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Schatten norm

by Yee Wei Law - Sunday, 19 March 2023, 9:47 PM
 

For , suppose its singular values are .

Then, the Schatten -norm of , where , is defined as [Ber09, Proposition 9.2.3]:

When , we have the Schatten 1-norm, which is also called the trace norm or nuclear norm [Ber09, p. 549]:

When , we have the Frobenius norm:

When , we have the spectral norm:

References

[Ber09] D. R. Bernstein, Matrix Mathematics: Theory, Facts, and Formulas, 2nd ed., Princeton University Press, 2009.

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Schmidt decomposition

by Yee Wei Law - Wednesday, 14 June 2023, 10:11 PM
 

References

[Cho22] M.-S. Choi, A Quantum Computation Workbook, Springer Cham, 2022. https://doi.org/10.1007/978-3-030-91214-7.

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Secret key rate

by Yee Wei Law - Thursday, 23 March 2023, 12:44 PM
 

The secret key rate is the fraction of secure key bits produced per protocol round, where a round is the transmission of a quantum state through the quantum channel [Gra21, p. 38].

The secret key rate generally depends on the total number of rounds performed.

The asymptotic secret key rate (often just asymptotic key rate) is the secret key rate when is assumed to simplify analysis.

  • Assuming direct reconciliation, the asymptotic key rate of any quantum key distribution (QKD) protocol with one-way error correction is lower-bounded by the Devetak-Winter rate [DW05]:

    where denotes quantum mutual information, , and are the random variables representing Alice’s, Bob’s and Eve’s raw key bits.

  • An intuitive interpretation of the Devetak-Winter rate: the fraction of secret bits generated per round of using the protocol is equal to the amount of information shared by Alice and Bob, , minus the amount of information that Eve has on Alice’s part of the key, [Wol21, p.145].

Assuming is not realistic, and the security of a QKD protocol has to be analysed assuming a finite and generally finite resources.

Analysis of the secret key rate and associated security properties of a QKD protocol is called finite-key analysis [TLGR12], to be covered in the future.

References

[DW05] I. Devetak and A. Winter, Distillation of secret key and entanglement from quantum states, Proceedings: Mathematical, Physical and Engineering Sciences 461 no. 2053 (2005), 207–235.
[Gra21] F. Grasselli, Quantum Cryptography: From Key Distribution to Conference Key Agreement, Quantum Science and Technology, Springer Cham, 2021. https://doi.org/10.1007/978-3-030-64360-7.
[TLGR12] M. Tomamichel, C. C. W. Lim, N. Gisin, and R. Renner, Tight finite-key analysis for quantum cryptography, Nat Commun 3 no. 634 (2012). https://doi.org/10.1038/ncomms1631.
[Wol21] R. Wolf, Quantum Key Distribution: An Introduction with Exercises, Springer, Cham, 2021. https://doi.org/10.1007/978-3-030-73991-1.

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Security of quantum key distribution (QKD): overview

by Yee Wei Law - Tuesday, 29 August 2023, 5:25 PM
 

Methods for analysing the security of quantum key distribution (QKD) schemes/protocols are still being developed [PAB+20, PR22].

  • These methods combine existing cryptographic notions and techniques with quantum information theory.
  • Depending on the implementation of the QKD scheme, physical laws governing the implementation (e.g., quantum-optical laws) also play a role.
  • Compared to computational security, which can be reduced to complexity-theoretic reasoning based on the Turing machine, methods for analysing the security of QKD schemes are thus more involved and, in terms of the transdisciplinary effort required, more challenging.

The security of a QKD scheme is typically analysed in terms of the level of success of 1️⃣ individual attacks, 2️⃣ collective attacks and 3️⃣ coherent attacks; in 🔼 increasing order of power given to the adversary [Wol21, Sec. 5.3.1].

  • Individual and collective attacks are usually considered in order to simplify the security analysis, but it is necessary to also consider coherent attacks in order to prove the security of a QKD scheme.
  • We can analyse the different types of attacks by the way 😈 Eve interacts with 👩 Alice’s signals and how Eve processes the information she gets in this way.
  • General procedure for extracting information from a quantum system:

    1. 😈 Eve attaches an ancilla system in the predefined state to the quantum state transmitted by 👩 Alice; and are density-matrix representations of quantum states.

      Please spend time understanding pure state, mixed state and density matrix.

    2. 😈 Eve then performs a unitary operation on the composite system, which leaves the state of the ancilla system in the form:

      where is the Kronecker operator and denotes partial trace.

      Please spend time understanding reduced density matrix and partial trace.

The security of a QKD scheme is also often analysed in terms of composability, short for universally composable security.

  • Informally, we say a QKD scheme is composable if the key it produces is almost as good as if it were distributed with an ideal key distribution protocol [Van06, Sec. 12.2.6].
  • A cryptographic primitive, which is secure when used with an ideally secret key, must still be secure if used with a QKD-distributed key.
  • Composability is critical since QKD-derived secret keys are used in other applications, e.g., data encryption [JCM+22].

TODO: Trace distance criterion [PR22], security = correctness + secrecy [Gra21]. Asymptotic vs finite-key security analysis.

Security evaluation of practical QKD implementations involves evaluating the level of success of “quantum hacking” (i.e., side-channel attacks on QKD).

References

[Gra21] F. Grasselli, Quantum Cryptography: From Key Distribution to Conference Key Agreement, Quantum Science and Technology, Springer Cham, 2021. https://doi.org/10.1007/978-3-030-64360-7.
[JCM+22] N. Jain, H.-M. Chin, H. Mani, C. Lupo, D. S. Nikolic, A. Kordts, S. Pirandola, T. B. Pedersen, M. Kolb, B. Ömer, C. Pacher, T. Gehring, and U. L. Andersen, Practical continuous-variable quantum key distribution with composable security, Nature Communications 13 no. 1 (2022), 4740. https://doi.org/10.1038/s41467-022-32161-y.
[PAB+20] S. Pirandola, U. L. Andersen, L. Banchi, M. Berta, D. Bunandar, R. Colbeck, D. Englund, T. Gehring, C. Lupo, C. Ottaviani, J. L. Pereira, M. Razavi, J. Shamsul Shaari, M. Tomamichel, V. C. Usenko, G. Vallone, P. Villoresi, and P. Wallden, Advances in quantum cryptography, Advances in Optics and Photonics 12 no. 4 (2020), 1012–1236. https://doi.org/10.1364/AOP.361502.
[PR22] C. Portmann and R. Renner, Security in quantum cryptography, Rev. Mod. Phys. 94 no. 2 (2022), 025008. https://doi.org/10.1103/RevModPhys.94.025008.
[Sch10] S. Schauer, Attack Strategies on QKD Protocols, in Applied Quantum Cryptography (C. Kollmitzer and M. Pivk, eds.), Lect. Notes Phys. 797, Springer Berlin Heidelberg, 2010, pp. 71–95. https://doi.org/10.1007/978-3-642-04831-9_5.
[TL17] M. Tomamichel and A. Leverrier, A largely self-contained and complete security proof for quantum key distribution, Quantum 1 (2017), 14. https://doi.org/10.22331/q-2017-07-14-14.
[Van06] G. Van Assche, Quantum Cryptography and Secret-Key Distillation, Cambridge University Press, 2006. https://doi.org/10.1017/CBO9780511617744.
[Wol21] R. Wolf, Quantum Key Distribution: An Introduction with Exercises, Springer, Cham, 2021. https://doi.org/10.1007/978-3-030-73991-1.

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Separable vs entangled

by Yee Wei Law - Friday, 9 June 2023, 10:13 AM
 

The joint state, , of two quantum systems and is separable [WN17, Definition 2.1.1] if there exists a probability distribution , and sets of density matrices and such that

(1)

If such an expression for does not exist, is entangled [WN17, Definition 2.1.1].

Specifically, if is a pure state, then is separable if and only there exists and such that

Example 1: [WN17, Example 2.1.3]

The density matrix

is separable because it takes the form of Eq. (1).

Subsystems and are not entangled, but they are (classically) correlated.

Example 2: [WN17, Example 2.1.2]

The state is separable because

Example 3: [WN17, Example 2.1.1]
The Einstein-Podolsky-Rosen (EPR) pair is entangled because it cannot be expressed in the form of Eq. (1).
Example 4: [WN17, Example 2.1.4]

This example is meant to highlight the difference between the following two states:

where . is separable whereas is not.

Consider the outcomes of measuring subsystem in and in the standard basis and in the Hadamard basis.

Measuring subsystem of in the Hadamard basis:

Define the measurement operators to be and , where and . Clearly, and .

Using projective measurement, the post-measurement state conditioned on measurement outcome is

Let us work out the numerator and the denominator separately, starting with the numerator:

The preceding equality follows from these identities, which you will derive in the practical:

The denominator is:

The preceding computation is tedious but straightforward given the right tool (e.g., NumPy). Combining the results for the numerator and denominator, we get

Thus, upon measuring a on subsystem , subsystem can be in either or at equal probabilities; we say the reduced state on is maximally mixed.

Measuring subsystem of in the Hadamard basis:

The pair of and can be re-expressed as and , because ‘’ is orthogonal to ‘’ just as ‘’ is orthogonal to ‘’.

The preceding statement implies when is measured on subsystem , subsystem is in state as well.

Correlations in are thus stronger than those in .

References

[WN17] S. Wehner and N. Ng, Lecture Notes: edX Quantum Cryptography, CaltechDelftX: QuCryptox, 2017. Available at https://courses.edx.org/courses/course-v1:CaltechDelftX+QuCryptox+3T2018/pdfbook/0/.

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Singular value

by Yee Wei Law - Saturday, 11 March 2023, 3:16 PM
 

Singular values, like eigenvalues, are an intrinsic property of a matrix. Unsurprisingly, they can be defined in terms of eigenvalues:

Definition 1: Singular value [Ber09, Definition 5.6.1]

Suppose matrix has eigenvalues , where . Then, the singular values of are the nonnegative numbers:

Singular values can also be defined through the operation called singular value decomposition:

Definition 2: Singular value decomposition and singular values [Hog13, Sec. 5.6]

A singular value decomposition (SVD) of a matrix is the factorisation:

where , and are unitary.

The diagonal entries of , namely , are the singular values of .

The columns of are the left singular vectors of .

The columns of are the right singular vectors of .

References

[Ber09] D. R. Bernstein, Matrix Mathematics: Theory, Facts, and Formulas, 2nd ed., Princeton University Press, 2009.
[Hog13] L. Hogben (ed.), Handbook of Linear Algebra, 2nd ed., CRC Press, 2013. https://doi.org/10.1201/b16113.

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Spectral theorem and spectral decomposition

by Yee Wei Law - Saturday, 24 August 2024, 11:14 PM
 

Spectral theorem is one of the fundamental theorems of linear algebra [ABH09, Sec. 3.6].

Based on the spectral theorem, spectral decomposition is an essential tool in quantum theory [NC10, Box 2.2].

Multiple equivalent interpretations of the spectral theorem exist, e.g., [ABH09, Theorems 3.6.4 and 3.6.12].

The interpretation in Theorem 1 directly defines spectral decomposition, and is hence also called the spectral decomposition theorem.

Theorem 1: Spectral (decomposition) theorem [Hol13, Theorem 8.23; Zha11, Theorem 3.4; KLM07, Theorem 2.4.3]

An -square complex matrix is normal iff it is orthogonally diagonalisable or unitarily diagonalisable, i.e., there exists a unitary matrix such that

where

  • are the eigenvalues (i.e., spectrum) of ;
  • consists of the orthonormal eigenvectors of in its columns in the same order as .

In particular,

  • is positive semidefinite .
  • is Hermitian are real.
  • is unitary .

Any normal operator, , has an outer product representation [KLM07, Sec. 2.4; Mey00, p. 517]:

where

  • are the eigenpairs of ;
  • form an orthormal basis of the Hilbert space in which is defined.

The outer products are projectors that satisfy

  • the completeness relation: ; and
  • the orthonormality relation: , where is the Kronecker delta.
Example 1 [KLM07, Theorem 2.4.2, Example 2.4.4]

The Pauli-X matrix is a normal operator:

Manually or using NumPy, we can determine the eigenpairs of to be and . Thus,

References

[ABH09] M. A. Akcoglu, P. F. A. Bartha, and D. M. Ha, Analysis in Vector Spaces: A Course in Advanced Calculus, John Wiley & Sons, 2009. https://doi.org/10.1002/9781118164587.
[Hol13] J. Holt, Linear Algebra with Applications, W. H. Freeman and Company, 2013.
[KLM07] P. Kaye, R. Laflamme, and M. Mosca, An Introduction to Quantum Computing, Oxford University Press, 2007. Available at https://ebookcentral.proquest.com/lib/unisa/reader.action?docID=415080.
[Mey00] C. Meyer, Matrix Analysis and Applied Linear Algebra, SIAM, 2000. Available at http://portal.igpublish.com.eu1.proxy.openathens.net/iglibrary/obj/SIAMB0000114.
[NC10] M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information, 10th anniversary ed., Cambridge University Press, 2010. Available at http://mmrc.amss.cas.cn/tlb/201702/W020170224608149940643.pdf.
[Zha11] F. Zhang, Matrix Theory: Basic Results and Techniques, 2nd ed., Universitext, Springer New York, NY, 2011. https://doi.org/10.1007/978-1-4614-1099-7.

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SymPy

by Yee Wei Law - Wednesday, 2 August 2023, 3:21 PM
 

SymPy is a Python library for symbolic computing. In symbolic computing, we reason with symbols rather than numeric values.

When running SymPy in Google Colab, make sure you are using a WebKit-based browser such as Chrome or Edge.

The first thing to do when using SymPy is creating symbols.

There are seven ways to create a symbol.

Right from the beginning, it is crucial to know how to make assumptions in SymPy.

  • A really useful method for making assumptions is refine.