The DOI matches its Crossref record; 239 of the 239 references it cites were found in a registry or an index; 53 of the 89 sources its claims cite were checked for support.
Quantum non-Markovianity: characterization, quantification and detection
Open accessAbstract
We present a comprehensive and up-to-date review of the concept of quantum non-Markovianity, a central theme in the theory of open quantum systems. We introduce the concept of a quantum Markovian process as a generalization of the classical definition of Markovianity via the so-called divisibility property and relate this notion to the intuitive idea that links non-Markovianity with the persistence of memory effects. A detailed comparison with other definitions presented in the literature is provided. We then discuss several existing proposals to quantify the degree of non-Markovianity of quantum dynamics and to witness non-Markovian behavior, the latter providing sufficient conditions to detect deviations from strict Markovianity. Finally, we conclude by enumerating some timely open problems in the field and provide an outlook on possible research directions.
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Claims the paper makes
76 Own interpretation · 20 No basis given · 16 Couldn’t confirm · 12 Partly supported · 9 Sources not found · 5 Supported so far · 3 Supported · 3 No full text of the sources · 2 Not checked yet · 1 Not supported · 1 Own result · accuracy not yet measured
At least one cited source doesn’t support this claim or points to a different paper, and none confirmed it — see the verdicts below.
In the paper
“A different witness in terms of fidelity was previously proposed in [196]; however that was only able to detect deviations from time-homogeneous Markov processes, i.e. quantum dynamical semigroups where E (t 2 ,t 1 ) = E (t 2 -t 1 ) for every t 1 and t 2 .”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.1.2. Fidelity.
What the sources say
- Disputedcites [196] sourceExperimental — accuracy not yet measured
Section II (p. 2) explicitly defines this paper’s witness using Hilbert–Schmidt distance between covariance matrices, not fidelity. The mentions of fidelity in the results (pp. 4–5) concern how accurately initial states are represented, not a non-Markovianity witness. The copy does not establish the claimed time-homogeneous-only limitation. (LLM, conf=0.84)
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- Disputed
The sources with a verdict support this claim; the rest weren’t checked yet, have no full text we could read, or their verdicts aren’t rated yet.
In the paper
“Theorem 3.1 (Gorini-Kossakowski-Susarshan-Lindblad). An operator L t is the generator of a quantum Markov (or divisible) process if and only if it can be written in the form dρ(t) dt = L t [ρ(t)] = -i[H (t), ρ(t)] + k γ k (t) V k (t)ρ(t)V † k (t) - 1 2 {V † k (t)V k (t), ρ(t)} , (32 ) where H (t) and V k (t) are time-dependent operators, with H (t) self-adjoint, and γ k (t) 0 for every k and time”
§ Quantum non-Markovianity: characterization, quantification and detection > Theorem 3.1 (Gorini-Kossakowski-Susarshan-Lindblad).
What the sources say
- Not checked — no full text of the sourcecites [28] source
- Not matchedcites [29] Kossakowski A, 1972, Bull. Acad. Pol. Sci., Ser. Math. Astron. Phys.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Not checked — no full text of the sourcecites [31] source
- Not checked — no full text of the sourcecites [26] Rivas A, 2011, Open Quantum Systems. An Introduction source
- Supportedcites [32] Chruściński D, 2012, J. Phys. B: At. Mol. Opt. Phys. source
Section 4, Theorem 4.1 (page not available), states that a dynamical map is divisible if and only if its local generator has GKSL form at all times. Definition 4.1 equates divisibility with Markovian evolution, and equation (27) gives the GKSL form with a positive coefficient matrix, equivalent to nonnegative coefficients in a diagonal representation. (LLM, conf=0.94)
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Every source the paper cites for this claim was checked and supports it.
In the paper
“A completely positive dynamics E(ρ S ) can be seen as the reduced dynamics of some unitary evolution acting on an extended state with the form ρ S ⊗ ρ E , where ρ E is the same, independent of ρ S .”
§ Quantum non-Markovianity: characterization, quantification and detection > Theorem 3.5 (Stinespring [46]).
What the sources say
- Supportedcites [46] source
The Summary and Conclusions (p. 9) states that an arbitrary quantum map can be represented by a unitary operation on a larger system and identifies this as the Stinespring–Kraus dilation theorem. The Collision Model (p. 4) uses an initial product state ρ⊗ξ, evolves it unitarily, and obtains the system map by tracing out the reservoir; ξ is specified separately from the system state. (LLM, conf=0.86)
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In the paper
“However, as pointed out in [102,103], this interpretation in terms of information flowing between the system and the environment may be problematic if taken strictly, because it is possible to obtain quantum non-Markovian dynamics with the form E (t,t 0 ) (ρ) = i p i U i (t, t 0 )ρU † i (t, t 0 ).”
§ Quantum non-Markovianity: characterization, quantification and detection > 4.2. Optimization of the Helstrom matrix norm
What the sources say
- Partly supportedcites [102] Caruso F, 2012 sourceExperimental — accuracy not yet measured
Section C (pp. 50–51) describes information backflow from the environment as a signature of non-Markovian dynamics. It does not say that this interpretation is problematic or give a non-Markovian dynamics example of the stated random-unitary form. The unitary rotations discussed in §4 (p. 23) are instead an example of Markovian correlated noise. (LLM, conf=0.82)
- Supportedcites [103] source
Section VII, Eq. (17), gives the system dynamics as a probabilistic mixture of unitary transformations and explains that it can be modeled with a classical environment without back-action. Section VIII identifies the model’s dynamics as non-Markovian. Together, these passages support the caution against interpreting non-Markovianity strictly as information flowing back from the environment. (LLM, conf=0.92)
hedgedconditional- Asserted: passes — The authors put it forward as true.
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- Partly supported
The cited sources that could be checked couldn’t confirm or dispute this claim.
In the paper
“Accardi, Frigerio and Lewis proved in their seminal paper [66] that this definition of a Markovian process implies our definition 3.2 (rewritten in the Heisenberg picture).”
§ Quantum non-Markovianity: characterization, quantification and detection > 3.4. Comparison with other definitions of quantum Markovianity
What the sources say
- Could not verifycites [66] sourceExperimental — accuracy not yet measured
The discussion of Markovian dynamics in “A. Non-Markovian dynamics” (p. 1) concerns master equations and semigroups, not an algebraic definition implying divisibility. Searches for Accardi, Frigerio and Lewis, divisibility, and the Heisenberg picture found no passage in our copy addressing the claimed proof. (LLM, conf=0.77)
boosteddirect- Asserted: passes — The authors put it forward as true.
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In the paper
“While very efficient under certain conditions, there are some non-Markovian processes that cannot be witnessed by the trace distance, for example those where the non-Markovianity is encoded just in the 'non-unital part' of the dynamics [185].”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.1.1. Trace distance and the BLP quantifier.
What the sources say
- Could not verifycites [185] Wissmann S, 2013 sourceExperimental — accuracy not yet measured
The Introduction (p. 1) says increases in trace distance can quantify dynamical memory effects, while the Conclusions (p. 8) discuss limits on detecting initial system–environment correlations. Neither passage addresses non-Markovianity encoded only in the non-unital part of the dynamics; searches for that term and for processes trace distance cannot witness found no such discussion in our copy. (LLM, conf=0.76)
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In the paper
“However, the reverse implication fails to hold, i.e. not every dynamics fulfilling equation ( 55) satisfies theorem 3.4 (e.g.[34,79,80]).”
§ Quantum non-Markovianity: characterization, quantification and detection > 3.4. Comparison with other definitions of quantum Markovianity
What the sources say
- Supportedcites [34] source
Section II (pp. 1–2) defines the divisibility-based Markovian condition and explicitly says that, although divisibility implies the BLP condition, the converse is generally false. The Introduction (p. 1) says the paper provides a BLP-Markovian evolution that is indivisible. (LLM, conf=0.94)
- Not matchedcites [79] Breuer H-P, 2012, J. Phys. B: At. Mol. Opt. Phys.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Supportedcites [80] source
Section IV (p. 4) states that every divisible map is Markovian under the BLP trace-distance criterion, but that the converse is generally false. Section III (p. 3) identifies divisibility as the RHP Markovian condition. The source thus supports the attributed failure of the reverse implication, though it does not refer to the citing paper’s theorem number. (LLM, conf=0.95)
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In the paper
“This definition is a particular case of definition 3.2.”
§ Quantum non-Markovianity: characterization, quantification and detection > 3.4. Comparison with other definitions of quantum Markovianity
What the sources say
- Supportedcites [33] source
The paper defines non-Markovianity by an increase in trace distance for some pair of initial states (p. 2). It also states that divisible processes have non-increasing trace distance and that processes non-Markovian by its criterion must be non-divisible (pp. 1–2). This supports the claimed particular-case relationship, not an equivalence between the definitions. (LLM, conf=0.86)
- Not matchedcites [77] Ji U C, 2010, Commun. Stoch. Anal.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Supportedcites [78] source
The paper defines its measure through increases in trace distance between pairs of initial states (Construction of the measure, p. 2). It then states that all divisible maps have monotonically decreasing trace distance, so a process non-Markovian by this measure must be nondivisible (Divisible maps, p. 3). It conjectures that the two notions are not equivalent (Other approaches to non-Markovianity, p. 7). (LLM, conf=0.91)
plaindirect- Asserted: passes — The authors put it forward as true.
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- Supported
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In the paper
“Given a family {E (t,t 0 ) , t t 0 }, the basic idea of this measure is to quantify how much non-completely positive the intermediate dynamics {E (t,t 1 ) , t t 1 t 0 } is for every time t 1 .”
§ Quantum non-Markovianity: characterization, quantification and detection > 4.3. The RHP measure
What the sources say
- Partly supportedcites [106] Liu B-H, 2014 sourceExperimental — accuracy not yet measured
Section I identifies divisibility of the dynamical map as a characterization of non-Markovianity and cites Rivas, Huelga, and Plenio for it. But it does not say that their measure quantifies departures of intermediate maps from complete positivity; the measure this paper develops instead uses increases in trace distance (Section I, Eqs. 1–2). (LLM, conf=0.78)
plaindirect- Asserted: passes — The authors put it forward as true.
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The cited sources that could be checked couldn’t confirm or dispute this claim.
In the paper
“D Q A [ρ SA (t)] = Q E (t,0) ,(158)”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.2.3. Quantum discord.
What the sources say
- Could not verifycites [230] sourceExperimental — accuracy not yet measured
The discussion on pp. 2–3 describes discord during partial decoherence of a measurement model, but does not state the claimed equality. Searches for the pure-dephasing model and Q E(t,0) found no passage that establishes it; the formulas in our copy are not decoded. (LLM, conf=0.73)
- Could not verifycites [245] sourceExperimental — accuracy not yet measured
The paper’s discussion of entanglement monotones and logarithmic negativity (Abstract; Notations and definitions, page unavailable) does not address the stated equality. Searches for quantum discord and the pure dephasing model found no passage in our copy supporting it. (LLM, conf=0.76)
plaincorrelational- Asserted: passes — The authors put it forward as true.
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- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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- Could not verify
- It may state two separate claims as one. 53
- Probably not a substantial claim. 31
- The quote only partly states this claim. 2
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“Despite the fact that the most studied models to describe such a dynamics result in Markovian master equations of the form of equation (32) [25,[39][40][41], it is well-known that the exact dynamics of an open quantum system is essentially non-Markovian [22,24,26,42].”
§ Quantum non-Markovianity: characterization, quantification and detection > 3.3.2. Memoryless environment.
What the sources say
- Not matchedcites [22] Davies E B, 1976, Quantum Theory of Open Systems
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Partly supportedcites [24] Breuer H-P, 2002, The Theory of Open Quantum Systems sourceExperimental — accuracy not yet measured
The Introduction (p. 1) describes Markovian master equations as arising from an approximation, and §5 (p. 12) describes a formally exact equation for reduced dynamics with a memory kernel. These passages support the distinction between exact dynamics and commonly used Markovian descriptions, but the paper says non-Markovian treatment is necessary under conditions such as stronger coupling; it does not state that every open system’s exact dynamics is non-Markovian. (LLM, conf=0.84)
- Not checked — no full text of the sourcecites [26] Rivas A, 2011, Open Quantum Systems. An Introduction source
- Not checked — no full text of the sourcecites [42] source
boosted- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- Not matched
The sources with a verdict support this claim; the rest weren’t checked yet, have no full text we could read, or their verdicts aren’t rated yet.
In the paper
“Complete positive maps are much easier to characterize than maps that are merely positive [19,20]; they admit the so-called Kraus representation,”
§ Quantum non-Markovianity: characterization, quantification and detection > Definition 3.1 (P-divisible process).
What the sources say
- Supportedcites [19] source
Theorem 1 gives a matrix-sum representation for completely positive maps, the form the claim calls a Kraus representation. The paper also describes their structure as tractable and says the general structure of positive maps is very complicated (Additional information). (LLM, conf=0.93)
- Not checked — no full text of the sourcecites [20] source
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
- Supported
The cited sources that could be checked couldn’t confirm or dispute this claim.
In the paper
“Recently, Chruściński and Maniscalco have proposed a hierarchical way to assess non-Markovianity [129].Their approach, based on the concept of k-divisibility, is interesting as it provides a way to define some kind of maximally non-Markovian dynamics.”
§ Quantum non-Markovianity: characterization, quantification and detection > 4.5. Hierarchical k-divisibility degrees
What the sources say
- Could not verifycites [129] sourceExperimental — accuracy not yet measured
The Introduction (p. 1) discusses proposals to measure non-Markovianity, and Section V (pp. 7–8) examines decoherence factors in a cavity system. Searches for Chruściński and Maniscalco and for k-divisibility found no passage in our copy addressing their proposed hierarchy. (LLM, conf=0.76)
plain- Asserted: passes — The authors put it forward as true.
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- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Could not verify
The cited sources that could be checked couldn’t confirm or dispute this claim.
In the paper
“Indeed, Chruściński and Maniscalco propose to give the name 'maximally non-Markovian dynamics' to those that D I ND(1) = 1, and consequently D I ND(2) = D I ND(3) = . . . = D I ND(d) = 1.”
§ Quantum non-Markovianity: characterization, quantification and detection > 4.5. Hierarchical k-divisibility degrees
What the sources say
- Could not verifycites [129] sourceExperimental — accuracy not yet measured
The Introduction (p. 1) mentions proposals for measuring non-Markovianity, and Section V (pp. 7–8) examines it through a cavity’s decoherence factor. Neither those passages nor searches for Chruściński and Maniscalco, “maximally non-Markovian,” and D_ND(k) address the proposed criterion or the claimed implication. (LLM, conf=0.76)
plaindirect- Asserted: passes — The authors put it forward as true.
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- Load-bearing: fails — The paper’s conclusions may not depend on it.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
- Could not verify
The cited sources that could be checked couldn’t confirm or dispute this claim.
In the paper
“Actually, it can be proven [127] that f (t) = d 2 g(t) (see equation ( 68)), so this quantity is proportional to N I RHP , equation (69), N I γ = d 2 N I RHP . (94”
§ Quantum non-Markovianity: characterization, quantification and detection > 4.4. Decay rates measures
What the sources say
- Could not verifycites [127] sourceExperimental — accuracy not yet measured
The Summary (p. 7) discusses divisibility-based measures of non-Markovianity, but does not state a relationship between f(t) and g(t) or between the two cited measures. Searches for those expressions, the factor d², and the RHP measure found no passage establishing the claimed equations in our copy. (LLM, conf=0.72)
boosteddirect- Asserted: passes — The authors put it forward as true.
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- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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It may state two separate claims as one.
- Could not verify
The sources with a verdict support this claim; the rest weren’t checked yet, have no full text we could read, or their verdicts aren’t rated yet.
In the paper
“Secondly, if the 'Born-Markov' approximation is combined with the secular approximation (i.e.neglecting fast oscillating terms in the evolution equation), a valid quantum dynamical semigroup is obtained [24-26, 39, 41], and then the dynamics can be certainly called Markovian.”
§ Quantum non-Markovianity: characterization, quantification and detection > 3.4. Comparison with other definitions of quantum Markovianity
What the sources say
- Supportedcites [24] Breuer H-P, 2002, The Theory of Open Quantum Systems source
Section 3 (p. 6) describes the weak-coupling quantum optical master equation as Markovian and identifies the rotating-wave approximation as essential to its derivation. Section 3 (p. 5) says that the resulting Markovian dynamics has the quantum dynamical semigroup property. The paper specifies timescale conditions under which these approximations apply. (LLM, conf=0.87)
- Not checked — no full text of the sourcecites [25] Alicki R, 2007, Quantum Dynamical Semigroups and Applications source
- Not checked — no full text of the sourcecites [26] Rivas A, 2011, Open Quantum Systems. An Introduction source
- Not checked — no full text of the sourcecites [39] source
- Not matchedcites [41] Gardiner C W, 2004, Quantum Noise
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
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It may state two separate claims as one.
- Supported
The cited sources that could be checked couldn’t confirm or dispute this claim.
In the paper
“Since E (t,t 0 ) is completely positive for any final time t, it is easy to prove that N + k ( ˜ , I ) |N - k ( ˜ , I )| [129], therefore D I ND(k) ∈ [0, 1] for all k.”
§ Quantum non-Markovianity: characterization, quantification and detection > 4.5. Hierarchical k-divisibility degrees
What the sources say
- Could not verifycites [129] sourceExperimental — accuracy not yet measured
Appendix C (p. 10) derives the reduced density matrix by tracing over the bath, while Section V (pp. 7–8) discusses a decoherence factor as a probe of non-Markovianity. Searches for complete positivity, the N⁺ₖ and N⁻ₖ inequality, and the claimed [0,1] bound found no passage in our copy addressing that assertion. (LLM, conf=0.70)
plaindirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
- Could not verify
The sources with a verdict support this claim; the rest weren’t checked yet, have no full text we could read, or their verdicts aren’t rated yet.
In the paper
“Firstly, despite the fact that the 'Born-Markov' approximation leads to master equations with time-independent coefficients, they do not always define a valid quantum dynamical semigroup [84,85].This is because they break complete positivity.Thus, these models should not be referred as 'Markovian' in the strict sense, as a Markovian process must preserve the positivity of any state, or of any”
§ Quantum non-Markovianity: characterization, quantification and detection > 3.4. Comparison with other definitions of quantum Markovianity
What the sources say
- Supportedcites [84] Whitney R S, 2008, J. Phys. A: Math. Theor. source
Section 1.2 (pp. 3–4) says the usual time-independent-parameter assumption in Bloch–Redfield master equations can violate positivity on timescales comparable to the environment’s memory time. Section 2 (p. 5) identifies complete positivity as a requirement for a valid dynamical semigroup, while the Conclusions (p. 13) reserve ‘strictly Markovian’ evolution for zero memory time. The claim’s ‘not always’ qualification is important: the paper does not say every time-independent equation fails. (LLM, conf=0.86)
- Not checked — no full text of the sourcecites [85] source
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
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It may state two separate claims as one.
- Supported
The sources with a verdict support this claim; the rest weren’t checked yet, have no full text we could read, or their verdicts aren’t rated yet.
In the paper
“Later on, the problem to decide whether a quantum channel is Markovian was shown to be very hard in the complexity theory sense [98,99]; however it has been analyzed for small-sized systems in [77] and [100].”
§ Quantum non-Markovianity: characterization, quantification and detection > 4.1. Geometric measures
What the sources say
- Could not verifycites [98] sourceExperimental — accuracy not yet measured
The Preliminaries (p. 1) define Markovian evolution, and the Discussion (p. 4) says the method can quantify non-Markovianity through environmental memory. Neither passage addresses the complexity of deciding whether a quantum channel is Markovian or reports an analysis of that decision problem for small systems; searches for those points found no such passage. (LLM, conf=0.76)
- Not checked — no full text of the sourcecites [99] source
- Not matchedcites [77] Ji U C, 2010, Commun. Stoch. Anal.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Supportedcites [100] source
The p. 2 discussion states that any instance of the NP-complete 1-IN-3SAT problem can be transformed into a quantum Markovianity problem, establishing the complexity-theory claim. The sentence cites other works for the analysis of small-sized systems; that part is not attributed to this source. (LLM, conf=0.93)
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- Could not verify
The cited sources that could be checked couldn’t confirm or dispute this claim.
In the paper
“in [127] Hall, Cresser, Li and Andersson proposed a measure of non-Markovianity focused on properties of the generator.”
§ Quantum non-Markovianity: characterization, quantification and detection > 4.4. Decay rates measures
What the sources say
- Could not verifycites [127] sourceExperimental — accuracy not yet measured
The Introduction (p. 1) discusses generators as background, while the Summary (p. 7) describes the paper’s approach in terms of stochastic and dynamical maps. Searches for Hall, Cresser, Li and Andersson and for a generator-based non-Markovianity measure found no passage saying that those authors proposed the measure. (LLM, conf=0.74)
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Could not verify
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“E (t,t 1 ) ⊗ 1 1 (| |) 1 = 1 iff E (t,t 1 ) is CP, > 1 otherwise.”
§ Quantum non-Markovianity: characterization, quantification and detection > 4.3. The RHP measure
What the sources say
- Partly supportedcites [19] sourceExperimental — accuracy not yet measured
Theorem 2 (in the copy’s Abstract section; page not provided) states that a map is completely positive if and only if its associated matrix is positive. The copy does not state the claimed trace-norm criterion or establish the normalization needed for a threshold of 1. (LLM, conf=0.86)
- Not checked — no full text of the sourcecites [20] source
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- Partly supported
The cited sources that could be checked couldn’t confirm or dispute this claim.
In the paper
“There are already studies showing its usefulness to prepare steady entangled states [122], to enhance the achievable resolution in quantum metrology [259,260] or to assist certain tasks in quantum information and computation [208,261,262].”
§ Quantum non-Markovianity: characterization, quantification and detection > 6. Conclusion and outlook
What the sources say
- Could not verifycites [122] sourceExperimental — accuracy not yet measured
The discussion of non-Markovian behavior (p. 3) uses entanglement decay between a qubit and an ancilla as a measure; it does not describe preparing a steady entangled state. The Conclusion (p. 4) concerns a single-qubit channel and its non-Markovian effects. Searches for steady entangled states and their preparation found no passage addressing the attributed use. (LLM, conf=0.76)
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Could not verify
The cited sources that could be checked couldn’t confirm or dispute this claim.
In the paper
“For the quantum discord, note that the state (140) belongs to the subclass known as 'X-states', for which the optimization problem in equation ( 152) can be efficiently solved [230,245].”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.2.3. Quantum discord.
What the sources say
- Could not verifycites [230] sourceExperimental — accuracy not yet measured
The discussion on p. 2 describes minimizing quantum discord over measurements, but does not address X-states or whether that optimization can be efficiently solved for them. Searches for X-states and efficient optimization found no passage establishing the claim; the cited paper also cannot identify the citing document’s state (140). (LLM, conf=0.72)
- Could not verifycites [245] sourceExperimental — accuracy not yet measured
The paper’s Abstract discusses entanglement measures and the difficulty of optimizing some of them, but not quantum discord. Searches for quantum discord and X-states found no passage identifying the citing paper’s state (140) as an X-state or addressing efficient solution of its equation (152). (LLM, conf=0.76)
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- Could not verify
The cited sources that could be checked couldn’t confirm or dispute this claim.
In the paper
“Another interesting proposed way to witness non-Markovianity was suggested by Lorenzo et al in [211].”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.1.6. Bloch volume measure.
What the sources say
- Could not verifycites [211] sourceExperimental — accuracy not yet measured
The Information back-flow section (p. 3) discusses a non-Markovianity measure based on quantum channel capacity, not a Bloch-volume witness. Searches for Lorenzo, Bloch volume, and the volume of accessible states found no passage in our copy addressing the attributed proposal. (LLM, conf=0.75)
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Could not verify
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“There are already studies showing its usefulness to prepare steady entangled states [122], to enhance the achievable resolution in quantum metrology [259,260] or to assist certain tasks in quantum information and computation [208,261,262].”
§ Quantum non-Markovianity: characterization, quantification and detection > 6. Conclusion and outlook
What the sources say
- Could not verifycites [259] Borrelli M, 2014 sourceExperimental — accuracy not yet measured
The Conclusions report that a non-Markovianity measure is sensitive to a phase transition, and Section III describes Ramsey interferometry as a way to study the crystal. Neither passage says that non-Markovian processes enhance achievable resolution in quantum metrology; searches for metrological precision and resolution found no such result in our copy. (LLM, conf=0.76)
- Supportedcites [260] source
The paper’s summary (p. 1) says that non-Markovian dynamics enables quantum-correlated probes to outperform uncorrelated probes using otherwise identical resources. The Conclusions (p. 4) likewise report a metrological advantage under non-Markovian dephasing, supporting the citing sentence’s broad claim about enhanced resolution. (LLM, conf=0.92)
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Could not verify
The cited sources that could be checked couldn’t confirm or dispute this claim.
In the paper
“However, similarly to the trace distance, since the volume of accessible states is independent of the affine vector c (t,t 0 ) , it is not sensitive to dynamics where the non-Markovianity is encoded in c (t,t 0 ) .More concretely, it can be shown that the volume of accessible states only detects non-Markovian dynamics such that Tr[M (t,t 0 ) ] > 0 [127].”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.1.6. Bloch volume measure.
What the sources say
- Could not verifycites [127] sourceExperimental — accuracy not yet measured
The Abstract (p. 1) and Summary (p. 7) discuss non-Markovianity using intermediate dynamical maps, divisibility, and trace distance, but do not address a Bloch-volume witness. Searches for the affine vector c(t,t0), accessible-state volume, and the condition Tr[M(t,t0)] > 0 found no passage supporting the claimed limitations. (LLM, conf=0.76)
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- Could not verify
The cited sources that could be checked couldn’t confirm or dispute this claim.
In the paper
“Interestingly, | det[M (t,t 0 ) ]| describes the change in volume of the set of states accessible through the evolution [211], so that Markovian evolutions reduce (or leave invariant) the volume of accessible states.”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.1.6. Bloch volume measure.
What the sources say
- Could not verifycites [211] sourceExperimental — accuracy not yet measured
The Information back-flow section (p. 3) says that quantum channel capacity decreases monotonically under Markovian dephasing, but it does not identify that quantity with the volume of accessible states. Searches for det M and accessible-state volume found no passage explaining a determinant as a volume change or stating the claimed volume result. (LLM, conf=0.75)
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- Could not verify
The cited sources support only part of this claim — read each source’s verdict below.
In the paper
“For instance, some quantitative relations between non-Markovianity effects and criticality and phase transitions [175, 256-258], Loschmidt echo [174, 175, 256], symmetry breaking [166], and Zeno and anti-Zeno effects [150] have already been described.”
§ Quantum non-Markovianity: characterization, quantification and detection > 6. Conclusion and outlook
What the sources say
- Supportedcites [175] source
The cited paper is attached to the criticality/phase-transition and Loschmidt-echo parts of the sentence, not to symmetry breaking or Zeno effects. It establishes a relationship between a non-Markovianity measure and the Loschmidt echo (p. 2) and reports that the measure pinpoints the critical field in its qubit–Ising-chain model (p. 4). (LLM, conf=0.93)
- Could not verifycites [256] de Vicente J I, 2014 sourceExperimental — accuracy not yet measured
The Introduction (p. 2) and Conclusions (pp. 18–19) discuss nonlocality as a resource theory and its measures, not non-Markovianity. Searches for the claimed links and for “Loschmidt echo” found no passage in our copy addressing criticality, symmetry breaking, or Zeno/anti-Zeno effects. (LLM, conf=0.77)
- Partly supportedcites [257] sourceExperimental — accuracy not yet measured
On p. 4, the paper links the Loschmidt echo to non-Markovianity and reports that non-Markovianity varies with the critical parameter α in a trapped Fermi gas. It does not report a relation to phase transitions. The citing sentence assigns symmetry breaking and Zeno/anti-Zeno effects to other references, not this paper. (LLM, conf=0.84)
- Supportedcites [258] source
The paper’s introduction (p. 1) reports that its non-Markovianity measure is sensitive to the critical point of a Coulomb crystal’s structural phase transition. Its conclusions (p. 4) likewise say that changes in the measure pinpoint criticality. The citing sentence assigns separate citations to the Loschmidt echo, symmetry breaking, and Zeno effects; those are not claims attributed to this source here. (LLM, conf=0.91)
- Could not verifycites [174] sourceExperimental — accuracy not yet measured
The Introduction (p. 1) and Conclusion (p. 10) describe a relation between non-Markovian information flow and geometric phase, not the listed phenomena. The “critical point” in the time-local master-equation section (p. 6) is a parameter point where information flow begins to increase, not a physical phase transition. Searches for Loschmidt echo, symmetry breaking, and Zeno or anti-Zeno effects found no passage addressing them. (LLM, conf=0.76)
- Could not verifycites [166] sourceExperimental — accuracy not yet measured
The citation to this paper, [150], is attached specifically to Zeno and anti-Zeno effects. Our copy discusses non-Markovianity in random-telegraph and colored-noise channels (Conclusions, p. 7), but the passages returned for searches of “Zeno anti-Zeno effects” and the other listed links do not address a relation to Zeno or anti-Zeno effects. (LLM, conf=0.75)
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
- Supported
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“ρ(t) = E (t,0) [ρ(0)] = ρ 11 ρ 12 R(t) ρ 21 R(t) ρ 22 , with R(t) = e -2 t 0 γ (s) ds .(107) Note that 0 R(t) 1.”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.1.1. Trace distance and the BLP quantifier.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Note that for p(t) = 1/2, the coherences vanish completely.”
§ Quantum non-Markovianity: characterization, quantification and detection > 4.3. The RHP measure
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“g(t) = p (t) 1 -2p(t) - p (t) 1 -2p(t) = 0 i f p (t) 1 -2p(t) 0, - 2p (t) 1 -2p(t) if p (t) 1 -2p(t) < 0. (86”
§ Quantum non-Markovianity: characterization, quantification and detection > 4.3. The RHP measure
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“F [ρ 1 (t), ρ 2 (t)] = Tr ρ 1 (t)ρ 2 (t) ρ 1 (t) 2 = 1 -R 2 (t), (113 ) so that dF [ρ 1 (t), ρ 2 (t)] dt = -2R (t)R(t) = 4γ (t)R 2 (t). (114”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.1.2. Fidelity.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“D 1 [ρ 1 (t), ρ 2 (t)] = 1 2 ρ 1 (t) -ρ 2 (t) 1 = 0 R(t) R(t) 0 1 = |R(t)| = R(t), (108 ) so that σ (ρ 1 , ρ 2 , t) = dD 1 [ρ 1 (t), ρ 2 (t)] dt = -2γ (t)R(t). (109”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.1.1. Trace distance and the BLP quantifier.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“For instance, γ (t) = sin(t) or γ (t) = tan(t) in t ∈ [0, 2π ]. Interestingly, for these two examples the Choi-Jamiolkowski measure equation ( 71) provides different values.We obtain D I NM(RHP) = 0.758 and D I NM(RHP) = 0.803, for γ (t) = sin(t) and γ (t) = tan(t), respectively.”
§ Quantum non-Markovianity: characterization, quantification and detection > 4.5. Hierarchical k-divisibility degrees
plaincorrelational- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Note that since the eigenvalues γ j (t) are independent of the basis, this form is unique (up to degeneracy).”
§ Quantum non-Markovianity: characterization, quantification and detection > 4.4. Decay rates measures
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“I QFI (t) = k γ k (t)J k (t), (129 ) J k (t) := -Tr ρ θ (t)[L(ρ θ , t), V k (t)] † [L(ρ θ , t), V k (t)] 0. (130”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.1.4. Quantum Fisher information.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Therefore, both quantities have the same derivative dC ea E (t,0) dt = dQ E (t,0) dt = -γ (t)R(t) log 1 + R(t) 1 -R(t) , (143 ) which can be positive only for non-Markovian dynamics, γ (t) < 0.”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.1.5. Capacity measures.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“D 2 B (ρ θ , ρ θ+δθ ) = 1 4 R 2 (t)(δθ ) 2 + O[(δθ ) 3 ] ⇒ F(ρ θ ) = R 2 (t). (131 ) Thus, the quantum Fisher information flow, equation (127), is I QFI (t) = γ (t)J (t), with J (t) = -4R 2 (t), (132”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.1.4. Quantum Fisher information.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“On the other hand, since the maps {E (t 2 ,t 1 ) , t 2 t 1 t 0 } are supposed to describe some quantum evolution, they are linear (there is no experimental evidence against this fact [15][16][17]).”
§ Quantum non-Markovianity: characterization, quantification and detection > 3.2. Definition in terms of one-point probabilities: divisibility
What the sources say
- Not matchedcites [15][16][17]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- Not matched
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Since the Markov condition (1) implies that P(x 3 , t 3 |x 2 , t 2 ; x 1 , t 1 ) = P(x 3 , t 3 |x 2 , t 2 ), by taking the sum over x 2 and dividing both sides by P(x 1 , t 1 ), we arrive at P(x 3 , t 3 |x 1 , t 1 ) = x 2 ∈X P(x 3 , t 3 |x 2 , t 2 )P(x 2 , t 2 |x 1 , t 1 ), (4) which is called the Chapman-Kolmogorov equation.”
§ Quantum non-Markovianity: characterization, quantification and detection > Definition 2.1 (Markov process).
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Thus, note that when q = 0 or q = 1, we immediately obtain zero probability of wrongly identifying the true state.”
§ Quantum non-Markovianity: characterization, quantification and detection > Theorem 3.2.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“For instance, it follows immediately from (1) that for a Markov process E(x n , t n |x n-1 , t n-1 ; . . .; x 0 , t 0 ) = E(x n , t n |x n-1 , t n-1 ), for all {t n t n-1 . . .t 0 } ⊂ I,”
§ Quantum non-Markovianity: characterization, quantification and detection > Definition 2.1 (Markov process).
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“That is not the case for a P-divisible dynamics where just the positivity of E (t 2 ,t 1 ) is required instead of the complete positivity.”
§ Quantum non-Markovianity: characterization, quantification and detection > Theorem 3.4.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one. The quote only partly states this claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“a very nice property is that divisibility may be defined in the quantum case without any explicit mention of measurement processes.”
§ Quantum non-Markovianity: characterization, quantification and detection > 3.2. Definition in terms of one-point probabilities: divisibility
hedged- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In the case of non-Markovian processes, T (x 2 , t 2 |x 1 , t 1 ) might be not well defined for t 1 = t 0 .Nevertheless, if the matrix P(x 1 , t 1 |x 0 , t 0 ) is invertible for every t 1 , then T (x 2 , t 2 |x 1 , t 1 ) can be written in terms of well-defined quantities.”
- Also stated in: § 2.2. Transition matrices · hedged
“However, condition equation ( 12) may be not fulfilled, which prevents any interpretation of T (x 2 , t 2 |x 1 , t 1 ) as a conditional probability and therefore manifests the non-Markovian character of such a stochastic process.”
§ Quantum non-Markovianity: characterization, quantification and detection > 2.2. Transition matrices
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one. The quote only partly states this claim.
- Also stated in: § 2.2. Transition matrices · hedged
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Corollary 2.1.Consider a Markov process {X(t), t ∈ I }, then for any t 3 t 2 t 1 t 0 , the transition matrix satisfies the properties x 2 ∈X T (x 2 , t 2 |x 1 , t 1 ) = 1,(11) T (x 2 , t 2 |x 1 , t 1 ) 0, (12 ) T (x 3 , t 3 |x 1 , t 1 ) = x 2 ∈X T (x 3 , t 3 |x 2 , t 2 )T (x 2 , t 2 |x 1 , t 1 ). (13 )”
§ Quantum non-Markovianity: characterization, quantification and detection > 2.2. Transition matrices
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Then the minimum (averaged) probability to give the wrong answer turns out to be P min (fail) = 1 -w(x) 1 2 , (18 )”
§ Quantum non-Markovianity: characterization, quantification and detection > 2.3. Contractive property
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“Concretely, in order to keep the positivity of density matrices in the presence of entanglement with another extra system, we must impose complete positivity instead of positivity [14,[21][22][23][24][25][26].”
§ Quantum non-Markovianity: characterization, quantification and detection > Definition 3.1 (P-divisible process).
What the sources say
- Not matchedcites [14,[21][22][23][24][25][26]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
boosteddirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim.
- Not matched
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“However, it is not immediately apparent that the memoryless property present in the classical case is also present in the quantum domain.There are at least two ways to visualize this property, which is hidden in definition 3.2.”
§ Quantum non-Markovianity: characterization, quantification and detection > 3.3. Where is the memoryless property in quantum Markovian processes?
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
In the paper
“Since for a closed quantum system the evolution is given by some two-parameter family of unitary operators U(t 1 , t 0 ), which fulfil U(t 2 , t 0 ) = U(t 2 , t 1 )U (t 1 , t 0 ) (see for instance [38]), the evolution of a closed quantum system is trivially Markovian.”
§ Quantum non-Markovianity: characterization, quantification and detection > 3.3.2. Memoryless environment.
What the sources say
- Not matchedcites [38] Galindo A, 1990, Quantum Mechanics
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
- Not matched
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Note that the apparent lack of memory in an unbiased case does not imply a general memoryless property; it only manifests in a general biased case (and taking into account possible ancillary systems).”
§ Quantum non-Markovianity: characterization, quantification and detection > 3.4. Comparison with other definitions of quantum Markovianity
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The 1-divisible processes are the P-divisible processes as introduced in definition 3.1, and the 0-divisible processes are processes where E (t 2 ,t 1 ) is not a positive operator for some t 1 and t 2 t 1 .”
§ Quantum non-Markovianity: characterization, quantification and detection > 4.5. Hierarchical k-divisibility degrees
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: fails — The paper’s conclusions may not depend on it.
- Based: passes — The paper gives a reason why it holds.
Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
In the paper
“Among several properties, the fidelity is monotonic under complete positive maps E, F [E(ρ 1 ), E(ρ 2 )] F (ρ 1 , ρ 2 ), (112 ) reaching the equality if and only if the completely positive map is unitary E(•) = U(•)U † [189].”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.1.2. Fidelity.
What the sources say
- Not checked — no full text of the sourcecites [189] source
plaindirect- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
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In the paper
“Moreover, as k increases, so does the dimension of the space in the optimization problem equation ( 100), and hence it is clear that 0 D I ND(1) . . . D I ND(d) 1.(101)”
§ Quantum non-Markovianity: characterization, quantification and detection > 4.5. Hierarchical k-divisibility degrees
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In the paper
“Therefore, g(t) = f (t), as expected in this case since d = 2.”
§ Quantum non-Markovianity: characterization, quantification and detection > 4.4. Decay rates measures
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Thus, the fidelity is monotonically increasing for Markovian evolutions, and therefore it may be used to witness non-Markovian behavior.”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.1.2. Fidelity.
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In the paper
“This quantity lies between 0 and 1 and is positive if and only if the process is non-Markovian, therefore it is a degree of non-Markovianity.”
§ Quantum non-Markovianity: characterization, quantification and detection > 4.1. Geometric measures
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“Therefore, if the dimension of the quantum system is d, a k-divisible process with k d is what in this work has been called a divisible or Markovian process (see definition 3.2).”
§ Quantum non-Markovianity: characterization, quantification and detection > 4.5. Hierarchical k-divisibility degrees
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In the paper
“If these maps are completely positive (CP) for all t 1 , the time evolution is Markovian (definition 3.2).”
§ Quantum non-Markovianity: characterization, quantification and detection > 4.3. The RHP measure
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“For Markovian evolution, γ -(t) 0, γ z (t) 0 for all t and g(t) = f (t) = 0.”
§ Quantum non-Markovianity: characterization, quantification and detection > 4.4. Decay rates measures
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In the paper
“Therefore, if σ (ρ 1 , ρ 2 , t) > 0 for some t, then γ (t) < 0, and the dynamics is non-Markovian.”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.1.1. Trace distance and the BLP quantifier.
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“In this equation, only D I ND(d) is a degree of non-Markovianity as defined in this work.The other quantities are zero for non-Markovian but k-divisible (k < d) dynamics.”
§ Quantum non-Markovianity: characterization, quantification and detection > 4.5. Hierarchical k-divisibility degrees
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“In other words, this approach does not distinguish between non-Markovian and Markovian equations of motion with time-dependent coefficients.Moreover, this problem persists in the classical limit.”
§ Quantum non-Markovianity: characterization, quantification and detection > 3.4. Comparison with other definitions of quantum Markovianity
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Hence, if the fidelity decreases at some time t, then γ (t) < 0 and the dynamics is non-Markovian.”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.1.2. Fidelity.
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“As commented in section 3.3.1,this phenomenon denotes the existence of memory effects in the dynamics, as an increase of information at time t + with respect to t suggests that the system is 'remembering' its original state at t + .”
§ Quantum non-Markovianity: characterization, quantification and detection > 4.2. Optimization of the Helstrom matrix norm
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In the paper
“Despite the conceptually clear meaning of D I NM(g) , it suffers from an important drawback, as it is very hard to compute in practice because of the involved optimization process.In fact, note that the set of Markovian maps M is non-convex [96], which makes the problem computationally intractable as the dimension of the system grows.”
§ Quantum non-Markovianity: characterization, quantification and detection > 4.1. Geometric measures
What the sources say
- Not checked — no full text of the sourcecites [96] source
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In the paper
“Note that, in general [see an exception in the point (iii) below], we need to know the complete dynamical map {E (t,t 0 ) , t t 0 } to compute the function g(t).The standard way to obtain it is resorting to process tomography.”
§ Quantum non-Markovianity: characterization, quantification and detection > 4.3. The RHP measure
What the sources say
- Not matchedcites [14] Nielsen M A, 2000, Quantum Computation and Quantum Information
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaindirect- Asserted: passes — The authors put it forward as true.
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- Not matched
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In the paper
“This dynamics describes the process where the nondiagonal elements (coherences) of ρ change the sign with a probability p(t), and with a probability 1-p(t) the qubit remains in the same state ρ.”
§ Quantum non-Markovianity: characterization, quantification and detection > 4.3. The RHP measure
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“It should be noted however, that this definition does not coincide with the definition adopted in this review and, in our view, suffers from certain drawbacks.The semigroup law equation ( 50) is just a particular case of the two-parameter composition law equation ( 29), which encompasses the case of time-inhomogeneous Markovian processes.”
§ Quantum non-Markovianity: characterization, quantification and detection > 3.4. Comparison with other definitions of quantum Markovianity
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“As in the case of geometric measures, the main drawback of the quantity N I H is the difficult optimization process, which makes this measure rather impractical.”
§ Quantum non-Markovianity: characterization, quantification and detection > 4.2. Optimization of the Helstrom matrix norm
plaindirect- Asserted: passes — The authors put it forward as true.
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Nevertheless, note that the quantum Fisher information provides just a lower bound to the error on θ , and in fact there are cases where this bound is not achievable.”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.1.4. Quantum Fisher information.
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Going back to the problem of witnessing non-Markovianity, the quantum Fisher information is also monotonically decreasing under Markovian dynamics, as it cannot increase under completely positive maps.”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.1.4. Quantum Fisher information.
plaindirect- Asserted: passes — The authors put it forward as true.
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The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“This is a purification of the maximally mixed state, which is the one that solves the optimization problem in equations (137) and ( 138) for this model as commented in example 5.5.”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.2.3. Quantum discord.
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“Therefore, the quantum relative entropy between any two states is monotonically decreasing with time in a Markovian process, and any increment of it at some time instant reveals the non-Markovian character of the dynamics.”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.1.3. Quantum relative entropies.
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“The quantum coherent information is monotonic under completely positive maps 7 : I c (ρ, E 2 E 1 ) I c (ρ, E 1 ),(136) and the same equation is satisfied for I (ρ, E).”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.1.5. Capacity measures.
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“Quantum Markovian processes have been defined by taking the divisibility approach, which allows us to circumvent the problem of constructing a hierarchy of probabilities in quantum mechanics.”
§ Quantum non-Markovianity: characterization, quantification and detection > 6. Conclusion and outlook
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Since 0 R(t) 1, everything multiplying γ (t) in the above equation is negative.Hence, an increment in the quantum relative entropy at some t implies γ (t) < 0 and non-Markovianity.”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.1.3. Quantum relative entropies.
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“Note, however, that the existence of quantitative connections between quantum discord and completely positive maps remains controversial [237][238][239][240][241][242][243].”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.2.3. Quantum discord.
What the sources say
- Not matchedcites [237][238][239][240][241][242][243]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The entanglement assisted capacity C ea sets a bound on the amount of classical information that can be transmitted along the dynamical process described by E (t,t 0 ) when the sender at t 0 and the receiver at t are allowed to share an unlimited amount of entanglement.Similarly, the capacity Q provides the limit to the rate at which quantum information can be reliably sent by the quantum process”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.1.5. Capacity measures.
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In addition, any dynamics that is Markovian according to the semigroup definition is also Markovian according to the divisibility definition, and hence Markovian according to the BLP definition.”
§ Quantum non-Markovianity: characterization, quantification and detection > 6. Conclusion and outlook
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“The divisibility definition is essentially equivalent to the algebraic one (see main text).”
§ Quantum non-Markovianity: characterization, quantification and detection > 6. Conclusion and outlook
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“In [208] Bylicka et al have suggested the use of capacity measures to detect non-Markovianity.”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.1.5. Capacity measures.
What the sources say
- Not matchedcites [208] Hao X, 2013, J. Phys. A: Math. Theor.
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
- Not matched
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“and I QFI (t) > 0 for some t denotes γ (t) < 0 and non-Markovianity.”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.1.4. Quantum Fisher information.
plaincorrelational- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Quantum Markovian processes have been defined by taking the divisibility approach, which allows us to circumvent the problem of constructing a hierarchy of probabilities in quantum mechanics.”
§ Quantum non-Markovianity: characterization, quantification and detection > 6. Conclusion and outlook
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
In the paper
“Recall that the quantum discord between two quantum systems is a non-symmetric measure of correlations, so that it is not the same to measure the quantum discord between S and A, with respect to A, as with respect to S.”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.2.3. Quantum discord.
What the sources say
- Not matchedcites [228][229][230]
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
- Not matched
The paper relies on this without support.
A classical random variable can be sampled without changing its later statistics, while measuring a quantum system disturbs its state and changes later outcomes.
“The problem arises because we can sample a classical random variable without affecting its posterior statistics; however, in order to 'sample' a quantum system, we need to perform measurements, and these measurements disturb the state of the system and affect the subsequent outcomes.”
The authors state this as an assumption.
A definition of Markovianity ought to be independent of the measurement scheme used to verify it.
“This is very inconvenient as the definition of Markovianity should be independent of what is required to verify it.”
The authors state this as an assumption.
The inverse E^{-1}(t1,t0) exists, so that the intermediate map E(t,t1) is well defined by equation (65).
“At the moment, we shall assume that E -1 (t 1 ,t 0 ) does exist (we will come back to this point later on), so that E (t,t 1 ) is well defined by equation (65).”
The authors state this as an assumption.
In the collisional model, the environment state ρ_E is the same for every collision.
“where ρ E is the state of the environment assumed to be the same for every collision”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The check wasn’t confident either way, so it gives no verdict — never disputed.
In the citing paper
“Recent work has demonstrated that the retention of quantum dynamics in biological systems is intricately connected with environmental fluctuations taking place at biologically relevant length and time scales [51–55].”
From The origins of quantum biology (2018) · cites this paper as [55]
Experimental — accuracy not yet measured
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Quantum non-Markovianity: characterization, quantification and detection > 1. Introduction
“Section 4 gives a detailed review of different measures of non-Markovianity and section 4 describes different approaches in order to construct witnesses able to detect non-Markovian dynamics.”
§ Quantum non-Markovianity: characterization, quantification and detection > 4. Measures of quantum non-Markovianity
“Thus, the present section is devoted to the quantification problem whereas the detection of non-Markovian dynamics by witnesses is left to section 5.”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Quantum non-Markovianity: characterization, quantification and detection > 4.4. Decay rates measures
“Since Markovian dynamics is characterized by generators with the form of equation (32), in [127] Hall, Cresser, Li and Andersson proposed a measure of non-Markovianity focused on properties of the generator.”
§ Quantum non-Markovianity: characterization, quantification and detection > References
“127. Usha Devi A R, Rajagopal A K, Sudha and Rendell R W 2012 J. Quantum Inform. Sci. 2 47-54 128. Hall M J W, Cresser J D, Li L and Andersson E 2014 Phys. Rev. A 89 042120”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Quantum non-Markovianity: characterization, quantification and detection > 5.1.4. Quantum Fisher information.
“F(ρ θ ) := 4 lim δθ→0 D B (ρ θ , ρ θ +δθ ) δθ 2 . (121”
§ Quantum non-Markovianity: characterization, quantification and detection > 5.1.4. Quantum Fisher information.
“D 2 B (ρ θ , ρ θ +δθ ) = 1 4 F(ρ θ )(δθ ) 2 + O[(δθ ) 3 ],(120)”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Quantum non-Markovianity: characterization, quantification and detection > 1. Introduction
“Section 4 gives a detailed review of different measures of non-Markovianity and section 4 describes different approaches in order to construct witnesses able to detect non-Markovian dynamics.”
§ Quantum non-Markovianity: characterization, quantification and detection > 5. Witnesses of Quantum non-Markovianity
“## 5. Witnesses of Quantum non-Markovianity”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Quantum non-Markovianity: characterization, quantification and detection > 4.3. The RHP measure
“A computationally simpler measure of non-Markovianity was introduced in [106] by Rivas, Huelga and Plenio.”
§ Quantum non-Markovianity: characterization, quantification and detection > References
“106. Liu B-H, Wißmann S, Hu X-M, Zhang C, Huang Y-F, Li C-F, Guo G-C, Karlsson A, Piilo J and Breuer H-P 2014 Locality, universality of quantum memory effects (arXiv:1403.4261) 107. Rivas A, Huelga S F and Plenio M B 2010 Phys. Rev. Lett. 105 050403”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Quantum non-Markovianity: characterization, quantification and detection > Theorem 3.5 (Stinespring [46]).
“## Theorem 3.5 (Stinespring [46]).”
§ Quantum non-Markovianity: characterization, quantification and detection > References
“46. Ziman M and Bužek V 2005 Phys. Rev. A 72 022110 47. Stinespring W F 1955 Proc. Am. Math. Soc. 6 211”
Citations
Cites
Horodecki, R., Horodecki, P., Horodecki, M., & Horodecki, K. (2009) · Reviews of Modern Physics · doi:10.1103/revmodphys.81.865 · Cited by 10,118
Horn, R. A., & Johnson, C. R. (1991) · Cambridge University Press eBooks · doi:10.1017/cbo9780511840371 · Cited by 9,324
On the generators of quantum dynamical semigroups
Lindblad, G. (1976) · Communications in Mathematical Physics · doi:10.1007/bf01608499 · Cited by 7,548
Completely positive dynamical semigroups of N -level systems
Gorini, V., Kossakowski, A., & Sudarshan, E. C. G. (1976) · Journal of Mathematical Physics · doi:10.1063/1.522979 · Cited by 3,998
Quantum Discord: A Measure of the Quantumness of Correlations
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Cited by
Sorted by citations
Colloquium : Quantum coherence as a resource
Streltsov, A., Adesso, G., & Plenio, M. B. (2017) · Reviews of Modern Physics · doi:10.1103/revmodphys.89.041003 · Cited by 1,878
Self-citationColloquium : Non-Markovian dynamics in open quantum systems
Breuer, H., Laine, E.-M., Piilo, J., & Vacchini, B. (2016) · Reviews of Modern Physics · doi:10.1103/revmodphys.88.021002 · Cited by 1,459
“For a review on recent results on quantum non-Markovianity see Rivas et al. (2014).”
BackgroundInfluentialChitambar, E., & Gour, G. (2019) · Reviews of Modern Physics · doi:10.1103/revmodphys.91.025001 · Cited by 1,442
“…(Bartlett et al., 2007), quantum thermodynamics (Goold et al., 2016; Gour et al., 2015b), coherence (Streltsov et al., 2016), nonlocality (Brunner et al., 2014), non-Gaussianity (Weedbrook et al., 2012a), non-Markovianity (Rivas et al., 2014), and quantum correlations (Modi et al., 2012)).”
BackgroundDynamics of non-Markovian open quantum systems
de Vega, I., & Alonso, D. (2017) · Reviews of Modern Physics · doi:10.1103/revmodphys.89.015001 · Cited by 1,294
Quantum Fisher information matrix and multiparameter estimation
Liu, J., Yuan, H., Lu, X.-M., & Wang, X. (2020) · Journal of Physics A: Mathematical and Theoretical · doi:10.1088/1751-8121/ab5d4d · Cited by 802
Metadata
- Authors
- Rivas, Á., Huelga, S. F., & Plenio, M. B.
- Affiliations
- Universidad Complutense de Madrid (ES)
- Universität Ulm (DE)
- ORCID
- Plenio, Martin B
- Venue
- Reports on Progress in Physics
- Publisher
- IOP Publishing
- Published
- 2014-08-22
- Volume
- 77
- Issue
- 9
- Pages
- 094001
- ISSN
- 0034-4885 (ISSN-L)
- 1361-6633
- OpenAlex
- W2031255223
- Semantic Scholar
- e26ce9e5754399e0a1752aedeb6663b48376f572
- Cited by
- 1,035
- References
- 206
- Citations by year
- 2026: 96 · 2025: 103 · 2024: 89 · 2023: 69 · 2022: 76 · 2021: 107 · 2020: 107 · 2019: 94 · 2018: 79 · 2017: 83 · 2016: 63 · 2015: 58 · 2014: 11
- FWCI
- 35.62
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- Locations
- Reports on Progress in Physics (publishedVersion)
- arXiv (Cornell University) (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Advanced Thermodynamics and Statistical Mechanics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Keywords
- quantum non-Markovianity, open quantum systems, divisibility, memory effects, quantum dynamics
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