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Shared entanglement plus one or two qubits lets Alice and Bob beat classical random-access codes, and the best win rate certifies Alice's encoding unitaries.

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2026-07-14 12:59 UTC pith:BXP6RV2M

load-bearing objection Solid analytic extension of entanglement-assisted RACs: tight 4 o1/4 o2 optima with unitary certification, plus clean upper bounds for 5 o l and n o n-2. the 1 major comments →

arxiv 2607.10273 v1 pith:BXP6RV2M submitted 2026-07-11 quant-ph

Certified quantum supremacy in entanglement-assisted prepare-measure random-access-code

Rajdeep Paul , Prabuddha Roy , A. K. Pan This is my paper
classification quant-ph PACS 03.67.Hk03.67.Mn03.65.Ud
keywords entanglement-assisted prepare-measurerandom-access codesemi-device-independentquantum supremacyself-testing of unitariesdense codingquantum communication advantage
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper shows that when Alice and Bob already share a fixed amount of entanglement, Alice can encode an n-bit string by acting only on her half of the shared state and then send only l < n qubits to Bob; Bob measures the whole system and tries to guess a randomly chosen bit of the string. For the 4-to-1 and 4-to-2 games the authors obtain exact optimal success probabilities (approximately 0.853 and 0.933) that exceed both the best classical strategies and the best ordinary quantum prepare-and-measure strategies that do not use pre-shared entanglement. Reaching the 4-to-1 optimum forces the shared state to be maximally entangled and forces Alice's four encoding unitaries to be a specific mutually anti-commuting set; thus the numerical win rate itself certifies the encoding operations. Upper bounds are also derived for the 5-to-l family and for the general n-to-(n-2) family, all of which still beat the corresponding classical limits. The results therefore turn a communication game into a self-testing protocol for unitary encodings while simultaneously quantifying the communication advantage of entanglement assistance.

Core claim

In the entanglement-assisted prepare-and-measure random-access-code setting the optimal quantum success probability for the 4-to-1 game is exactly 1/2 + 1/(2 sqrt(2)) and for the 4-to-2 game is 1/2 + sqrt(3)/4; both values strictly surpass the classical and standard quantum bounds, and the 4-to-1 optimum is attained only when the shared two-qubit state is maximally entangled and Alice's unitaries form a mutually anti-commuting triple (together with the identity).

What carries the argument

The correlation functional I_{n to l} assembled from the signed sums of Alice's encoded states (the operators M or N) and then bounded by rewriting it as a weighted sum of dichotomic observables whose weights are scaled Frobenius norms; the convex inequality that forces those weights equal, together with the maximisation of a handful of pairwise traces, yields the tight quantum values.

Load-bearing premise

The simultaneous attainment of equality in the convex weight-equalisation step and in four independent pairwise-trace maximisations is assumed to be compatible with one and the same set of pure mutually orthogonal two-qubit states that also form complete bases.

What would settle it

An explicit two-qubit shared state and four unitary encodings that produce a success probability strictly larger than 1/2 + 1/(2 sqrt(2)) for the 4-to-1 game, or a numerical optimisation over three-qubit states that exceeds 1/2 + sqrt(3)/4 for the 4-to-2 game, would falsify the claimed optima.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The numerical win rate of a 4-to-1 entanglement-assisted RAC becomes a device-independent certificate that Alice performed a concrete set of anti-commuting unitaries.
  • Any classical or entanglement-free quantum strategy for 4-bit random access with one or two bits of communication is strictly weaker than the entanglement-assisted optimum.
  • The same analytic technique supplies concrete upper bounds for every 5-to-l and every n-to-(n-2) entanglement-assisted RAC that already beat the classical limits.
  • Self-testing of multi-qubit unitaries can be reduced to the observation of a single scalar success probability in a communication game.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same norm-and-trace technique is likely to give tight optima for other small n-to-l pairs (e.g., 6-to-2 or 6-to-3) once the corresponding bases of mutually orthogonal states can be enumerated.
  • Because the optimum certifies both the shared state and Alice's unitaries, the protocol can be repurposed as a semi-device-independent certification of dense-coding-type operations without full tomography.
  • Noise robustness of the self-testing statement remains open; a modest experimental demonstration with present-day entangled-photon sources would already be informative.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. The paper introduces a family of entanglement-assisted prepare-and-measure random-access codes (PMRACs) in the semi-device-independent setting. Alice and Bob share a fixed-dimension entangled state; Alice encodes an n-bit string by local CPTP maps (unitaries) on her share and transmits l < n qubits; Bob performs a joint measurement to recover a randomly chosen bit. For the 4 o1 and 4 o2 cases the authors derive the optimal quantum success probabilities analytically (1/2 + 1/(2√2) ≈ 0.853 and 1/2 + √3/4 ≈ 0.933) by rewriting the figure of merit as a correlation of dichotomic operators, bounding it with the scaled Frobenius norm and a convex inequality, and saturating the bound with explicit states and measurements. These values exceed both the corresponding classical RACs (even with more communicated bits) and the best-known standard quantum PMRACs. The 4 o1 optimum is shown to certify a maximally entangled shared state, Alice’s mutually anti-commuting unitaries, and Bob’s observables (Corollary 1, Proposition 1, Theorem 1). Upper bounds are obtained for the 5 o l (l = 1,2,3) games and for the general n o n-2 family, again demonstrating quantum advantage.

Significance. The work supplies clean, fully analytical demonstrations of quantum supremacy in a hybrid entanglement-assisted communication task together with a self-testing statement for unitary encodings that follows directly from optimality. The technical engine—recasting the success probability as a sum of scaled-Frobenius-normalized correlations and applying a convex bound—is elegant, reusable, and free of free parameters. Explicit saturating constructions (including a three-qubit GHZ state for 4 o2) and the certification theorems constitute concrete, checkable contributions to the SDI toolkit. The n o n-2 extension, while only an upper bound, already recovers the tight 3 o1 and 4 o2 values and therefore provides a solid foundation for further generalisations. These results are of clear interest for quantum communication, device-independent certification, and foundational studies of prepare-and-measure scenarios.

major comments (1)
  1. Appendix A (Eqs. A8–A13) and Sec. III.A (Eqs. 12–13): the derivation of I_{4 o1}^opt = 32√2 simultaneously requires equality in the convex inequality (all ω_y equal) and the four independent trace maximisations Tr[M_4,1^1 M_4,3^1] = au = 4 (and cyclic). While the explicit two-qubit strategy given after Eq. (17) and in Eq. (18) saturates the bound, the manuscript does not prove that these maximisations are always compatible for every set of pure, mutually orthogonal states that form complete bases. A short uniqueness (or local-unitary equivalence) argument would make the self-testing claim of Theorem 1 fully rigorous rather than example-dependent.
minor comments (5)
  1. Table I, 5 o1 row: the entry for the standard quantum PMRAC appears as “<0.5”, which is inconsistent with the classical value 0.69 and with known lower bounds. Please correct the numerical value (or the formatting) and cite the source of the bound.
  2. Sec. III.B and Appendix E.1: the statement that “no grand unitary transformation exists” for the 4 o2 example is interesting but left without a short proof or reference. A one-sentence argument (or an explicit check that the two orthonormal bases are not related by a local unitary on Alice’s side) would improve clarity.
  3. Fig. 2 caption and axis labels: the vertical axis is labelled “Success Probability P_{n o n-2}_Q” while the plotted classical curve is the known upper bound (1-1/2^n). Adding the classical formula to the caption would make the comparison self-contained.
  4. Throughout: the scaled Frobenius norm is defined with a dimension-dependent prefactor (1/2 for two qubits, 1/√8 for three qubits, au). A single unified definition placed once in Sec. II would avoid repeated redefinitions.
  5. References: the classical n o n-2 bound is attributed to [52]; a more precise pointer to the relevant theorem or page would help the reader.

Circularity Check

0 steps flagged

No significant circularity; optimal bounds and self-testing follow from success-probability definition plus explicit constructions, with only non-load-bearing self-citations to the 3 o1 precursor.

full rationale

The central claims (tight optima for 4 o1 and 4 o2, upper bounds for 5 o l and n o n-2, and the self-testing statements of Corollary 1 / Theorem 1 / Proposition 1) are obtained by rewriting the success probability (Eq. 2) as a correlation functional I, applying the scaled Frobenius norm and the convex inequality ∑ω_y ≤ √(n ∑ ω_y^{2}), maximizing the resulting traces under the mutual-orthogonality and completeness conditions that follow from the dichotomic observables, and saturating the bound with an explicit shared state, Alice unitaries and Bob observables (after Eq. 17 and App. E.1). No free parameters are fitted to data; the simultaneous maximality of the four independent traces (App. A) is verified by construction rather than assumed by fiat. The only self-citations ([31] for the 3 o1 case and standard RAC literature) supply numerical baselines or the trivial dense-coding limit; they are not used as uniqueness theorems or as hidden premises that force the new optima. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The work rests entirely on standard quantum mechanics (finite-dimensional Hilbert spaces, completely-positive maps, projective measurements) together with the semi-device-independent dimension constraint that is conventional in the RAC literature. No numerical parameters are fitted and no new physical entities are postulated; the operators M, N, L are simply convenient rewritings of the success probability.

axioms (4)
  • domain assumption Quantum states are density operators on C^{2^l} ⊗ C^2; Alice's encodings are CPTP maps (in practice unitaries); Bob's measurements are projective dichotomic observables.
    Invoked from the definition of the entanglement-assisted PMRAC (Sec. II.B, Eq. 2) onward; standard quantum information theory.
  • domain assumption The Hilbert-space dimension of the system Alice communicates is at most 2^l (semi-device-independent assumption).
    Stated in the abstract and Sec. I; required for the SDI framework and for the comparison with classical l-bit RACs.
  • standard math The scaled Frobenius norm ||O|| = (1/d) √ Tr[O†O] (d = 2 or 4 or 8 according to the number of qubits) normalizes dichotomic observables so that max Tr[M B] = ||M|| when B = M.
    Used throughout Sec. III-V and Apps. A-H to convert the success probability into a sum of norms; conventional in quantum correlation inequalities.
  • standard math Convex inequality ∑_y ω_y ฺ √(n ∑_y ω_y^{2}) with equality iff all ω_y are equal.
    Applied in Eqs. (12), (24), (29), (47) etc. to obtain a single scalar upper bound; elementary Cauchy-Schwarz.

pith-pipeline@v1.1.0-grok45 · 49525 in / 3174 out tokens · 43987 ms · 2026-07-14T12:59:40.017023+00:00 · methodology

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read the original abstract

We develop a family of semi-device-independent (SDI) entanglement-assisted prepare-measure (PM) communication games involving two parties, within the $n\rightarrow l$ random-access code (RAC) framework where the sender Alice holds a n-bit string and communicates $l<n$ bits or qubits to the receiver Bob. In contrast to the standard quantum PMRAC, here the parties share a prior entanglement, and Alice applies quantum operations on her sub-system to encode her inputs and sends to Bob. We first consider the $4\rightarrow l$ entanglement-assisted PMRAC with $l=1$ and $2$ and derive the optimal quantum success probabilities using an elegant analytical technique. We demonstrate quantum supremacy over both classical RACs and conventional quantum PMRACs. Moreover, we exhibit that the optimal quantum advantage allows one to certify Alice's unitary operations. We then derive an upper bound on the quantum success probabilities for $5\rightarrow l$ entanglement-assisted PMRAC with $l=1,2$ and $3$. Further, we extend the demonstration of quantum advantage for $n\rightarrow n-2$ case where n is arbitrary.

Figures

Figures reproduced from arXiv: 2607.10273 by A. K. Pan, Prabuddha Roy, Rajdeep Paul.

Figure 1
Figure 1. Figure 1: FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The Graph depicts how the quantum success probabil [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

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Reference graph

Works this paper leans on

57 extracted references · 2 linked inside Pith

  1. [1]

    Alice’s unitary operators are U0000 =112, U1100 =i Q 1, U0011 =−i P 1 and U 1111 =−i R 1 are mutually anti- commuting. 5

  2. [2]

    The first step relies on two key elements: the mutually orthogonality con- dition satisfied by the states as stated in Eq

    There exists grand unitaries U 1 G = −112+U0011√ 2 , U 2 G = 112+U1100√ 2 and U3 G = 112+U1100−U0011+U1111 2 which gives (U1 G)†{ρ0000, ρ1111, ρ0011, ρ1100}U1 G → {ρ0001, ρ1110, ρ0010, ρ1101} (U2 G)†{ρ0000, ρ1111, ρ0011, ρ1100}U2 G → {ρ0100, ρ0111, ρ1011, ρ1000}(17) (U3 G)†{ρ0000, ρ1111, ρ0011, ρ1100}U3 G → {ρ0101, ρ0110, ρ1010, ρ1001} Proof.The proof is ...

    2021

  3. [3]

    11 TABLE II

    The relation- ship among the various Tr ρxρx′ ’s are summarized in the following table. 11 TABLE II. Relations among all the statesρ, that is, the quantities Tr ρxρx′ for allx,x ′ ∈[15]. ρ0000 ρ0001 ρ0010 ρ0011 ρ0100 ρ0101 ρ0110 ρ0111 ρ1000 ρ1001 ρ1010 ρ1011 ρ1100 ρ1101 ρ1110 ρ1111 ρ0000 1 1/2 1/2 0 1/2 1/4 1/4 0 1/2 1/4 1/4 0 0 0 0 0 ρ0001 1/2 1 0 1/2 1/...

  4. [4]

    Hence, the grand unitary is derived as follows U3 G = 112 +i(Q 1 +P 1 −R 1) 2 ≡ 112 +U 1100 −U 0011 +U 1111 2 (D34) 15 Appendix E: Detailed derivation of optimal quantum success probability for4→2entanglement-assisted PMRAC In the 4→2 entanglement-assisted PMRAC game, the quantum success probability from Eq. (2) in the main text can be explicitly written ...

  5. [5]

    4+ 1 4 Tr h N1 5,1N1 5,3 +N 1 5,1N1 5,2 +N 1 5,1N1 5,4 +N 1 5,3N1 5,2 +N 1 5,3N1 5,4 +N 1 5,2N1 5,4 i# 1 2 (G9) µ2 =

    An illustrative 4→2 example of states and measurements that attains the optimal success probability Let, the joint state shared by Alice and Bob is a GHZ state of the following form, ρ0000 = 1 8 112 ⊗112 ⊗112 +σ z ⊗σ z ⊗112 +σ z ⊗112 ⊗σ z +112 ⊗σ z ⊗σ z +σ x ⊗σ x ⊗σ x −σ x ⊗σ y ⊗σ y −σ y ⊗σ x ⊗σ y −σ y ⊗σ y ⊗σ x (E15) 17 where Alice holds first two qubits...

    1975

  6. [6]

    Pawłowski, T

    M. Pawłowski, T. Paterek, D. Kaszlikowski, V . Scarani, A. Win- ter, and M. ´Zukowski, Nature461, 1101 (2009)

    2009

  7. [7]

    Brassard, H

    G. Brassard, H. Buhrman, N. Linden, A. A. M ´ethot, A. Tapp, and F. Unger, Phys. Rev. Lett.96, 250401 (2006)

    2006

  8. [8]

    Brunner, M

    N. Brunner, M. Navascu´es, and T. V´ertesi, Phys. Rev. Lett.110, 150501 (2013)

    2013

  9. [9]

    Wehner, M

    S. Wehner, M. Christandl, and A. C. Doherty, Phys. Rev. A78, 062112 (2008)

    2008

  10. [10]

    Gallego, N

    R. Gallego, N. Brunner, C. Hadley, and A. Ac ´ın, Phys. Rev. Lett.105, 230501 (2010)

    2010

  11. [11]

    J. I. de Vicente, Phys. Rev. A95, 012340 (2017)

    2017

  12. [12]

    A. K. Pan and S. S. Mahato, Phys. Rev. A102, 052221 (2020)

    2020

  13. [13]

    Ambainis, A

    A. Ambainis, A. Nayak, A. Ta-Shma, and U. Vazirani, inPro- ceedings of the Thirty-First Annual ACM Symposium on Theory of Computing (STOC ’99)(ACM Press, New York, NY , USA,

  14. [14]

    Ambainis, D

    A. Ambainis, D. Leung, L. Mancinska, and M. Ozols, Quantum random access codes with shared randomness (2008)

    2008

  15. [16]

    Li, Z.-Q

    H.-W. Li, Z.-Q. Yin, Y .-C. Wu, X.-B. Zou, S. Wang, W. Chen, G.-C. Guo, and Z.-F. Han, Phys. Rev. A84, 034301 (2011)

    2011

  16. [17]

    H.-W. Li, M. Pawłowski, Z.-Q. Yin, G.-C. Guo, and Z.-F. Han, Phys. Rev. A85, 052308 (2012)

    2012

  17. [18]

    Pawłowski and N

    M. Pawłowski and N. Brunner, Phys. Rev. A84, 010302 (2011)

    2011

  18. [19]

    Woodhead and S

    E. Woodhead and S. Pironio, Phys. Rev. Lett.115, 150501 (2015)

    2015

  19. [20]

    Tavakoli, J

    A. Tavakoli, J. m. k. Kaniewski, T. V ´ertesi, D. Rosset, and N. Brunner, Phys. Rev. A98, 062307 (2018)

    2018

  20. [21]

    Farkas and J

    M. Farkas and J. m. k. Kaniewski, Phys. Rev. A99, 032316 (2019)

    2019

  21. [22]

    Mironowicz and M

    P. Mironowicz and M. Pawłowski, Phys. Rev. A100, 030301 (2019)

    2019

  22. [23]

    Tavakoli, M

    A. Tavakoli, M. Smania, T. V´ertesi, N. Brunner, and M. Bouren- nane, Science Advances6, eaaw6664 (2020)

    2020

  23. [24]

    Smania, P

    M. Smania, P. Mironowicz, M. Nawareg, M. Pawłowski, A. Ca- bello, and M. Bourennane, Optica7, 123 (2020)

    2020

  24. [25]

    Miklin, J

    N. Miklin, J. J. Borkała, and M. Pawłowski, Physical Review Research2, 033014 (2020)

    2020

  25. [26]

    E. S. G ´omez, S. G ´omez, P. Gonz ´alez, G. Ca ˜nas, J. F. Barra, A. Delgado, G. B. Xavier, A. Cabello, M. Kleinmann, T. V´ertesi, and G. Lima, Phys. Rev. Lett.117, 260401 (2016)

    2016

  26. [27]

    G ´omez, A

    S. G ´omez, A. Mattar, E. S. G´omez, D. Cavalcanti, O. J. Far´ıas, A. Ac´ın, and G. Lima, Phys. Rev. A97, 040102 (2018)

    2018

  27. [28]

    A. K. Pan, Phys. Rev. A104, 022212 (2021)

    2021

  28. [29]

    ˇSupi´c, M

    I. ˇSupi´c, M. J. Hoban, L. D. Colomer, and A. Ac´ın, New Journal of Physics22, 073006 (2020)

    2020

  29. [30]

    Tavakoli, J

    A. Tavakoli, J. Kaniewski, T. V ´ertesi, D. Rosset, and N. Brun- ner, Phys. Rev. A98, 062307 (2018)

    2018

  30. [31]

    Tavakoli, Physical review letters125 15, 150503 (2020)

    A. Tavakoli, Physical review letters125 15, 150503 (2020)

    2020

  31. [32]

    Lunghi, J

    T. Lunghi, J. B. Brask, C. C. W. Lim, Q. Lavigne, J. Bowles, A. Martin, H. Zbinden, and N. Brunner, Phys. Rev. Lett.114, 150501 (2015)

    2015

  32. [33]

    Mohan, A

    K. Mohan, A. Tavakoli, and N. Brunner, New J. Phys.21, 083034 (2019)

    2019

  33. [34]

    A. S. S., S. Mukherjee, and A. K. Pan, Phys. Rev. A107, 012411 (2023)

    2023

  34. [35]

    R. Paul, S. Sasmal, and A. K. Pan, Phys. Rev. A110, 012444 (2024)

    2024

  35. [36]

    R. Paul, P. Roy, and A. K. Pan, Phys. Rev. A112, 062216 (2025)

    2025

  36. [37]

    J. I. D. Vicente, Physical Review A95, 10.1103/Phys- RevA.95.012340 (2017)

    doi:10.1103/phys- 2017

  37. [38]

    A. S. Holevo, Problemy Peredachi Informatsii9, 3 (1973), english translation: Problems of Information Transmission, 9(3):177–183 (1973)

    1973

  38. [39]

    Tavakoli, J

    A. Tavakoli, J. Pauwels, E. Woodhead, and S. Pironio, PRX Quantum2, 040357 (2021)

    2021

  39. [40]

    Ambainis, D

    A. Ambainis, D. Leung, L. Mancinska, and M. Ozols, Quan- tum random access codes with shared randomness (2009), arXiv:0810.2937 [quant-ph]

    Pith/arXiv arXiv 2009

  40. [41]

    E. F. Galv ˜ao, Phys. Rev. A65, 012318 (2001)

    2001

  41. [42]

    Hayashi, K

    M. Hayashi, K. Iwama, H. Nishimura, R. Raymond, and S. Ya- mashita, New J. Phys.8, 129 (2006)

    2006

  42. [43]

    Tavakoli, A

    A. Tavakoli, A. Hameedi, B. Marques, and M. Bourennane, Phys. Rev. Lett.114, 170502 (2015)

    2015

  43. [44]

    Chailloux, I

    A. Chailloux, I. Kerenidis, S. Kundu, and J. Sikora, New J. Phys.18, 045003 (2016)

    2016

  44. [45]

    Pawłowski and M

    M. Pawłowski and M. ˙Zukowski, Phys. Rev. A81, 042326 (2017)

    2017

  45. [46]

    Hameedi, D

    A. Hameedi, D. Saha, P. Mironowicz, M. Pawłowski, and M. Bourennane, Phys. Rev. A95, 052345 (2017)

    2017

  46. [47]

    R. W. Spekkens, D. H. Buzacott, A. J. Keehn, B. Toner, and G. J. Pryde, Phys. Rev. Lett.102, 010401 (2009)

    2009

  47. [48]

    Hameedi, A

    A. Hameedi, A. Tavakoli, B. Marques, and M. Bourennane, Phys. Rev. Lett.119, 220402 (2017)

    2017

  48. [49]

    Pauwels, S

    J. Pauwels, S. Pironio, E. Woodhead, and A. Tavakoli, Phys. Rev. Lett.129, 250504 (2022)

    2022

  49. [50]

    Pauwels, A

    J. Pauwels, A. Tavakoli, E. Woodhead, and S. Pironio, New Journal of Physics24, 063015 (2022)

    2022

  50. [51]

    Piveteau, J

    A. Piveteau, J. Pauwels, E. Håkansson, S. Muhammad, M. Bourennane, and A. Tavakoli, Nature Communications13, 7878 (2022)

    2022

  51. [52]

    R. K. Singh, S. Sasmal, and A. K. Pan, Phys. Rev. A112, 032404 (2025)

    2025

  52. [53]

    Mukherjee and A

    S. Mukherjee and A. K. Pan, Phys. Rev. A104, 062214 (2021)

    2021

  53. [54]

    Imamichi and R

    T. Imamichi and R. Raymond, inAsian Quantum Information Symposium (AQIS)(2018)

    2018

  54. [55]

    J. F. Doriguello and A. Montanaro, Quantum5, 402 (2021)

    2021

  55. [56]

    Farkas, N

    M. Farkas, N. Miklin, and A. Tavakoli, Quantum9, 1643 (2025)

    2025

  56. [57]

    Nayak, Optimal lower bounds for quantum automata and random access codes (1999), arXiv:quant-ph/9904093 [quant- ph]

    A. Nayak, Optimal lower bounds for quantum automata and random access codes (1999), arXiv:quant-ph/9904093 [quant- ph]

    Pith/arXiv arXiv 1999

  57. [58]

    Horodecki, P

    R. Horodecki, P. Horodecki, and M. Horodecki, Physics Letters A200, 340 (1995)

    1995