On algebraic analysis of Baker-Campbell-Hausdorff formula for Quantum Control and Quantum Speed Limit

Go Kato; Koji Maruyama; Masaki Owari

REVIEW 3 major objections 5 minor 38 references

On algebraic analysis of Baker-Campbell-Hausdorff formula for Quantum Control and Quantum Speed Limit

T0 review · 3 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A distance derived from the Baker-Campbell-Hausdorff formula gives a tight, algebra-aware lower bound on quantum control time.

desk verdict A genuinely new Lie-algebra-aware control-time bound, but the whole result rests on an intricate technical estimate in Appendix F that no referee should take on faith. read the letter →

arxiv 2411.13155 v1 pith:UIH2ATXM submitted 2024-11-20 quant-ph

Go Kato , Masaki Owari , Koji Maruyama This is my paper · ORCID

classification quant-ph

keywords quantumcontrolspeedlimitBaker-Campbell-HausdorffformuladynamicalLiealgebratimelowerboundFrobeniusnormunitarymetricgroupgeometry

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes a lower bound on the time needed to implement a target unitary operation with a given set of control Hamiltonians. The bound is: for any control $H(t)$ that generates $U_{\rm target}$ at time $T$, there is a Hermitian generator $C[T]$ lying in the dynamical Lie algebra $L(\{iH_m\})$ of the available controls, with $e^{-iC[T]}=U_{\rm target}$ and $\|C[T]\|_F \le \int_0^T \|H(t)\|_F\,dt$. Because the Frobenius norm of $H(t)$ is at most its maximum, this yields $T \ge c/\max_t\|H(t)\|_F$, where $c$ is the smallest Frobenius norm of any generator of $U_{\rm target}$ inside $L$. The bound is tight when the control is a single time-scaled generator, and as far as the authors can compare, it is never worse than the Mandelstam-Tamm bound expressed through the Choi representation, and it can be strictly better. A reader should care because standard speed limits measure state-space distance and can be loose when the available Hamiltonians cannot realize the geometric 'shortest path'; this bound carries the algebraic structure of the controls themselves.

What carries the argument

The load-bearing object is the metric $d$ on the unitaries generated by the dynamical Lie algebra $L(\{iH_m\})$, defined by the minimal Frobenius norm of a logarithm in the algebra. It is a genuine metric because of Lemma 1, whose triangle inequality is the engine of every subsequent bound. The proof of Lemma 1 uses the Baker-Campbell-Hausdorff formula as a local averaging tool: the two evolutions are split into many small pieces, adjacent pieces are combined with the BCH product, and an iterative averaging process forces the pieces to coalesce into one generator while keeping the total product equal to the original one; the key estimate $\|M(A,B)\|_F^2 \le 2\|A\|_F^2+2\|B\|_F^2-\|A-B\|_F^2$ controls each merging step. This machinery converts a local BCH identity into a global operator triangle inequality, and then Theorem 1 follows by discretizing $H(t)$ and applying Lemma 1 to successive short-time evolutions.

What would settle it

Numerically search pairs of anti-Hermitian matrices $A,B$ with $\|A\|_F+\|B\|_F$ around $2\pi$ and minimize $\|C\|_F$ subject to $C\in L(\{A,B\})$ and $e^C=e^A e^B$; a single pair whose minimum exceeds $\|A\|_F+\|B\|_F$ would falsify Lemma 1 and collapse Theorem 1.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1: whenever a time-dependent Hamiltonian $H(t)$ built from a fixed set $H_m$ produces a unitary $U_{\rm target}$ at time $T$, the target can also be generated by a single anti-Hermitian element $iC[T]$ of the dynamical Lie algebra $L(\{iH_m\})$, and the Frobenius norm of $C[T]$ is bounded by the time-integrated Frobenius norm of $H(t)$. The proof runs through a new metric $d(U_1,U_2)=\min\{\|C\|_F: e^C=U_1U_2^{-1},\, C\in L\}$ on the unitaries generated by $L$. The key step is Lemma 1, an operator triangle inequality for the metric: for anti-Hermitian $A,B$ there is $C\in L(\{A,B\})$ with $e^C=e^A e^B$ and $\|C\|_F\le \|A\|_F+\|B\|_F$, proved by fine subdivision of the evolutions and repeated BCH synthesis. The metric respects the curved, algebra-constrained structure of operator space, so the resulting control-time lower bound does not rely on shortcuts that leave the reachable Lie group. In comparisons through the Choi representation the bound is at least as large as the Mandelstam-Tamm bound, and in a three-level example it is substantially closer to the actual minimal time.

Load-bearing premise

The proof rests on Lemma 1: for any two anti-Hermitian operators $A,B$, a single generator $C$ in the Lie algebra they generate can reproduce $e^A e^B$ with Frobenius norm no larger than $\|A\|_F+\|B\|_F$; if this operator triangle inequality ever fails, the main control-time bound collapses.

Editorial extensions

If this is right

For any target unitary reachable with controls $H_m$, the minimal control time satisfies $T \ge c/\max_t\|H(t)\|_F$, with $c$ the minimal Frobenius-norm generator in $L(\{iH_m\})$; equality is achieved for a single generator scaled in time, so the bound is tight.
The metric $d$ satisfies the same comparison inequality used in Nielsen-style complexity bounds: $d(U_A(T),U_B(T)) \le \int_0^T \|H_A(t)-H_B(t)\|_F\,dt$, so it can replace the norm-of-difference distance in gate-complexity arguments.
Because $\|U_A(T)-U_B(T)\|_F \le d(U_A(T),U_B(T))$, any lower bound obtained with $d$ is at least as strong as the corresponding bound from the norm-difference inequality, and strict improvements occur when the algebra restricts the available logarithms.
Expressed through the Choi representation, the bound is never below the Mandelstam-Tamm bound for the same operation, and in the paper's three-level example it is strictly closer to the true control time.
The bound refines the stabilizer-based bound of Lee et al.: for any $V$ commuting with all control fields, $T \ge d(U(T),VU(T)V^\dagger)/\|[H_0,V]\|_F \ge \|[U(T),V]\|_F/\|[H_0,V]\|_F$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The BCH-averaging construction likely extends to other unitarily invariant norms, which would make the control-time lower bound applicable when experimental constraints are set by operator-norm, trace-norm, or energy bounds rather than the Frobenius norm; the paper does not pursue this.
The metric $d$ is essentially the geodesic distance of a left-invariant metric determined by the algebra $L$, so time-optimal control could be read as a geodesic problem in a curved space whose curvature encodes the noncommutativity of the Hamiltonians; the authors do not draw this geodesic picture explicitly.
For few-qubit systems one could compute $c$ numerically for a target unitary and compare $c/\max\|H\|_F$ against the best controls found by optimal-control solvers; the paper's comparison is restricted to one three-level example, so a numerical scan would show how often the bound is tight.
Because $d$ satisfies the triangle inequality and is unitarily invariant, it can serve as a parameter-free complexity measure for quantum circuits built from a fixed set of Hamiltonians, potentially linking minimum control time to circuit depth; this connection is not made in the paper.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. First, this paper introduces a metric on the unitary group that depends explicitly on the dynamical Lie algebra of the available controls, and uses it to prove a lower bound on control time. In the Choi-representation comparison, that bound is provably at least as strong as the Mandelstam-Tamm bound, and the paper gives a concrete example where it is much tighter. Second, the entire program hangs on Lemma 1, the triangle inequality for this metric, whose proof is long, technical, and not something I could fully verify. The reader's CONDITIONAL verdict is the right one.

What is genuinely new here: the metric d and the BCH-based proof of its triangle inequality. The resulting bound is tight for a single-generator control, and the comparison with Mandelstam-Tamm is sound, not just numerically suggestive. The example showing the improvement near the identity is also helpful. The paper is honest about scope: it compares only with the Frobenius-norm version of other bounds, and it flags where the comparison is not direct.

The soft spot is exactly where the reader and the stress-test note point. Lemma 1 depends on Eq. (46), an inequality for the BCH series proved in Appendix F using trace identities and a positivity bound. If any sign in the h-coefficients is wrong, or if the trace identities do not extend to the full BCH expansion, the triangle inequality fails and Theorem 1 collapses. The averaging/subdivision argument in Appendix G is also delicate, especially the limit r -> infinity. I did not find an obvious error, and the paper itself is appropriately careful about the anti-Hermiticity requirement, but this is not a proof I would certify without independent checking. The comparison with Lee et al. is only carried out for the Frobenius norm, which is a real but minor limitation.

Who is this for? People working on quantum speed limits, quantum control, and geometric approaches to gate complexity. It deserves a serious referee, not a desk reject. The referee's main job is to verify Appendix F, especially the derivation of Eq. (46) and the convergence bounds. If that estimate survives, this is a solid contribution. If not, the main theorem collapses. I would bring it to a reading group and would cite it, but only with a 'proof as claimed, pending independent check of Appendix F' caveat.

Referee Report

3 major / 5 minor

Summary. This paper derives a lower bound on the time needed to implement a target unitary under a time-dependent Hamiltonian drawn from a fixed set of control Hamiltonians. The main result, Theorem 1, states that for any such target unitary U(target) there exists a Hermitian generator C[T] in the dynamical Lie algebra L({iH_m}) with exp(-iC[T]) = U(target) and ||C[T]||_F ≤ ∫_0^T ||H(t)||_F dt, which implies the control-time bound T ≥ c / max_t ||H(t)||_F, where c is the minimal Frobenius norm of a logarithm of U(target) in L. The proof rests on a BCH-based triangle inequality for a metric on unitaries (Lemma 1), proved by subdividing the evolution and applying a small-norm estimate (Eq. (46)) proved in Appendix F. The authors compare their bound with the Mandelstam-Tamm bound through the Choi representation and give a numerical example in which their bound is tighter.

Significance. If the proof is correct, the result is a genuinely Lie-algebra-aware quantum speed limit for unitary operations, giving a tight lower bound in simple cases and improving on state-based QSLs in the examples shown. The paper is self-contained, contains no fitted parameters, and makes a falsifiable quantitative claim (Eq. (10)) that can be tested on explicit control problems. The main ideas are novel in the quantum-control literature, even though the underlying Lie-group geometry is classical. The chief risk is that the central Lemma 1 is proved by a very long and intricate BCH analysis, and the comparison with the Mandelstam-Tamm bound contains a step that is not fully justified.

major comments (3)

[§III.A, Eq. (23)] The chain of inequalities leading to Corollary 2 uses the equality dev(C̃[T]) = dev(C[T]) without proof. From e^{-iC̃} = e^{-i∫ε} e^{-iC} it does not follow that C̃ - C is a scalar multiple of the identity; the logarithms can differ by integer multiples of 2π in individual eigenvalues. The equality is only valid if C̃ is chosen as C + (∫ε)I, but the paper does not show that the generator produced by Theorem 1 for H(t)+ε(t)I can be taken in that form. Since Corollary 2 and the subsequent claim that the present bound is tighter than the Mandelstam-Tamm bound depend on this step, the authors should add a lemma establishing the equivariance of the construction under adding f(t)I to the Hamiltonian, or otherwise justify the branch choice.
[Appendix F, Lemmas 5 and Eq. (F1)-(F12)] The proof of Eq. (46) is the load-bearing technical estimate, and it relies on the trace identities (F5)-(F9) and on rearranging infinite series. The paper should state explicitly why the rearrangements in Eq. (F4) are valid, i.e., why the absolute convergence of the f-, g-, and h-series in the operator norm justifies the term-by-term trace manipulation for anti-Hermitian A and B. In addition, the definition of Δ in Eq. (F1) must be read as Δ = (log 2 / 6) δ for Eq. (F12) to hold; if Δ = (log 2)/(6δ) were intended, Eq. (F12) would be false. Please clarify the notation and confirm the intended definition, since the subsequent positivity bound (F10)-(F11) depends on it.
[Appendix G, Eqs. (G1)-(G2)] The proof of Lemma 1 does not treat the case where A or B is zero. In that case the division numbers m_a or m_b in Eqs. (G1)-(G2) can be zero, which breaks the subsequent construction (e.g., the requirement that m_a be odd). Since Lemma 1 is stated for arbitrary anti-Hermitian A and B, the proof should handle the zero case separately or define m_a = max(1, ceil(...)). This is a local technical gap, but it occurs in the proof of the paper's central lemma.

minor comments (5)

[Throughout] There are numerous typographical errors, including 'Fuculty', 'opeartor', 'Appnendix', 'Phys. Rrev.', and 'Phys. Rew.'. The manuscript should be carefully proofread.
[Eq. (15)] The metric d is defined with a minimum, but the existence of the minimum is not discussed. For anti-Hermitian logarithms the set of admissible C with bounded norm is discrete, so the minimum is attained, but this should be stated explicitly, especially because the paper emphasizes non-compact examples in Appendix B.
[§II, after Eq. (8)] The sentence 'The LHS of Eq. (8) can be replaced with d(U(target)_T, I)' is confusingly written: the notation U(target)_T is not defined and the subscript seems to conflict with the control time T. Please clarify.
[Figure 1] The figure caption refers to line styles and colors ('red dot-dashed line', 'green dashed line') but the printed rendering may not preserve colors. Please ensure the figure is readable in grayscale, or add distinct markers.
[References] Several reference entries contain typos in journal names (e.g., 'Phys. Rrev.', 'Phys. Rew.') and inconsistent spellings such as 'Mandelstamm' vs. 'Mandelstam'. These should be corrected.

Circularity Check

0 steps flagged · score 1.0 of 10

No material circularity: the control-time bound is derived from a self-contained BCH-based triangle inequality, and the only self-citation is a peripheral discussion pointer.

full rationale

The paper's central chain is Theorem 1 -> Lemma 1 -> BCH formula plus the technical estimate Eq. (46). None of these steps assumes the conclusion. Theorem 1 asserts the existence of a generator C[T] in the dynamical Lie algebra with norm bounded by the integrated Hamiltonian norm; Eq. (10) then follows by taking c as the minimum norm of any such generator. This is a standard variational argument, not a fitted parameter renamed as a prediction: c is defined from the target unitary and the Lie algebra, and the inequality c <= ||C[T]|| is a consequence of the definition of minimum, not an input. The metric d defined in Eq. (15) is a definition, and the triangle inequality is proved in Lemma 1 using the BCH formula, a subdivision argument, and Eq. (46); it is not assumed. The comparison with Mandelstam-Tamm and Margolus-Levitin bounds is an external mathematical comparison, not a circular use of the theorem. Appendix D uses Theorem 1 in the proof of Lemma 3, but Theorem 1 itself is proved in Appendix H solely from Lemma 1, so this is a legitimate forward reference rather than a circular dependence. The only self-citation, Ref. [32] by the same authors, appears in Section V as a peripheral remark about algebra structure in limited controls and plays no role in the proof of the main inequalities. The skeptical concern about Eq. (46) is a correctness or verification risk, not a circularity, because the estimate is derived from trace identities and convergence bounds rather than assumed from the target result.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claim uses no fitted parameters. The proof relies on the BCH formula and standard functional analysis. No new physical entities are postulated.

assumptions (4)

standard math Baker-Campbell-Hausdorff series converges when ||A||_op + ||B||_op < log 2
Stated as Theorem 2 and used throughout the proof of Lemma 1.
domain assumption Control Hamiltonians are Hermitian, piecewise continuous, bounded, and their modulation functions are finite
Sec II, Eqs (3) and (4). Standard in quantum control.
domain assumption The dynamical Lie algebra L is well-defined and finite-dimensional
Sec II, Eq (5); all Hamiltonians act on a finite-dimensional Hilbert space.
standard math Frobenius norm is unitarily invariant and satisfies standard inequalities
Used throughout, e.g., Eq (46) and Appendix F.

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Pith. "Pith review of On algebraic analysis of Baker-Campbell-Hausdorff formula for Quantum Control and Quantum Speed Limit." pith.science (2026). https://pith.science/paper/UIH2ATXM

@misc{pith2026241113155,
  author       = {Pith},
  title        = {Pith review of: On algebraic analysis of Baker-Campbell-Hausdorff formula for Quantum Control and Quantum Speed Limit},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UIH2ATXM}},
  note         = {Machine review of arXiv:2411.13155}
}

read the original abstract

The necessary time required to control a many-body quantum system is a critically important issue for the future development of quantum technologies. However, it is generally quite difficult to analyze directly, since the time evolution operator acting on a quantum system is in the form of time-ordered exponential. In this work, we examine the Baker-Campbell-Hausdorff (BCH) formula in detail and show that a distance between unitaries can be introduced, allowing us to obtain a lower bound on the control time. We find that, as far as we can compare, this lower bound on control time is tighter (better) than the standard quantum speed limits. This is because this distance takes into account the algebraic structure induced by Hamiltonians through the BCH formula, reflecting the curved nature of operator space. Consequently, we can avoid estimates based on shortcuts through algebraically impossible paths, in contrast to geometric methods that estimate the control time solely by looking at the target state or unitary operator.

Figures

Figures reproduced from arXiv: 2411.13155 by the authors.

**Figure 2.** FIG. 2: Intuitive picture of the evolution of the sequence [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

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Works this paper leans on

38 extracted references · 35 canonical work pages

[1]

M. R. Frey, Quantum Inf. Process. 15, 3919 (2016)

work page 2016
[2]

Deﬀner and S

S. Deﬀner and S. Campbell, J. Phys. A: Math. Theor. 50, 453001 (2017)

2017
[3]

Mandelstam and I

L. Mandelstam and I. Tamm, J. Phys. 9, 249 (1985)

work page 1985
[4]

Margolus and L

N. Margolus and L. B. Levitin, Physica D 120, 188-195 (1998)

work page 1998
[5]

M. A. Nielsen, M. R. Dowling, M. Gu, and A. C. Doherty, Science 311, 1133-1135 (2006)

work page 2006
[6]

M. A. Nielsen, M. R. Dowling, M. Gu, and A. C. Doherty, Phys. Rev. A 73, 062323 (2006)

work page 2006
[7]

M. R. Dowling and M. A. Nielsen, Quantum Information & Computation 8, 861 (2008)

work page 2008
[8]

Arenz, B

C. Arenz, B. Russell1, D. Burgarth and H. Rabitz, New J. Phys. 19, 103015 (2017)

work page 2017

Show all 38 references

[9]

J. Lee, C. Arenz, H. Rabitz and B. Russell, New J. Phys. 20, 063002 (2018)

2018
[10]

Carlini, A

A. Carlini, A. Hosoya, T. Koike, and Y. Okudaira, Phys. Rev. Lett. 96, 060503 (2006)

2006
[11]

Carlini, A

A. Carlini, A. Hosoya, T. Koike, and Y. Okudaira, Phys. Rev. A 75, 042308 (2007)

2007
[12]

D’Alessandro, ”Introduction to Quantum Control and Dynamics” , (Taylor and Francis, Boca Raton, 2008)

D. D’Alessandro, ”Introduction to Quantum Control and Dynamics” , (Taylor and Francis, Boca Raton, 2008)

2008
[13]

Lloyd, A.J

S. Lloyd, A.J. Landahl, J.J.E. Slotine, Phys. Rev. A 69, 012305 (2004)

2004
[14]

Burgarth, K

D. Burgarth, K. Maruyama, M. Murphy, S. Montangero, T. C alarco, F. Nori and M.B. Plenio, Phys. Rev. A 81, 040303(R) (2010)

2010
[15]

Electron Spin Resonance (ESR) Based Quantum Computing

K. Maruyama and D. Burgarth, Gateway schemes of quantum control for spin networks in “Electron Spin Resonance (ESR) Based Quantum Computing”, Eds. T. Takui, L. Berliner a nd G. Hanson, Springer, New York, pp. 167-192 (2016)

2016
[16]

A. D. Boozer, Phys. Rev. A 85, 012317 (2012)

2012
[17]

P. M. Poggi, Phys. Rev. A 99, 042116 (2022)

2022
[18]

P. M. Poggi, Anales AF A 31 (1), 29-3899 (2019)

2019
[19]

Boscain, M

U. Boscain, M. Sigalotti, and D. Sugny, Phys. Rrev. X Quant. 2, 030203 (2021)

2021
[20]

Koike, Philos Trans A

T. Koike, Philos Trans A. 380, 20210273 (2022)

2022
[21]

Khaneja, R

N. Khaneja, R. Brockett, S. Glaser, Phys. Rew. A , 63, 032308 (2001)

2001
[22]

Khaneja, S

N. Khaneja, S. Glaser, R. Brockett, Phys. Rew. A , 65, 032301 (2001)

2001
[23]

Caneva, M

T. Caneva, M. Murphy, T. Calarco, R. Fazio, S. Montanger o, V. Giovannetti, and G. E. Santoro1, Phys. Rev. Lett. , 103, 240501 (2009)

2009
[24]

H. F. Chau, Quantum Inform. Comput. , 11 721 (2011)

2011
[25]

K.-Y. Lee, H. F. Chau, arXiv:1202.1899 (2012)

2012 arXiv
[26]

Russell, S

B. Russell, S. Stepney, J. Phys. A: Math. Theor. 48, 115303 (2015)

2015
[27]

Sevitz, N

S. Sevitz, N. Mirkin, D. A. Wisniacki, arXiv:2208.0436 2 (2022)

2022
[28]

Jozsa, J

R. Jozsa, J. Mod. Opt. 41, 2315 (1994)

1994
[29]

Anandan and Y

J. Anandan and Y. Aharonov, Phys. Rev. Lett. 65, 1697 (1990)

1990
[30]

Bengtsson and K

I. Bengtsson and K. ˙Zyczkowski, Geometry of Quantum States: An Introduction to Quantum Enta nglement (Cambridge University Press, Cambridge, UK, 2017)

2017
[31]

Deﬀner and E

S. Deﬀner and E. Lutz, Phys. Rev. Lett. 111, 010402 (2013)

2013
[32]

G. Kato, M. Owari and K. Maruyama, Ann. Phys. 412, 168046 (2019)

2019
[33]

Deﬀner and E

S. Deﬀner and E. Lutz, J. Phys. A: Math. Theor. 46, 335302 (2013)

2013
[34]

Biagi, A

S. Biagi, A. Bonﬁglioli, and M. Matone, Linear and Multilinear Algebra 68, 1310-1328 (2020)

2020
[35]

At least one of the ratios between eigenvalues of H0 are assumed to be irrational, as well as ∫ T 0 f (t)dt ⁄= 0 ( ∀T > 0), in order to avoid for exp( −i ∫ f (t)dtH0) to return to the identity operator
[36]

(11) in [31] may be better replaced with trace norm of H(t)ρ(t) , which is equal to√ ⟨H(t)2⟩ for a pure state ρ = |ψ ⟩⟨ψ |, leading to the integrand in the RHS of Eq

The time-averaged energy Eτ in Eq. (11) in [31] may be better replaced with trace norm of H(t)ρ(t) , which is equal to√ ⟨H(t)2⟩ for a pure state ρ = |ψ ⟩⟨ψ |, leading to the integrand in the RHS of Eq. (20)
[37]

If not, an oﬀset should be added to make the lowest eigenv alue to be zero
[38]

Appendix A: A remark on Theorem 1 We shall show thatC[T ] in Theorem 1 cannot always be a continuous function of T with a speciﬁc example

A motivation behind this relation is, minimizing the Fr obenius norm of a Hamiltonian over ε after shifting all the energy eigenvalues by the same amount ε gives the deviation of the Hamiltonian. Appendix A: A remark on Theorem 1 We shall show thatC[T ] in Theorem 1 cannot alw...

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

M. R. Frey, Quantum Inf. Process. 15, 3919 (2016)

work page 2016

[2] [2]

Deﬀner and S

S. Deﬀner and S. Campbell, J. Phys. A: Math. Theor. 50, 453001 (2017)

2017

[3] [3]

Mandelstam and I

L. Mandelstam and I. Tamm, J. Phys. 9, 249 (1985)

work page 1985

[4] [4]

Margolus and L

N. Margolus and L. B. Levitin, Physica D 120, 188-195 (1998)

work page 1998

[5] [5]

M. A. Nielsen, M. R. Dowling, M. Gu, and A. C. Doherty, Science 311, 1133-1135 (2006)

work page 2006

[6] [6]

M. A. Nielsen, M. R. Dowling, M. Gu, and A. C. Doherty, Phys. Rev. A 73, 062323 (2006)

work page 2006

[7] [7]

M. R. Dowling and M. A. Nielsen, Quantum Information & Computation 8, 861 (2008)

work page 2008

[8] [8]

Arenz, B

C. Arenz, B. Russell1, D. Burgarth and H. Rabitz, New J. Phys. 19, 103015 (2017)

work page 2017