Quantum non-Markovianity: characterization, quantification and detection

May combine two claims.

The exact dynamics of an open quantum system is essentially non-Markovian, even though many commonly studied models yield Markovian master equations.

“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].”

• cites [22] Davies E B, 1976, Quantum Theory of Open SystemsNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [24] Breuer H-P, 2002, The Theory of Open Quantum Systems sourcePartly supported

  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)

  Experimental — accuracy not yet measured
• cites [26] Rivas A, 2011, Open Quantum Systems. An Introduction sourceNot checked — no full text of the source
• cites [42] sourceNot checked — no full text of the source

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Completely positive maps are easier to characterize than merely positive maps and have a Kraus representation.

“Complete positive maps are much easier to characterize than maps that are merely positive [19,20]; they admit the so-called Kraus representation,”

• cites [19] sourceSupported
• cites [20] sourceNot checked — no full text of the source

Probably not a substantial claim: may not pass S3 (load-bearing).

Chruściński and Maniscalco proposed a hierarchical way to assess non-Markovianity based on k-divisibility.

“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.”

• cites [129] sourceCould not verify

  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)

  Experimental — accuracy not yet measured

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Chruściński and Maniscalco propose calling dynamics maximally non-Markovian when D^I_ND(1)=1, which implies D^I_ND(2)=...=D^I_ND(d)=1.

“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.”

• cites [129] sourceCould not verify

  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)

  Experimental — accuracy not yet measured

May combine two claims.

It can be proven that f(t)=d^2 g(t), so N^I_γ = d^2 N^I_RHP.

“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”

• cites [127] sourceCould not verify

  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)

  Experimental — accuracy not yet measured

May combine two claims.

Combining the Born-Markov approximation with the secular approximation yields a valid quantum dynamical semigroup, so the resulting dynamics can be called Markovian.

“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.”

• cites [24] Breuer H-P, 2002, The Theory of Open Quantum Systems sourceSupported
• cites [25] Alicki R, 2007, Quantum Dynamical Semigroups and Applications sourceNot checked — no full text of the source
• cites [26] Rivas A, 2011, Open Quantum Systems. An Introduction sourceNot checked — no full text of the source
• cites [39] sourceNot checked — no full text of the source
• cites [41] Gardiner C W, 2004, Quantum NoiseNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

Since E(t,t0) is completely positive for any final time t, N^+_k( ̃,I) ≥ |N^-_k( ̃,I)|, so D^I_ND(k) lies in [0,1] for all k.

“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.”

• cites [129] sourceCould not verify

  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)

  Experimental — accuracy not yet measured

May combine two claims.

Born-Markov master equations with time-independent coefficients do not always define a valid quantum dynamical semigroup because they break complete positivity, so they should not be called Markovian in the strict sense.

“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”

• cites [84] Whitney R S, 2008, J. Phys. A: Math. Theor. sourceSupported
• cites [85] sourceNot checked — no full text of the source

May combine two claims.

Deciding whether a quantum channel is Markovian is very hard in the complexity-theory sense, though it has been analyzed for small-sized systems.

“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].”

• cites [98] sourceCould not verify

  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)

  Experimental — accuracy not yet measured
• cites [99] sourceNot checked — no full text of the source
• cites [77] Ji U C, 2010, Commun. Stoch. Anal.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [100] sourceSupported

Probably not a substantial claim: may not pass S3 (load-bearing).

Hall, Cresser, Li and Andersson proposed a non-Markovianity measure focused on properties of the generator.

“in [127] Hall, Cresser, Li and Andersson proposed a measure of non-Markovianity focused on properties of the generator.”

• cites [127] sourceCould not verify

  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)

  Experimental — accuracy not yet measured

May combine two claims.

The trace norm of the Choi-Jamiołkowski matrix of E(t,t1) equals 1 if E(t,t1) is completely positive and is greater than 1 otherwise.

“E (t,t 1 ) ⊗ 1 1 (| |) 1 = 1 iff E (t,t 1 ) is CP, > 1 otherwise.”

• cites [19] sourcePartly supported

  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)

  Experimental — accuracy not yet measured
• cites [20] sourceNot checked — no full text of the source

Probably not a substantial claim: may not pass S3 (load-bearing).

Non-Markovian processes have been shown useful to prepare steady entangled states.

“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].”

• cites [122] sourceCould not verify

  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)

  Experimental — accuracy not yet measured

May combine two claims.

The state (140) belongs to the class of X-states, for which the optimization problem in equation (152) can be efficiently solved.

“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].”

• cites [230] sourceCould not verify

  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)

  Experimental — accuracy not yet measured
• cites [245] sourceCould not verify

  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)

  Experimental — accuracy not yet measured

Probably not a substantial claim: may not pass S3 (load-bearing).

Lorenzo et al suggested a Bloch volume witness of non-Markovianity.

“Another interesting proposed way to witness non-Markovianity was suggested by Lorenzo et al in [211].”

• cites [211] sourceCould not verify

  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)

  Experimental — accuracy not yet measured

Probably not a substantial claim: may not pass S3 (load-bearing).

Non-Markovian processes have been shown useful to enhance the achievable resolution in quantum metrology.

“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].”

• cites [259] Borrelli M, 2014 sourceCould not verify

  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)

  Experimental — accuracy not yet measured
• cites [260] sourceSupported

May combine two claims.

The Bloch volume witness is not sensitive to non-Markovianity encoded in the affine vector c(t,t0), and it only detects non-Markovian dynamics such that Tr[M(t,t0)]>0.

“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].”

• cites [127] sourceCould not verify

  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)

  Experimental — accuracy not yet measured

May combine two claims.

|det M(t,t0)| describes the change in volume of accessible states, so Markovian evolutions reduce or leave invariant that volume.

“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.”

• cites [211] sourceCould not verify

  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)

  Experimental — accuracy not yet measured

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

Links have already been reported between non-Markovianity and criticality/phase transitions, the Loschmidt echo, symmetry breaking, and Zeno/anti-Zeno effects.

“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.”

• cites [175] sourceSupported
• cites [256] de Vicente J I, 2014 sourceCould not verify

  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)

  Experimental — accuracy not yet measured
• cites [257] sourcePartly supported

  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)

  Experimental — accuracy not yet measured
• cites [258] sourceSupported
• cites [174] sourceCould not verify

  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)

  Experimental — accuracy not yet measured
• cites [166] sourceCould not verify

  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)

  Experimental — accuracy not yet measured

May combine two claims.

For the pure dephasing master equation, the integrated map gives R(t)=exp(-2∫_0^t γ(s)ds) and 0≤R(t)≤1.

“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.”

Probably not a substantial claim: may not pass S3 (load-bearing).

For that map, when p(t)=1/2, the coherences vanish completely.

“Note that for p(t) = 1/2, the coherences vanish completely.”

May combine two claims.

For that map, g(t) is zero when p'(t)/(1-2p(t)) is nonnegative and equals -2p'(t)/(1-2p(t)) when that ratio is negative.

“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”

May combine two claims.

For the pure dephasing model and the two initial states, fidelity F=1-R^2(t) and dF/dt=4γ(t)R^2(t).

“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”

May combine two claims.

For initial states ρ1=1/2[[1,1],[1,1]] and ρ2=1/2[[1,-1],[-1,1]], the trace distance is R(t), and σ=-2γ(t)R(t).

“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”

May combine two claims.

For the master equation dρ/dt=γ(t)(σ_z ρσ_z -ρ) with γ(t) periodic on I, the Choi-Jamiołkowski measure gives D_NM(RHP)=0.758 for γ(t)=sin(t) and 0.803 for γ(t)=tan(t) on [0,2π].

“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.”

May combine two claims.

The eigenvalues γ_j(t) are independent of the basis, so the generator's diagonal form is unique up to degeneracy.

“Note that since the eigenvalues γ j (t) are independent of the basis, this form is unique (up to degeneracy).”

May combine two claims.

For a master equation, I_QFI(t)=sum_k γ_k(t)J_k(t), with J_k(t)≤0.

“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”

May combine two claims.

For that model, C_ea and Q have the same derivative, which can be positive only for non-Markovian dynamics with γ(t)<0.

“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.”

May combine two claims.

For the phase-shifted family under pure dephasing, F(ρθ)=R^2(t), and I_QFI(t)=γ(t)J(t) with J(t)=-4R^2(t).

“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”

May combine two claims.

Quantum evolution maps are linear, and there is no experimental evidence against this.

“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]).”

• cites [15][16][17]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Classified as probably not a substantial claim.

Markov processes satisfy the Chapman-Kolmogorov equation.

“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.”

Probably not a substantial claim: may not pass S3 (load-bearing).

When the prior probability q is zero or one, the probability of wrongly identifying the true state is zero.

“Thus, note that when q = 0 or q = 1, we immediately obtain zero probability of wrongly identifying the true state.”

Classified as probably not a substantial claim.

For a Markov process, the conditional expectation of a future value given all past values equals the conditional expectation given only the immediately preceding value.

“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,”

May combine two claims. The quote only partly states this claim.

For P-divisible dynamics, only positivity of E(t2,t1) is required rather than complete positivity, and the non-Markovian waiting advantage does not occur.

“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.”

Classified as probably not a substantial claim.

Divisibility can be defined for quantum systems without explicitly referring to measurement processes.

“a very nice property is that divisibility may be defined in the quantum case without any explicit mention of measurement processes.”

May combine two claims. The quote only partly states this claim.

For non-Markovian processes, if the initial conditional probability matrix is invertible, transition matrices can be expressed through well-defined quantities and satisfy composition and normalization, but positivity may fail, preventing their interpretation as conditional probabilities.

“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.”

May combine two claims.

For a Markov process, the transition matrices satisfy normalization, positivity and the composition law.

“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 )”

Probably not a substantial claim: may not pass S3 (load-bearing).

In classical two-distribution discrimination, the minimum average probability of error is given by one minus the L1 norm of w divided by two.

“Then the minimum (averaged) probability to give the wrong answer turns out to be P min (fail) = 1 -w(x) 1 2 , (18 )”

Classified as probably not a substantial claim.

When a system is entangled with another system, preserving positivity of density matrices requires complete positivity rather than mere positivity.

“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].”

• cites [14,[21][22][23][24][25][26]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

The memoryless property of classical Markov processes is also present in quantum Markovian processes, although it is not immediately visible in the divisibility definition.

“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.”

May combine two claims.

A closed quantum system evolves by a two-parameter family of unitary operators satisfying U(t2,t0)=U(t2,t1)U(t1,t0), so its evolution is trivially Markovian.

“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.”

• cites [38] Galindo A, 1990, Quantum MechanicsNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

An apparent lack of memory in the unbiased two-state discrimination case does not imply a general memoryless property; memory effects can appear in the general biased case and when ancillary systems are included.

“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).”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The 1-divisible processes are P-divisible, and 0-divisible processes are those where E(t2,t1) is not positive for some t1 and t2≥t1.

“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 .”

May combine two claims.

Fidelity is monotonic under completely positive maps, with equality if and only if the map is unitary.

“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].”

• cites [189] sourceNot checked — no full text of the source

May combine two claims.

As k increases, the optimization dimension increases, so 0 ≤ D^I_ND(1) ≤ ... ≤ D^I_ND(d) ≤ 1.

“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)”

Probably not a substantial claim: may not pass S3 (load-bearing).

For that model, g(t)=f(t) because d=2.

“Therefore, g(t) = f (t), as expected in this case since d = 2.”

May combine two claims.

Therefore fidelity is monotonically increasing for Markovian evolutions and can witness non-Markovian behavior.

“Thus, the fidelity is monotonically increasing for Markovian evolutions, and therefore it may be used to witness non-Markovian behavior.”

Classified as probably not a substantial claim. May combine two claims.

The geometric measure of non-Markovianity lies between 0 and 1 and is positive exactly when the process is non-Markovian, so it is a degree of non-Markovianity.

“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.”

Classified as probably not a substantial claim.

If the system dimension is d, a k-divisible process with k≥d is divisible or Markovian.

“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).”

Classified as probably not a substantial claim.

If the intermediate maps E(t,t1) are completely positive for all t1, the time evolution is Markovian.

“If these maps are completely positive (CP) for all t 1 , the time evolution is Markovian (definition 3.2).”

May combine two claims.

For Markovian evolution in that model, γ_-(t)≥0 and γ_z(t)≥0 for all t, and g(t)=f(t)=0.

“For Markovian evolution, γ -(t) 0, γ z (t) 0 for all t and g(t) = f (t) = 0.”

May combine two claims.

If σ>0 at some time, then γ(t)<0 and the dynamics is non-Markovian.

“Therefore, if σ (ρ 1 , ρ 2 , t) > 0 for some t, then γ (t) < 0, and the dynamics is non-Markovian.”

Classified as probably not a substantial claim. May combine two claims.

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.

“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.”

May combine two claims.

The semigroup definition fails to distinguish non-Markovian from Markovian equations of motion with time-dependent coefficients, and this problem persists in the classical limit.

“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.”

May combine two claims.

If fidelity decreases at some time, then γ(t)<0 and the dynamics is non-Markovian.

“Hence, if the fidelity decreases at some time t, then γ (t) < 0 and the dynamics is non-Markovian.”

May combine two claims.

An increase of information at a later time relative to an earlier time indicates memory effects, suggesting that the system is remembering its original state.

“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 + .”

May combine two claims.

The geometric measure is very hard to compute because of the optimization over Markovian maps, and the set of Markovian maps is non-convex, making the problem computationally intractable as system dimension grows.

“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.”

• cites [96] sourceNot checked — no full text of the source

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

In general, computing g(t) requires knowing the complete dynamical map {E(t,t0)}, which is obtained experimentally by process tomography.

“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.”

• cites [14] Nielsen M A, 2000, Quantum Computation and Quantum InformationNot matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

The qubit map E(t,0)(ρ)=[1-p(t)]ρ+p(t)σ_z ρσ_z describes a process in which coherences change sign with probability p(t) and the state remains unchanged with probability 1-p(t).

“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.”

May combine two claims.

The semigroup definition of quantum Markovianity does not coincide with the divisibility-based definition and is only a special case of the two-parameter composition law.

“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.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The main drawback of N_I^H is the difficult optimization process, which makes the measure impractical.

“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.”

Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

The quantum Fisher information provides only a lower bound to the error on θ, and there are cases where this bound is not achievable.

“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.”

May combine two claims.

Quantum Fisher information is monotonically decreasing under Markovian dynamics because it cannot increase under completely positive maps.

“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.”

May combine two claims.

The maximally entangled state is a purification of the maximally mixed state, which solves the optimization problem in equations (137) and (138) for this model.

“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.”

May combine two claims.

Therefore, in a Markovian process the relative entropy between any two states decreases monotonically with time, and any increase reveals non-Markovianity.

“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.”

May combine two claims.

The quantum coherent information is monotonic under completely positive maps, and the same holds for mutual information I(ρ,E).

“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).”

Classified as probably not a substantial claim.

The paper defines quantum Markovian processes through divisibility.

“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.”

May combine two claims.

Since 0≤R(t)≤1, the factor multiplying γ(t) is negative, so an increase in relative entropy implies γ(t)<0 and non-Markovianity.

“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.”

Probably not a substantial claim: may not pass S3 (load-bearing).

Whether there are quantitative links between discord and completely positive maps is still debated.

“Note, however, that the existence of quantitative connections between quantum discord and completely positive maps remains controversial [237][238][239][240][241][242][243].”

• cites [237][238][239][240][241][242][243]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Classified as probably not a substantial claim. May combine two claims.

The entanglement-assisted capacity C_ea bounds the classical information transmittable through the process when sender and receiver share unlimited entanglement; Q bounds the reliable quantum information rate for a single use.

“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”

May combine two claims.

If a dynamics is Markovian under the semigroup definition, it is also Markovian under the divisibility definition and therefore under the BLP definition.

“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.”

Probably not a substantial claim: may not pass S3 (load-bearing).

The divisibility-based definition of quantum Markovianity is essentially the same as the algebraic definition.

“The divisibility definition is essentially equivalent to the algebraic one (see main text).”

Probably not a substantial claim: may not pass S3 (load-bearing).

Bylicka et al suggested using capacity measures to detect non-Markovianity.

“In [208] Bylicka et al have suggested the use of capacity measures to detect non-Markovianity.”

• cites [208] Hao X, 2013, J. Phys. A: Math. Theor.Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

May combine two claims.

I_QFI(t)>0 for some t denotes γ(t)<0 and non-Markovianity.

“and I QFI (t) > 0 for some t denotes γ (t) < 0 and non-Markovianity.”

Classified as probably not a substantial claim.

Using divisibility avoids the need to build a hierarchy of probabilities in quantum mechanics.

“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.”

Classified as probably not a substantial claim.

The discord between two quantum systems is an asymmetric correlation measure, so the discord of S with A relative to A is not the same as relative to S.

“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.”

• cites [228][229][230]Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.