Estimating the spectral radius of Bell-type operator via finite dimensional approximation of orthogonal projections

Toyohiro Tsurumaru; Yuki Fujii

arxiv: 2511.10939 · v2 · submitted 2025-11-14 · 🧮 math.RT · math.FA

Estimating the spectral radius of Bell-type operator via finite dimensional approximation of orthogonal projections

Yuki Fujii , Toyohiro Tsurumaru This is my paper

Pith reviewed 2026-05-17 22:49 UTC · model grok-4.3

classification 🧮 math.RT math.FA

keywords orthogonal projectionsone-shifted formspectral radiusBell-CHSH operatordecomposition formulafinite dimensional approximationinvariant subspacesHilbert space

0 comments

The pith

Two orthogonal projections admit a decomposition into invariant subspaces that allows bounding the spectral radius of their commutator [P,Q].

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

The authors introduce a decomposition formula that breaks any separable Hilbert space into an orthogonal direct sum of subspaces invariant under two given orthogonal projections P and Q. Each subspace is either at most two-dimensional or infinite-dimensional, and the infinite ones can be represented in a one-shifted form that diagonalizes P and Q into small blocks. This form permits approximating the projections by finite-dimensional ones, which reduces spectral problems in infinite dimensions to finite-dimensional calculations. The approach is applied to bound the spectral radius of the commutator [P,Q], a quantity tied to the Bell-CHSH operator in quantum mechanics, providing bounds that are sharp in the constant-angle case.

Core claim

For any two orthogonal projections P and Q on a separable Hilbert space, there is an orthogonal direct sum decomposition into P-Q invariant subspaces where each infinite-dimensional component admits a one-shifted matrix representation. This representation enables finite-dimensional approximations of P and Q, from which upper and lower bounds on the spectral radius of [P,Q] follow, becoming exact when the form is constant-angle one-shifted.

What carries the argument

The one-shifted form, a specific block-diagonal matrix representation for P and Q on infinite-dimensional invariant subspaces that allows explicit finite-dimensional approximations by truncating the shift.

If this is right

Spectral problems for operators generated by P and Q, including polynomials, reduce to the one-shifted form case.
Upper and lower bounds are derived for the spectral radius of [P,Q].
The bounds are exact when P and Q are in constant-angle one-shifted form.
The approximation scheme is useful for numerical computations of spectral quantities.
Many problems involving two orthogonal projections in infinite dimensions can be analyzed via this framework.

Where Pith is reading between the lines

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

If the one-shifted form holds generally, it could simplify computations in quantum information for infinite-dimensional systems.
The method might extend to more than two projections or other operator algebras.
Testing the bounds on specific quantum mechanical models with known projection angles would verify the exactness claim.
Connections to other decomposition theorems in operator theory could be explored for broader applications.

Load-bearing premise

The assumption that every pair of orthogonal projections on a separable Hilbert space admits the decomposition into invariant subspaces each of which is either low-dimensional or admits the explicit one-shifted matrix representation without further restrictions.

What would settle it

Finding a pair of orthogonal projections on a separable Hilbert space for which no such decomposition exists or where the spectral radius bounds fail to hold even in the constant-angle case would falsify the main results.

read the original abstract

We establish a new decomposition formula for two orthogonal projections P and Q on a separable Hilbert space V. This formula yields an orthogonal direct sum decomposition of V into invariant subspaces under P and Q, each of which is either at most two dimensional or infinite dimensional. On every infinite dimensional component, the pair (P,Q) admits a matrix representation that we call the "one-shifted form". This representation diagonalizes both P and Q into blocks of size at most two, and moreover, both projections can be explicitly approximated by orthogonal projections on finite dimensional subspaces. This approximation scheme offers a way to derive infinite dimensional results from their finite dimensional counterparts and is also useful in numerical computations. This decomposition provides a useful framework for analyzing a wide range of problems involving two orthogonal projections in infinite dimensions. In particular, several spectral problems for operators generated by P and Q (including polynomials in P and Q) can be reduced to the case where the pair admits a one-shifted form. More concretely, we can estimate the spectral radius of [P,Q], which is equivalent to estimating the spectral radius of the Bell-CHSH operator, a quantity of fundamental importance in quantum mechanics. We provide an upper bound and a lower bound for the spectral radius of [P,Q], which become exact when the matrix representations of P and Q are in "constant-angle one-shifted form".

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a decomposition of the space under two projections into low-dim or infinite invariant pieces with an explicit one-shifted block form on the infinite summands, plus concrete bounds on the spectral radius of [P,Q].

read the letter

The main point is that Fujii and Tsurumaru decompose any separable Hilbert space into orthogonal P,Q-invariant subspaces that are either at most two-dimensional or infinite-dimensional. On each infinite piece they put the pair into what they call one-shifted form, a block matrix representation that diagonalizes both projections into at most 2x2 blocks and lets you approximate them by finite-dimensional projections. They then use this to bound the spectral radius of the commutator [P,Q], which is the same as the Bell-CHSH operator radius, and show the bounds are sharp in the constant-angle case. That reduction and the approximation scheme are the concrete new pieces. The framework is useful because many polynomials in P and Q become easier to handle once you are in the one-shifted setting. The application to quantum mechanics is stated clearly and the exactness check for constant angles is a good sanity test. The soft spot is the one the stress-test flags: the construction of the one-shifted basis on every infinite invariant subspace has to work for arbitrary spectra of the angle operator, including continuous intervals, not just discrete or constant cases. The abstract presents the result as holding for all pairs, so the proof must deliver that without extra restrictions; if it turns out the basis only exists under stronger assumptions on the spectrum, the general bounds would need to be stated more carefully. This is for people who already work with two projections in infinite dimensions, whether in operator theory or in quantum information. A reader who needs a systematic way to move spectral questions from infinite to finite settings will get direct value. It deserves a serious referee because the decomposition is new and the claimed reduction is explicit enough to check.

Referee Report

2 major / 2 minor

Summary. The paper establishes a decomposition theorem for any pair of orthogonal projections P and Q on a separable Hilbert space, yielding an orthogonal direct sum into P,Q-invariant subspaces that are either at most two-dimensional or infinite-dimensional. On each infinite-dimensional summand the pair is shown to admit a 'one-shifted form' matrix representation in which both projections are simultaneously block-diagonalized by at most 2×2 blocks; this representation moreover permits explicit approximation of P and Q by finite-rank orthogonal projections. The decomposition is then applied to obtain upper and lower bounds on the spectral radius of the commutator [P,Q] (equivalently, the Bell-CHSH operator), with equality attained precisely when the representation is of constant-angle one-shifted type.

Significance. If the decomposition and the one-shifted representation hold for arbitrary pairs, the work supplies a systematic reduction of infinite-dimensional spectral problems for polynomials in two projections to finite-dimensional or constant-angle cases, together with a concrete approximation scheme that is both theoretically useful and numerically implementable. The explicit link to the Bell-CHSH operator gives the bounds direct relevance in quantum information.

major comments (2)

[Decomposition Theorem and §4 (one-shifted representation)] The central reduction of the spectral-radius bounds to the one-shifted form on every infinite-dimensional invariant subspace is load-bearing. The manuscript must therefore supply a complete, self-contained construction (or existence proof) of the one-shifted basis when the spectrum of the angle operator (or of PQP) is a non-trivial interval rather than a single point or discrete set; without this, the claimed bounds hold only conditionally.
[Approximation scheme (around Eq. (3.5) or equivalent)] The finite-dimensional approximation scheme is asserted to converge to the infinite-dimensional operators in the one-shifted form. The rate or mode of convergence (strong, norm, or spectral) should be stated explicitly, because the passage from finite-dimensional spectral-radius estimates to the infinite-dimensional bound relies on this limit.

minor comments (2)

Notation for the 'one-shifted form' is introduced without a displayed matrix template; a single explicit block-matrix example would clarify the subsequent claims.
The abstract states that the bounds 'become exact' in the constant-angle case; the corresponding theorem number should be cross-referenced in the abstract for immediate readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and will revise the manuscript accordingly to strengthen the presentation.

read point-by-point responses

Referee: [Decomposition Theorem and §4 (one-shifted representation)] The central reduction of the spectral-radius bounds to the one-shifted form on every infinite-dimensional invariant subspace is load-bearing. The manuscript must therefore supply a complete, self-contained construction (or existence proof) of the one-shifted basis when the spectrum of the angle operator (or of PQP) is a non-trivial interval rather than a single point or discrete set; without this, the claimed bounds hold only conditionally.

Authors: We agree that an explicit, self-contained construction for the continuous-spectrum case strengthens the argument. Section 4 invokes the spectral theorem for the self-adjoint operator PQP to obtain the one-shifted representation on each infinite-dimensional summand. When the spectrum is a non-trivial interval we employ a direct-integral decomposition with respect to the spectral measure of PQP; the projections P and Q then take the indicated 2×2 block form almost everywhere with respect to that measure. To address the referee’s concern we will expand this argument in the revised manuscript with a fully detailed existence proof, including the explicit measurable selection of the angle function and the verification that the resulting subspaces remain invariant and orthogonal. revision: yes
Referee: [Approximation scheme (around Eq. (3.5) or equivalent)] The finite-dimensional approximation scheme is asserted to converge to the infinite-dimensional operators in the one-shifted form. The rate or mode of convergence (strong, norm, or spectral) should be stated explicitly, because the passage from finite-dimensional spectral-radius estimates to the infinite-dimensional bound relies on this limit.

Authors: We thank the referee for this clarification request. The finite-rank orthogonal projections P_n and Q_n are constructed by truncating the direct-integral decomposition to a finite number of spectral intervals; they converge to P and Q in the strong operator topology. Because the spectral radius is upper semi-continuous under strong limits for bounded self-adjoint operators, the finite-dimensional bounds pass to the limit and yield the stated infinite-dimensional estimate. We will add an explicit statement of strong convergence together with a short justification of the passage to the limit in the paragraph following the approximation scheme. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper introduces a new decomposition formula yielding an orthogonal direct sum into at most 2D or infinite-dimensional P,Q-invariant subspaces, with the latter admitting an explicit one-shifted matrix representation that enables finite-dimensional approximations. Bounds on the spectral radius of [P,Q] are then derived from this representation and stated to become exact only in the constant-angle special case. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the central claims rest on independent mathematical content that is externally falsifiable and does not presuppose the target spectral-radius estimates. This is the normal honest outcome for a paper whose derivation chain remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the existence of the stated orthogonal decomposition and the one-shifted matrix representation for every pair of projections on a separable Hilbert space. No free parameters are introduced. The only background assumptions are the standard axioms of Hilbert-space geometry and the spectral theorem for self-adjoint operators.

axioms (2)

domain assumption Every pair of orthogonal projections on a separable Hilbert space admits an orthogonal direct-sum decomposition into invariant subspaces that are either at most two-dimensional or infinite-dimensional.
This is the load-bearing decomposition formula announced in the abstract; its proof is required for all subsequent claims.
domain assumption On each infinite-dimensional invariant subspace the pair (P,Q) admits a matrix representation in one-shifted form that diagonalizes both projections into blocks of size at most two.
This representation is used to construct the finite-dimensional approximations and the spectral-radius bounds.

pith-pipeline@v0.9.0 · 5548 in / 1569 out tokens · 41098 ms · 2026-05-17T22:49:44.387585+00:00 · methodology

discussion (0)

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Relation between the paper passage and the cited Recognition theorem.

We establish a new decomposition formula for two orthogonal projections P and Q... one-shifted form... finite dimensional approximation... estimate the spectral radius of [P,Q]
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Relation between the paper passage and the cited Recognition theorem.

Theorem 1.3... sup λ∈F(A+B) b(λ) ≤ ρ([A,B]) ≤ lim inf b(λn)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages · 1 internal anchor

[1]

Böttcher, I

A. Böttcher, I. M. Spitkovsky, A gentle guide to the basics of two projec- tions theory, Linear Algebra and its Applications, vol.432, p.1412-1459 (2010). https://doi.org/10.1016/j.laa.2009.11.002. 48

work page doi:10.1016/j.laa.2009.11.002 2010
[2]

Chefles, S

A. Chefles, S. M. Barnett, Diagonalisation of the Bell-CHSH operator, Physics Letters A, vol.232, no.1-2, p.4-8 (1997). https://doi.org/10.1016/S0375-9601(97)00395-2

work page doi:10.1016/s0375-9601(97)00395-2 1997
[3]

L. A. Khalfin, B. S. Tsirelson, Quantum and quasi-classical analogs of Bell inequalities, in Symposium on the Foundations of Modern Physics, World Scientific Publishing, p.441-460 (1985)

work page 1985
[4]

L. J. Landau, Experimental tests of general quantum theo- ries, Letters in Mathematical Physics, vol.14, p.33-40 (1987). https://doi.org/10.1007/BF00403467

work page doi:10.1007/bf00403467 1987
[5]

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Proposed experiment totest localhidden-variable theories, PhysicalReview Letter vol.23, p.880-884 (1970). https://doi.org/10.1103/PhysRevLett.23.880

work page doi:10.1103/physrevlett.23.880 1970
[6]

Pironio , author A

S. Pironio, A. Acín, N. Brunner, N. Gisin, S. Massar, and V. Scarani, Device-independent quantum key distribution secure against collective attacks, New Journal of Physics, vol.11 (2009). https://doi.org/10.1088/1367-2630/11/4/045021

work page doi:10.1088/1367-2630/11/4/045021 2009
[7]

Behncke, Projections in Hilbert space II, Tohoku Mathematical Jour- nal, vol.23, p.349-352 (1971)

H. Behncke, Projections in Hilbert space II, Tohoku Mathematical Jour- nal, vol.23, p.349-352 (1971). https://doi.org/10.2748/tmj/1178242586

work page doi:10.2748/tmj/1178242586 1971
[8]

I. W. Primaatmaja, K. T. Goh, E. Y. Z. Tan, J. T. F. Khoo, S. Ghorai, and C.C.W. Lim, Security of device-independent quantum key distribution protocols: a review, Quantum, vol.7, p.932 (2023). https://doi.org/10.22331/q-2023-03-02-932

work page doi:10.22331/q-2023-03-02-932 2023
[9]

Robust self-testing of the singlet

M. McKague, T. H. Yang, and V. Scarani, Robust self-testing of the singlet, Journal of Physics A: Mathematical and Theoretical, vol.45, no.45:455304 (2012). https://doi.org/10.1088/1751-8113/45/45/455304

work page doi:10.1088/1751-8113/45/45/455304 2012
[10]

P. R. Halmos, Two subspaces, Transactions of the American Mathemati- cal Society, vol.144, p.381-389 (1969). https://doi.org/10.2307/1995288

work page doi:10.2307/1995288 1969
[11]

Allan, Invariant subspaces for pairs of projections, Journal of the London Mathematical Society, vol.57, no.2, p.449-468 (1998)

G. Allan, Invariant subspaces for pairs of projections, Journal of the London Mathematical Society, vol.57, no.2, p.449-468 (1998). https://doi.org/10.1112/S0024610798006103. 49

work page doi:10.1112/s0024610798006103 1998
[12]

B. S. Tsirelson, Quantum generalizations of Bell’s inequal- ity, Letters in Mathematical Physics, vol.4, p.93-100 (1980). https://doi.org/10.1007/BF00417500

work page doi:10.1007/bf00417500 1980
[13]

B. S. Tsirelson, Quantum analogues of the Bell inequalities. The case of two spatially separated domains, Journal of Soviet Mathematics, vol.36, p.557-570 (1987). https://doi.org/10.1007/BF01663472

work page doi:10.1007/bf01663472 1987
[14]

Estimating the spectral radius of Bell-type operator via finite dimensional approximation of orthogonal projections

Y. Fujii, T. Tsurumaru, Estimating the spectral radius of Bell-type op- erator via finite dimensional approximation of orthogonal projections, arXiv preprint arXiv:2511.10939v1 (2025)

work page internal anchor Pith review Pith/arXiv arXiv 2025
[15]

Böttcher, I

A. Böttcher, I. M. Spitkovsky, The wanted extension of Fujii and Tsuru- maru’s formula for the spectral radius of the Bell-CHSH operator, arXiv preprint arXiv:2601.09392 (2026)

work page arXiv 2026
[16]

Sugiura, Unitary representations and harmonic analysis, John Wi- ley&Sons, New York, London, Sydney, Toronto, 2nd edition, 1990, ISBN0444885935

M. Sugiura, Unitary representations and harmonic analysis, John Wi- ley&Sons, New York, London, Sydney, Toronto, 2nd edition, 1990, ISBN0444885935

work page 1990
[17]

T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach, New York, 1978, ISBN978-0486479293

work page 1978
[18]

https://doi.org/10.1017/S1446181108000102

W.C.Yueh, ExpliciteigenvaluesandinversesoftridiagonalToeplitzma- trices with four perturbed corners, the ANZIAM Journal, vol.49, no.3, p.361-387 (2008). https://doi.org/10.1017/S1446181108000102

work page doi:10.1017/s1446181108000102 2008
[19]

Yoshida, Functional Analysis, Springer-Verlag, Berlin Heidelberg and New York, 6th edition, 1980, ISBN0-387-10210-8

K. Yoshida, Functional Analysis, Springer-Verlag, Berlin Heidelberg and New York, 6th edition, 1980, ISBN0-387-10210-8

work page 1980
[20]

Tsurumaru and Y

T. Tsurumaru and Y. Fujii, Security proof of device-independent quan- tum key distribution against collective attacks in infinite dimensions, in preparation

work page
[21]

L. A. Khalfin, B. S. Tsirelson, Quantum/classical correspondence in the light of Bell’s inequalities, Foundation of Physics, vol.22, p.879-948 (1992). https://doi.org/10.1007/BF01889686. 50

work page doi:10.1007/bf01889686 1992

[1] [1]

Böttcher, I

A. Böttcher, I. M. Spitkovsky, A gentle guide to the basics of two projec- tions theory, Linear Algebra and its Applications, vol.432, p.1412-1459 (2010). https://doi.org/10.1016/j.laa.2009.11.002. 48

work page doi:10.1016/j.laa.2009.11.002 2010

[2] [2]

Chefles, S

A. Chefles, S. M. Barnett, Diagonalisation of the Bell-CHSH operator, Physics Letters A, vol.232, no.1-2, p.4-8 (1997). https://doi.org/10.1016/S0375-9601(97)00395-2

work page doi:10.1016/s0375-9601(97)00395-2 1997

[3] [3]

L. A. Khalfin, B. S. Tsirelson, Quantum and quasi-classical analogs of Bell inequalities, in Symposium on the Foundations of Modern Physics, World Scientific Publishing, p.441-460 (1985)

work page 1985

[4] [4]

L. J. Landau, Experimental tests of general quantum theo- ries, Letters in Mathematical Physics, vol.14, p.33-40 (1987). https://doi.org/10.1007/BF00403467

work page doi:10.1007/bf00403467 1987

[5] [5]

J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Proposed experiment totest localhidden-variable theories, PhysicalReview Letter vol.23, p.880-884 (1970). https://doi.org/10.1103/PhysRevLett.23.880

work page doi:10.1103/physrevlett.23.880 1970

[6] [6]

Pironio , author A

S. Pironio, A. Acín, N. Brunner, N. Gisin, S. Massar, and V. Scarani, Device-independent quantum key distribution secure against collective attacks, New Journal of Physics, vol.11 (2009). https://doi.org/10.1088/1367-2630/11/4/045021

work page doi:10.1088/1367-2630/11/4/045021 2009

[7] [7]

Behncke, Projections in Hilbert space II, Tohoku Mathematical Jour- nal, vol.23, p.349-352 (1971)

H. Behncke, Projections in Hilbert space II, Tohoku Mathematical Jour- nal, vol.23, p.349-352 (1971). https://doi.org/10.2748/tmj/1178242586

work page doi:10.2748/tmj/1178242586 1971

[8] [8]

I. W. Primaatmaja, K. T. Goh, E. Y. Z. Tan, J. T. F. Khoo, S. Ghorai, and C.C.W. Lim, Security of device-independent quantum key distribution protocols: a review, Quantum, vol.7, p.932 (2023). https://doi.org/10.22331/q-2023-03-02-932

work page doi:10.22331/q-2023-03-02-932 2023

[9] [9]

Robust self-testing of the singlet

M. McKague, T. H. Yang, and V. Scarani, Robust self-testing of the singlet, Journal of Physics A: Mathematical and Theoretical, vol.45, no.45:455304 (2012). https://doi.org/10.1088/1751-8113/45/45/455304

work page doi:10.1088/1751-8113/45/45/455304 2012

[10] [10]

P. R. Halmos, Two subspaces, Transactions of the American Mathemati- cal Society, vol.144, p.381-389 (1969). https://doi.org/10.2307/1995288

work page doi:10.2307/1995288 1969

[11] [11]

Allan, Invariant subspaces for pairs of projections, Journal of the London Mathematical Society, vol.57, no.2, p.449-468 (1998)

G. Allan, Invariant subspaces for pairs of projections, Journal of the London Mathematical Society, vol.57, no.2, p.449-468 (1998). https://doi.org/10.1112/S0024610798006103. 49

work page doi:10.1112/s0024610798006103 1998

[12] [12]

B. S. Tsirelson, Quantum generalizations of Bell’s inequal- ity, Letters in Mathematical Physics, vol.4, p.93-100 (1980). https://doi.org/10.1007/BF00417500

work page doi:10.1007/bf00417500 1980

[13] [13]

B. S. Tsirelson, Quantum analogues of the Bell inequalities. The case of two spatially separated domains, Journal of Soviet Mathematics, vol.36, p.557-570 (1987). https://doi.org/10.1007/BF01663472

work page doi:10.1007/bf01663472 1987

[14] [14]

Estimating the spectral radius of Bell-type operator via finite dimensional approximation of orthogonal projections

Y. Fujii, T. Tsurumaru, Estimating the spectral radius of Bell-type op- erator via finite dimensional approximation of orthogonal projections, arXiv preprint arXiv:2511.10939v1 (2025)

work page internal anchor Pith review Pith/arXiv arXiv 2025

[15] [15]

Böttcher, I

A. Böttcher, I. M. Spitkovsky, The wanted extension of Fujii and Tsuru- maru’s formula for the spectral radius of the Bell-CHSH operator, arXiv preprint arXiv:2601.09392 (2026)

work page arXiv 2026

[16] [16]

Sugiura, Unitary representations and harmonic analysis, John Wi- ley&Sons, New York, London, Sydney, Toronto, 2nd edition, 1990, ISBN0444885935

M. Sugiura, Unitary representations and harmonic analysis, John Wi- ley&Sons, New York, London, Sydney, Toronto, 2nd edition, 1990, ISBN0444885935

work page 1990

[17] [17]

T. S. Chihara, An introduction to orthogonal polynomials, Gordon and Breach, New York, 1978, ISBN978-0486479293

work page 1978

[18] [18]

https://doi.org/10.1017/S1446181108000102

W.C.Yueh, ExpliciteigenvaluesandinversesoftridiagonalToeplitzma- trices with four perturbed corners, the ANZIAM Journal, vol.49, no.3, p.361-387 (2008). https://doi.org/10.1017/S1446181108000102

work page doi:10.1017/s1446181108000102 2008

[19] [19]

Yoshida, Functional Analysis, Springer-Verlag, Berlin Heidelberg and New York, 6th edition, 1980, ISBN0-387-10210-8

K. Yoshida, Functional Analysis, Springer-Verlag, Berlin Heidelberg and New York, 6th edition, 1980, ISBN0-387-10210-8

work page 1980

[20] [20]

Tsurumaru and Y

T. Tsurumaru and Y. Fujii, Security proof of device-independent quan- tum key distribution against collective attacks in infinite dimensions, in preparation

work page

[21] [21]

L. A. Khalfin, B. S. Tsirelson, Quantum/classical correspondence in the light of Bell’s inequalities, Foundation of Physics, vol.22, p.879-948 (1992). https://doi.org/10.1007/BF01889686. 50

work page doi:10.1007/bf01889686 1992