted schemes Demkowicz-Dobrzański and Maccone (2014), where the interplay between the multi-step unitary operations and the memory effects in the probes evolution has to be carefully treated.
Acknowledgements.
We acknowledge enlightening discussions with Bogna Bylicka. This work has been supported by Spanish Ministry National Plan FOQUS, Generalitat de Catalunya (SGR875), EU MSCA Individual Fellowship Q-METAPP, and EU FP7 IP project SIQS co-financed by the Polish Ministry of Science and Higher Education. This work has also received funding from the European Union’s Horizon 2020 research and innovation programme under the QUCHIP project GA no. 641039References
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Appendix A Dynamical maps – matrix and time-local master equation representations
In this section, we briefly recall the essential features of dynamical maps we use throughout our work and fix the notation. In particular, we focus on the matrix representation of the completely positive trace preserving (CPTP) linear maps, which is usually exploited to describe the dynamics of open quantum systems—see, e.g., Bengtsson and Życzkowski (2006); Ruskai et al. (2002); Andersson et al. (2007); Asorey et al. (2009); Chruściński et al. (2010); Smirne and Vacchini (2010).
a.1 Matrix representation of completely positive maps
Given a finite dimensional Hilbert space, , the set of linear operators on also forms a Hilbert space equipped with a Hilbert-Schmidt scalar product:
(16) |
where . Moreover, given the set of linear maps acting on , the Hilbert-Schmidt scalar product naturally induces a one-to-one correspondence between and the set of matrices. Explicitly, for any basis in that is orthonormal with respect to such scalar product, i.e., , one has
(17) |
for any and any linear map . In this way, any map is univocally associated with a matrix with elements . It is easy to see that the composition of two maps, , corresponds to the matrix product , and thus the inverse of the map is represented by the inverse matrix . Considering a basis such that and, for all , are orthonormal traceless self-adjoint operators, the map is trace-preserving if and only if its matrix representation can be written as
(18) |
where is a row vector made of 0s, is a column vector, and is a
In the case of qubit () maps, their matrix form has a simple geometrical interpretation, relying on the Bloch-ball representation of the qubit states. Given the basis on , where the s are the usual Puali matrices, any statistical operator on can be decomposed as
(19) |
where is the 3-dimensional real vector with elements and such that , while is a vector of Pauli matrices. Such decomposition defines the well-known one-to-one correspondence between the set of the statistical operators on and the 3 dimensional closed real ball of radius one centered at the origin, i.e., the Bloch ball. Moreover, any linear map with matrix representation as in Eq. (18) just yields
(20) |
In particular, the action of the map corresponds to an affine transformation of the Bloch ball, , where describes the translations, while describes: rotations, contractions and reflections about the three orthogonal axis (as can be seen after performing the singular value decomposition Ruskai et al. (2002)). In addition, the matrix representation allows for a clear geometrical characterisation of the complete positivity of qubit maps Ruskai et al. (2002). This is done by studying the positivity of the Choi matrix associated with the map Bengtsson and Życzkowski (2006). For any orthonormal basis in one can define the Choi matrix as
(21) |
with , so that the CP of is equivalent to the positivity of the matrix .
a.2 Time-local master equations of open quantum systems
Any dynamics of an open quantum systems may be generally described by a one-parameter family of CPTP maps , where the instance just corresponds to the initial time of the evolution. Thus the initial map must read
(22) |
while the system state at any later times is described by
(23) |
As discussed, can then be specified for any fixed time with help of its matrix representation in Eq. (18). Yet, we also consider the time-local master equation (TLME) satisfied by the state at any :
(24) |
Given a one-parameter family defining the dynamics, its corresponding TLME can be formally defined as Andersson et al. (2007); Chruściński et al. (2010); Smirne and Vacchini (2010):
(25) |
It is then clear that the matrix representation of the dynamical maps defined by Eq. (17) can be further exploited to get the matrix associated with the time-local generator reading:
(26) |
Applying Eq. (17) to , one may thus get an explicit form of the TLME, which for any trace- and hermiticity-preserving dynamics can be written as Gorini et al. (1976):
where is the Hamiltonian contribution and the are the (linear independent) Lindblad operators. The rates can be in general time-dependent and, importantly, one may deal with a well-defined CPTP evolution also in the presence of rates taking on negative values. In the next paragraphs, we will explicitly discuss the connection between the master equation and non-Markovianity in quantum dynamics. Customarily, one assumes that the time derivative of the dynamical map considered above always exists and is continuous, or equivalently that all the matrix elements of are functions. However, let us mention that there exist interesting dynamics exhibiting time instants at which the inverse of the dynamical map cannot be defined, typically due to some of the rates being divergent ^{1}^{1}1Owing to adequate constraints, the TLME may still be defined at such instances despite being then non-invertible Andersson et al. (2007).. Our analysis will also cover these dynamics.
Finally, let us clarify that when considering -qubit systems ( with ), we denote by a general linear operator in , while by an -fold tensor product of a given single-qubit operator , i.e., ( times). An analogous notation is used when describing linear maps from . In particular, given a unitary operator in , we denote by the unitary map (operator in ) defined as
(28) |
Lastly, let us note that given two linear maps and we denote their commutator and anti-commutator by and respectively.
a.3 Quantum Markovianity
Here, we briefly recall the distinction between Markovian and non-Markovian dynamics for open quantum systems, in particular w.r.t. the TLME in Eq. (A.2) and the notion of time-homogeneity; for a more detailed treatment the reader is referred to the recent reviews in Breuer et al. (2009); Rivas et al. (2010).
From a physical point of view, the dynamics of an open quantum system is Markovian if the memory effects due to its interaction with the environment can be neglected, typically due to a definite separation in the evolution time-scales of, respectively, the open system and the environment Breuer and Petruccione (2002). More precisely, quantum Markovianity can be formulated in terms of the divisibility properties of the dynamical maps . One defines the dynamics to be divisible if
(29) |
for any . Note that any dynamics such that exists at every time is divisible, as can be seen by simply setting , but is not in general a CP map. The linear maps are usually referred to as the propagators of the dynamics and they can be expressed in terms of the TLME in Eq. (A.2) as Rivas and Huelga (2012):
(30) |
where one has the identification . If the propagators depend only on the difference between their time arguments, i.e., for any , the dynamics is said to be time-homogeneous. One can easily see that this precisely corresponds to the case in which the TLME has constant coefficients. Furthermore, in this case, the dynamical maps satisfy
(31) |
for any , which is the well-known semigroup composition law. The most general form of the (bounded) generator of a semigroup of CPTP maps was characterized by Gorini, Kossakowski, Sudarshan and Lindblad in Gorini et al. (1976), and is given, in the finite dimensional case, by Eq. (A.2) with constant positive coefficients .
Quantum semigroup dynamics have been identified as the time-homogeneous Markovian dynamics in the quantum setting, both because of the analogy with the semigroup composition law for the transition probabilities of classical time-homogeneous Markovian stochastic processes, and because they describe satisfactorily the dynamics of open quantum systems when one can fully neglect the memory effects encoded into the environmental multi-time correlation functions Breuer and Petruccione (2002). A natural way to extend the definition of quantum Markovianity also to time-inhomogeneous dynamics is then to say that a given dynamics is Markovian when it is CP-divisible Rivas et al. (2010), i.e., not only Eq. (29) holds, but also the maps are CPTP for any . Furthermore, the dynamics generated by in Eq. (A.2) is CP-divisible if and only if all the rates are non-negative functions of time, i.e., for any and .
As a last remark, let us stress that different and non-equivalent definitions of Markovianity have been introduced. Nevertheless, the conclusions of this work do not depend on the definition exploited, as they only rely on the distinction between time-homogenous and time-inhomogeneous dynamics.
Appendix B Phase covariant dynamics
In this section, we show explicitly how to characterize the class of reduced dynamics due to identical independent and phase covariant (IIC) noise. First we derive the general form of the phase covariant map as given in Eq. (4) of the main text and then provide its corresponding TLME.
b.1 Phase covariant maps
The most general form of IIC maps could be obtained via the theory of covariant quantum channels Holevo (1993); *Holevo1996, which classifies the maps commuting with some group representation via the Choi-Jamiolkowski isomorphism Bengtsson and Życzkowski (2006). However, for the simple case of -covariant qubit channels, i.e., the phase covariant qubit channels, we can directly exploit the simpler tools provided by the matrix representation of dynamical maps that has been introduced in the previous section.
Let be a trace- and hermiticity- preserving linear map in and be a unitary map also in fixed by in , such that
(32) |
By Eq. (17), the matrix representation of is given by
(33) |
As clear from the previous section, commutation relation is equivalent to the same relation between the corresponding matrices . Taking the matrix representation given by Eq. (18), with real and , the vanishing commutator with the unitary-map matrix (33) requires
so that , , while
(34) |
with as defined in the main text and pictorially described in Fig. 2 therein. Finally, by multiplying and (that commute with one another) we obtain the matrix form of which is valid for each element of the -parametrised family of maps describing the dynamics stated in Eq. (4) of the main text:
(35) |
with .
For future convenience, let us note that the Choi matrix associated with the map described by Eq. (35) is given in the canonical basis (see Eq. (21)) by
(36) |
so that is CP if and only if:
(37) |
The above equations clearly imply that simply correspond to an additional rotation around the axis). For this class of dynamical maps, the unitary part does not affect the CP constraints (37), yet they will play an important role in the following analysis of App. C.3. Finally, by considering the eigenvectors of the Choi matrix (36), with , one may define the canonical Kraus operators that satisfy Bengtsson and Życzkowski (2006): (negative and without loss of generality we additionally restrict ,
(38) |
where
(39) |
In particular, note that typical qubit channels (see Bengtsson and Życzkowski (2006)) correspond to special instances of the map , i.e.: pure dephasing is recovered by setting , and considering ; for isotropic depolarisation (white local noise) ; whereas amplitude damping corresponds to and . , ,
The unital phase covariant transformations, i.e., ones that preserve identity,
In summary, given a phase covariant dynamics—a one-parameter family of CPTP maps for which at every time the map can be decomposed into a unitary -encoding and an -independent noise term , such that the two commute—we arrive at the general form of the evolution given by Eq. (35) for any time , i.e., Eq. (4) stated in the main text. As said, we assume throughout the work the matrix elements of dynamical maps considered to be smooth in , what is assured by and being real functions of class . Furthermore, the initial conditions
b.2 Time-local master equation of a phase covariant dynamics
The IIC dynamics can be equivalently characterized by investigating the TLME associated with the state in Eq. (23), as we explicitly show below.
Applying Eq. (25) to the family of maps specified by Eq. (35) and using Eq. (17) to obtain an explicit form of the TLME Smirne and Vacchini (2010), one ends up with Eq. (5) stated in the main text, i.e., Eq. (A.2) with the time-local (-dependent) generator:
(40) | |||||
In particular, one finds
(41) |
where we denote the derivative w.r.t. by .
Let us remark that Eq. (41) unambiguously defines the TLME except when or , for which in Eq. (25) does not exist. As both these parameters are guaranteed to be equal to identity at , can become singular only after finite duration of time. Thus, we may always assure that there exists a short enough range of timescales in which the TLME obeying Eq. (41) can be defined.
Crucially, reversing the argument, any TLME defined by the generator (40) with real, continuous and bounded functions , along with the initial condition , is uniquely solved by the one-parameter family of trace and hermiticity preserving linear maps fixed by Eq. (35). Since the coefficients of the master equations are assumed to be bounded, the generator (40) is bounded in norm for any , so that the solution of the master equation is unique and provided by the Dyson series Rivas and Huelga (2012):
(42) |
Consequently, the matrix elements of are given by the unique solution to the system of differential equations specified in Eq. (41) with initial conditions
(43) | |||||
If some of the coefficients in the TLME diverge at instants