Two-level control over quantum state creation via entangled equal-probability state

A.I.Zenchuk; E.B.Fel'dman; S.I.Doronin

REVIEW 2 major objections 6 minor 46 references

Two-level control over quantum state creation via entangled equal-probability state

T0 review · 2 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper establishes that, in a two-level control protocol, the phases of an equal-probability ancilla set how many state-creation unitaries the second controller can switch off, with the maximum reached exactly at zero bipartite…

desk verdict Solid concurrence formula and constraint counting, but the protocol's deterministic state-creation claim breaks on the outcome-dependent signs in Eq. (12). read the letter →

arxiv 2502.09124 v1 pith:KOEBLP3W submitted 2025-02-13 quant-ph

S.I.Doronin , E.B.Fel'dman , A.I.Zenchuk This is my paper · ORCID

classification quant-ph MSC 81P4081P68 PACS 03.67.-a03.67.Mn

keywords two-levelquantumcontrolequal-probabilitystateconcurrenceentanglementcontrolledunitaryoperatorsphaseconstraintsancillameasurementcreation

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 proposes a two-level scheme for preparing a desired quantum state of a system $S$, where the preparation is delegated through an auxiliary subsystem $R$. The first-level controller $M$ prepares $R$ in an equal-probability superposition with a chosen phase distribution, and the second-level controller $C$ applies $V$-operators to $R$ that switch off some of the unitary operators $U_k$ acting on $S$. The paper's central claim is that the phases of $R$ determine how many $U_k$ $C$ can switch off, and that this number is directly tied to the bipartite entanglement of $R$: imposing the independent phase constraints that create column dependencies in the phase matrix both enlarges $C$'s switching power and lowers the concurrence (an entanglement measure) of $R$. The maximum number of switchable operators, $(N_A-1)N_B$, is reached exactly when the concurrence vanishes and $R$ factorizes into its $A$ and $B$ parts.

What carries the argument

The load-bearing object is the equal-probability phase state $|\Psi_{\mathrm{eq.pr.}}\rangle_R = (1/\sqrt{2^n})\sum_k e^{i\varphi_k}|k\rangle$ together with the phase matrix $\Phi = \{e^{i\varphi_{k_A k_B}}\}$. The second-level operator $V_1$ acts on the $A$ part of $R$, and a row of $V_1$ can zero out amplitude coefficients $A_{l_A k_B}$; how many coefficients it can zero is governed by how many columns of $\Phi$ are linearly dependent on the first column. These dependencies are encoded in the phase differences $\varepsilon_{i_A j_A; i_B j_B}$ of Eq. (33), and the same quantities appear in the concurrence formula (50), $C = \frac{4}{N_A N_B}\sqrt{\sum \sin^2(\varepsilon/2)}$. Thus a single set of phase constraints does two jobs: it grants $C$ the power to switch off $U$-operators, and it reduces the entanglement in $R$, so the control-versus-entanglement relation is an identity rather than a coincidence.

What would settle it

Use the paper's own two-qubit example with the Case 1 phase choices and record the four outputs for the four measurement results on $R$: Eq. (45) gives $|10\rangle+|01\rangle$ for results 00 and 10, and $|10\rangle-|01\rangle$ for results 01 and 11. If the protocol promises a definite required state for any measurement result, this already settles that the promise fails unless the two Bell states are declared equivalent. Separately, compute the left and right sides of Eq. (50) for random phases numerically; any mismatch falsifies the concurrence formula.

Watch

Extended reading notes

Core claim

On its own terms, the paper's discovery is an exact equivalence between second-level control power and ancilla entanglement. Writing $R$ as an equal-probability state with amplitudes $e^{i\varphi_{k_A k_B}}/\sqrt{N_A N_B}$, an $n_A$-qubit operator $V_1$ controlled by $C$ can switch off at most $(N_A-1)N_B$ of the $N_A N_B$ unitaries $U_{l_A k_B}$. This maximum is attained precisely when the $(N_A-1)(N_B-1)$ independent phase constraints of Eq. (37) hold, and those same constraints are exactly the conditions under which the concurrence formula (50) vanishes, so the state of $R$ factorizes across the $A|B$ split. The paper derives the closed concurrence formula for any bipartite equal-probability state and illustrates the trade-off in a two-qubit control example in which the same circuit produces Bell states, separable states, or the initial ground state depending only on the phase constraints.

Load-bearing premise

The load-bearing premise is that after measuring $R$, the phase factors in front of the unitary terms do not matter, so any of the $2^n$ measurement outcomes is acceptable; but Eq. (12) shows these outcomes change relative signs between the surviving $U_j|\Psi\rangle$ terms, and Eq. (45) shows different outcomes give different Bell states, so a fixed target state is created deterministically only if sign-equivalent states are accepted or classical correction is added.

Editorial extensions

If this is right

A first-level controller $M$ can tune the phases of $R$ so that the second-level center $C$ is able to switch off up to $(N_A-1)N_B$ of the $N_A N_B$ unitaries; adding the full set of $(N_A-1)(N_B-1)$ independent constraints makes the ancilla factorize and the switching power maximal.
Each independent phase constraint reduces the maximum possible concurrence $C_{\max}(N_K)$, with sharp drops occurring when an entire column of the phase matrix becomes linearly dependent on the first column, selecting $N_A-1$ extra operators to switch off.
In the two-qubit example, the same circuit produces a maximally entangled Bell state, a separable state, or the original ground state depending only on the phase constraints, and in each regime the output appears for any measurement of $R$.
Formula (50) supplies a direct calibration: measuring the concurrence of $R$ after $W$ predicts how many $U$-operators the second controller can switch off, so the control capacity is experimentally readable from an entanglement measurement.

Reading between the lines

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

Beyond the paper: because Eq. (12) shows measurement outcomes change relative signs of the surviving $U_j|\Psi\rangle$ terms, a fixed target state is not created deterministically unless sign-equivalent outcomes are acceptable or feed-forward corrections are added; the Bell example yields $|10\rangle+|01\rangle$ for two outcomes and $|10\rangle-|01\rangle$ for the other two.
Beyond the paper: the sign freedom suggests a concrete extension, classical feed-forward from the $R$ measurement to $S$, that would make the protocol deterministic while preserving the phase-constraint control of which $U$'s are active.
Beyond the paper: the $C_{\max}(N_K)$ curves indicate a design rule for $W$: to maximize delegated control, prepare $R$ with zero $A$-$B$ entanglement, while nonzero entanglement deliberately reduces the second center's power, which could be useful as a hierarchy or policy mechanism.

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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

The math here is mostly solid, and there is one genuinely reusable result: the closed-form concurrence for bipartite equal-probability states, Eq. (50). I re-derived it from the trace identity and the counting works. The second useful piece is the constraint analysis around Eqs. (37)-(40) showing that column dependencies in the phase matrix simultaneously control how many operators can be switched off and how entangled the ancilla is. The two-qubit Bell example also checks out in all three cases, modulo a small index typo in Case 3. Credit is due: no fitting, no self-citation in the load-bearing argument, and the derivation is reproducible from the appendix.

The soft spot is the protocol's headline claim. The text after Eq. (12) says the measurement outcome affects only phase factors and therefore any outcome is acceptable. That is wrong. Eq. (12) has (-1)^{alpha.j} inside the sum, which changes relative signs between the surviving terms, not just an overall phase. The paper's own Bell example shows this: outcomes 00 and 10 give |10>+|01>, while outcomes 01 and 11 give |10>-|01>. These are orthogonal states, so the protocol does not deterministically create "the required quantum state." It either creates an outcome-dependent family or needs classical feed-forward corrections, which the paper does not discuss. This is not a nitpick; it is a load-bearing gap between the abstract and what the circuit actually does.

Two smaller issues. The abstract's "direct relation between entanglement and number of controlled operators" is a correspondence through shared phase constraints, not a functional relation. And the numerical figures rely on random maximization with only sample counts reported; the method is not fully specified. There is also no comparison with existing schemes like LCU or remote state preparation, which would help calibrate what is new here.

Bottom line: the ancillary mathematical result is correct and worth having, but the paper overclaims what the protocol delivers. It deserves a serious referee because the core derivations are sound and the formula is useful, but it needs revision before acceptance, starting with the measurement-outcome problem.

Referee Report

2 major / 6 minor

Summary. The manuscript proposes a two-level control scheme for quantum state creation. A main controller M prepares an equal-probability superposition of a register R with phases φ_k; a second controller C applies a unitary V_1 to a subset A of R, and the resulting amplitudes determine which of the operators U_{l_A k_B} acting on the target system S are switched off. The central technical result is the closed-form expression (50) for the generalized concurrence of an equal-probability bipartite state, derived in the Appendix, and the identification of the independent phase constraints (37)–(40) that simultaneously make the state factorizable and allow the maximal number (N_A−1)N_B of U-operators to be switched off. The two-qubit example in Sec. II C illustrates the three regimes: generic phases, one constraint (single U_1 applied), and full constraints (no U operator applied).

Significance. The derivation of Eq. (50) is sound and checkable: the trace computation in the Appendix is consistent, the counting of independent phase constraints agrees with the rank structure of the phase matrix, and the extremal equivalence (all constraints ⇔ factorization ⇔ zero concurrence ⇔ maximal switch-off) is correctly argued. The paper also provides valuable numerical support in Figs. 2–5 for the monotonic decrease of maximal concurrence with the number of imposed constraints. If the protocol issue described below is resolved, the entanglement–control relation established here is a useful and falsifiable contribution to the quantum-control literature.

major comments (2)

[Sec. II A, Eq. (12)] The claim that the measurement outcome over R 'effects only on the phase factors ahead of the unitary transformations' and that these phase factors are unimportant is incorrect. The factor (−1)^{α·j} changes the relative signs among the terms a_j U_j|Ψ>, which is physically relevant whenever more than one U_j term survives. The paper's own Eq. (45) shows this: outcomes 00 and 10 produce |10>+|01>, while outcomes 01 and 11 produce |10>−|01>, which are orthogonal states. The abstract's claim of creating 'the required quantum state' is therefore not supported; the protocol produces an outcome-dependent family of states. The authors should either add explicit classical feed-forward corrections based on the measurement outcome α (and show that such corrections can be implemented for the general set {U_j}), or reformulate the goal as creation of a state up to a known local unitary / within an outcome-tolerant family. The mathematical results on concurrence and switching are not affected, but the state-creation protocol needs revision.
[Sec. II B 2, after Eq. (31)] The assertion that 'if M_B columns are linearly dependent, we can put zero up to (N_A+M_B−2) terms' is not justified and appears internally inconsistent. For a fixed row l_A, the number of amplitude terms A_{l_A,k_B} that can be zeroed by a single row of V_1 is the size of a set of columns of the phase matrix whose rank is at most N_A−1; once the whole matrix has rank below N_A, one row can zero all N_B terms. The later discussion of the M_B=N_B case correctly states that a single equation then zeros all N_B amplitudes, which contradicts the count N_A+N_B−2. Please correct this statement or clarify what 'M_B' counts, since the gradual switch-off claim depends on it.

minor comments (6)

[Title] The title in the manuscript has a stray space in 'entangle d'; please correct to 'entangled'.
[Eq. (45)] The symbol α is used both for the measurement outcomes α_{j_1...j_{n(R)}} in Eq. (12) and for the scalar output amplitude α = e^{i(φ00−χ00+χ10)}/√2 in Eq. (45); please use a different symbol for one of them.
[Sec. II C, Case 3] The condition on φ_11 should read φ_11 = φ_10 − φ_00 + φ_01; the printed expression 'φ_11 = φ_10 − φ_00 + φ_10' appears to be a typo that affects the worked example.
[Fig. 1] Fig. 1 is referenced in the text but is not included in the manuscript body; please ensure the figure file is present in the final submission.
[Eq. (52)] The notation C(N_K, ε) is used in Eq. (52) before being defined; please specify that C(N_K, ε) is the concurrence (50) evaluated after imposing N_K constraints from list (40).
[General language] There are numerous English and typographical issues (e.g., 'effects only on the phase factors' should be 'affects only the phase factors' near Eq. (12)); a careful language edit is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the entanglement–control relation is a derived identity, not an assumed input.

full rationale

The derivation chain is self-contained and non-circular. The control count in Sec. II B 2 follows from the rank structure of the phase matrix Φ in Eqs. (29)–(31): the number of switchable operators U_{l_A k_B} is the number of amplitude equations that can be satisfied for fixed l_A, bounded by (N_A−1)N_B. The concurrence formula (50) is obtained in the Appendix from the externally defined generalized concurrence of Rungta et al. [40] (Eq. (49)), using the equal-probability amplitudes; no parameter is fitted and no external benchmark is assumed. The ‘remarkable’ coincidence of the zero-concurrence conditions (51) with the column-dependence constraints (33) is a proved identity: both sets are expressed in the same ε variables, but either quantity could be computed without assuming the other. Self-citations [27,28,32–35] appear only in the introduction and are not load-bearing. The only flagged limitation is the outcome-dependence of the state-creation protocol after Eq. (12), where the relative signs (−1)^{α·j} change the output state; this is a correctness concern about the protocol’s determinism, not a circular reduction of the derivation to its inputs.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central derivation rests on a standard external concurrence definition and elementary linear algebra of the phase matrix Phi (34). No parameters are fitted to data; the phases, V-elements, and constraint count are control knobs. The equal-probability restriction is a genuine scope assumption: both the concurrence formula (50) and the control count are specific to that family. The weakest structural premise, flagged in the paper itself, is that the R-measurement outcome affects only phase factors; see the report's weakest_assumption.

free parameters (3)

ancilla phases phi_{k_A k_B} = arbitrary reals; example uses phi_01 = phi_00, phi_11 = phi_10, phi_10 = pi + phi_00 + chi_10 - chi_00 (Case 1, Sec.
Control knobs of the main center M. Specific values are hand-picked in the worked example to produce Bell states; nothing is fitted to data.
elements of V_1 (a_{l_A k_A}, equivalently chi parameters) = V_1 = (1/sqrt(2)) [[e^{i(chi_10-chi_00)}, -1], [1, e^{-i(chi_10-chi_00)}]] (Eq. (43))
Second-level control operator. The specific form is chosen so condition (17) holds in Cases 2 and 3; unitarity constraints (14) are imposed.
number of imposed phase constraints N_K from list (40) = 0 to (N_A-1)(N_B-1)
The trade-off parameter: each constraint set makes one additional column of the phase matrix proportional to column 0, increasing the switchable-operator count and decreasing the concurrence. Purely a design parameter.

assumptions (5)

standard math Generalized concurrence C = sqrt(2(1 - Tr(rho_A^2))) is the entanglement measure for bipartite pure states (Eq. (49), cited from Rungta et al. [40]).
Adopted from external literature; the Appendix derivation of Eq. (50) starts from this definition.
domain assumption The auxiliary register R is prepared in an equal-probability state (1) with arbitrary real phases phi_k via operator W.
The entire analysis, including the concurrence formula and the control-counting arguments, is restricted to this state family (Sec. II, Eq. (1)).
domain assumption The V-operators are unitary (Eqs. (14) and (26)) and act on qubit registers.
Unitarity (V_1 V_1^dagger = I_A) is used in the switching-off construction and normalization arguments.
domain assumption The evolution is closed-system unitary with perfect gates; S and R start in ground states and C, M in excited states (Sec. II A before Eq. (3)).
No noise, decoherence, or gate errors are modeled; the initial-state choice is part of the protocol definition.
ad hoc to paper At most N_A - 1 rows of V_1 may solve zeroing equations, otherwise det(V_1) = 0.
A constraint on the construction stated at the end of Sec. II B 2; it bounds the maximal controllable count (N_A - 1) N_B.

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Pith. "Pith review of Two-level control over quantum state creation via entangled equal-probability state." pith.science (2026). https://pith.science/paper/KOEBLP3W

@misc{pith2026250209124,
  author       = {Pith},
  title        = {Pith review of: Two-level control over quantum state creation via entangled equal-probability state},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KOEBLP3W}},
  note         = {Machine review of arXiv:2502.09124}
}

abstract

We propose the scheme realizing the two-level control over the unitary operators $U_k$ creating the required quantum state of the system $S$. These operators are controlled by the superposition state of the auxiliary subsystem $R$ which is governed by two control centers. The first-level control center (main control) creates the equal-probability pure state of $R$ with certain distribution of phase factors that, in turn, govern the power of the second-level control center $C$ that applies the special $V$-operators to the same subsystem $R$ changing its state and thus controlling the applicability of $U_k$. In addition, the above phases are responsible for the entanglement in the subsystem $R$. We find the direct relation between this entanglement and the number of operators $U_k$ that can be controlled by $C$. The simple example of a two-level control system governing the creation of entangled state of the two-qubit system $S$ is presented.

Figures

Figures reproduced from arXiv: 2502.09124 by the authors.

**Figure 2.** FIG. 2: The dependence of [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗

**Figure 3.** FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p021_4.png]

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p021_5.png]

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Works this paper leans on

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One-qubit operator V1 Since all qubits in R are equivalent we consider the action of V1 (applied to the ﬁrst qubit of R and controlled by the ﬁrst qubit of C) on the state |Ψ eq.pr. ⟩R assuming that it is the one-qubit unitary operator in the form V1 = a00|0⟩⟨0| + a01|0⟩⟨1| + a10|1⟩⟨0| + a11|1⟩⟨1| (13) with unitarity conditions |a00|2 + |a01|2 = 1, a 00a∗...

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Thus, we split the subsystem R into the subsystems A and B having, respectively, nA and nB qubits, nA + nB = n(R)

Multi-qubit operator V1 We consider a particular nA-qubit operator V1 applied to the ﬁrst nA qubits of R and controlled by the ﬁrst qubit of C. Thus, we split the subsystem R into the subsystems A and B having, respectively, nA and nB qubits, nA + nB = n(R). Therefore, we change the subscript in Uk: Uk → UlAkB . It is also convenient to enumerate phases i...

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(58) The ﬁrst of Eqs.(38) yields ε01;11 = 0, and ε01;1jB = ε01;0jB, jB = 2,

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(60) The ﬁrst of Eqs.(38) yields ε01;1jB = 0, jB = 1 , 2, and ε01;1jB = ε01;2jB = ε01;0jB , jB = 3,

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Remote one-qubit-state control using the pure initial state of a two-qubit sender: Selective-region and eigenvalue creat ion, Phys.Rev.A

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E.B. Fel’dman, A.N. Pechen, A.I. Zenchuk, Complete str uctural restoring of transferred multi- qubit quantum state, Phys.Lett. A 413 (2021) 127605

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G.A.Bochkin, E.B.Fel’dman, I.D.Lazarev, A.N.Pechen , A.I.Zenchuk, Transfer of zero-order coherence matrix along spin-1/2 chain, Quant.Inf.Proc. 21, 261 (2022)

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P. Rungta, V. Buˇ zek, C.M. Caves, M. Hillery, and G. J. Mi lburn, Universal state inversion and concurrence in arbitrary dimensions, Phys.Rev.A 64, 042315 (2001) V. APPENDIX: DERIV A TION OF EQ.(50) FOR CONCURRENCE The object of our study is the pure state of an n(R)-qubit sy...

2001

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

One-qubit operator V1 Since all qubits in R are equivalent we consider the action of V1 (applied to the ﬁrst qubit of R and controlled by the ﬁrst qubit of C) on the state |Ψ eq.pr. ⟩R assuming that it is the one-qubit unitary operator in the form V1 = a00|0⟩⟨0| + a01|0⟩⟨1| + a10|1⟩⟨0| + a11|1⟩⟨1| (13) with unitarity conditions |a00|2 + |a01|2 = 1, a 00a∗...

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[2] [2]

Thus, we split the subsystem R into the subsystems A and B having, respectively, nA and nB qubits, nA + nB = n(R)

Multi-qubit operator V1 We consider a particular nA-qubit operator V1 applied to the ﬁrst nA qubits of R and controlled by the ﬁrst qubit of C. Thus, we split the subsystem R into the subsystems A and B having, respectively, nA and nB qubits, nA + nB = n(R). Therefore, we change the subscript in Uk: Uk → UlAkB . It is also convenient to enumerate phases i...

work page

[3] [3]

(58) The ﬁrst of Eqs.(38) yields ε01;11 = 0, and ε01;1jB = ε01;0jB, jB = 2,

One constraint: NK = 1, E1 : ε01;01 = 0. (58) The ﬁrst of Eqs.(38) yields ε01;11 = 0, and ε01;1jB = ε01;0jB, jB = 2, . . . , N B − 1. Then C(1, ε ) = 2 NB /radicaltp /radicalvertex /radicalvertex √ 2 NB− 1∑ jB =2 sin2 ε01;0jB 2 + NB− 2∑ iB=2 NB − 1∑ jB=iB +1 sin2 ε01;iB jB 2 , (59) where ε is the list in (53)

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(60) The ﬁrst of Eqs.(38) yields ε01;1jB = 0, jB = 1 , 2, and ε01;1jB = ε01;2jB = ε01;0jB , jB = 3,

Two constraints: NK = 2, E1, E 2 : ε01;0jB = 0, j B = 1, 2. (60) The ﬁrst of Eqs.(38) yields ε01;1jB = 0, jB = 1 , 2, and ε01;1jB = ε01;2jB = ε01;0jB , jB = 3, . . . , N B − 1. Then C(2, ε ) = 2 NB /radicaltp /radicalvertex /radicalvertex √ 3 NB− 1∑ jB =3 sin2 ε01;0jB 2 + NB− 2∑ iB=3 NB − 1∑ jB=iB +1 sin2 ε01;iB jB 2 . (61)

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, E k : ε01;0jB = 0, j B = 1,

k constraints: NK = k, E1, . . . , E k : ε01;0jB = 0, j B = 1, . . . , k, k ≤ NB − 1. (62) The ﬁrst of Eqs.(38) yields ε01;1jB = 0, jB = 1 , . . . , k , and ε01;iB jB = ε01;0jB , iB = 1 , . . . , k , jB = k + 1, . . . , N B − 1. Then C(k, ε ) = 2 NB /radicaltp /radicalvertex /radicalvertex √ (k + 1) NB− 1∑ jB=k+1 sin2 ε01;0jB 2 + NB − 2∑ iB=k+1 NB− 1∑ jB ...

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[6] [6]

We see that our curves Cmax(nB) for nA = 2 and 3 approach the appropriate theoretical maximums with an increase in nB. This is due to the fact that the 21 subsystem B plays the role of environment for the subsystem A and therefore an increase in nB leads to an increase in decoherence in A due to the interaction with environment, so that the state of A app...

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Khaneja, T

N. Khaneja, T. Reiss, C. Kehlet, T. Schulte-Herbr¨ uggen , and S.J. Glaser, Optimal control of coupled spin dynamics: design of NMR pulse sequences by gr adient ascent algorithms, J. Magn. Reson. 172(2), 296 (2005)

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Petruhanov and A.N

V.N. Petruhanov and A.N. Pechen, GRAPE optimization for open quantum systems with time-dependent decoherence rates driven by coherent and in coherent controls, J. Phys. A: Math. Theor. 56(30), 305303 (2023)

work page 2023