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Abstract algebra

by Yee Wei Law - Monday, 11 September 2023, 3:23 PM
 

See 👇 attachment or the latest source on Overleaf, on the topics of groups and Galois fields (pending).

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Ancilla

by Yee Wei Law - Wednesday, 7 June 2023, 1:01 PM
 

An ancilla (system) is an auxiliary quantum-mechanical system [ETS18].

Think of ancilla as something extra that is used to achieve some goal [Pre18].

References

[ETS18] ETSI, Quantum Key Distribution (QKD); Vocabulary, Group Report ETSI GR QKD 007 v1.1.1, December 2018. Available at https://www.etsi.org/deliver/etsi_gr/QKD/001_099/007/01.01.01_60/gr_qkd007v010101p.pdf.
[Pre18] J. Preskill, Lecture Notes for Ph219/CS219: Quantum Information Chapter 3, 2018. Available at http://theory.caltech.edu/~preskill/ph219/chap2_15.pdf.
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Axiomatic approach to quantum operations

by Yee Wei Law - Wednesday, 7 June 2023, 1:02 PM
 

In quantum information theory, the foundation of quantum cryptography, the concepts of general/generalised measurements and quantum channels are rooted in the concept of quantum operation [Pre18, Sec. 3.2.4].

  • A general measurement can be realised by entangling a system with a meter and performing a projective measurement on the meter.
  • A quantum channel arises if we measure the meter but completely forget the measurement outcome.

While the time evolution of a closed quantum system can be expressed using a single unitary operator, the time evolution of an open quantum system is much less straightforward [HXK20], and this is where the theory of quantum operations come in.

Here, the axiomatic approach in [NC10, Sec. 8.2.4] to defining quantum operations is discussed, starting with Definition 1.

Definition 1: Quantum operation [NC10, p. 367]

A quantum operation is a map from the set of density operators for the input space to the set of density operators for the output space , that satisfies these axioms:

Axiom 1

, with value between 0 and 1 inclusive, is the probability that the process represented by occurs, when is the initial state.

💡 Recall Eq. (1) in the discussion of density operator.

Axiom 2

is a convex-linear map on the set of density operators, i.e., for probabilities ,

Axiom 3

is a completely positive map, i.e., if maps density operators of system to density operators of system , then must be positive for any positive operator .

Furthermore, if we introduce an extra system of arbitrary dimensionality, then , where denotes the identity map on system , is positive for any positive operator on the combined system .

Justification of Axiom 1: In the absence of measurement, by itself completely describes the quantum operation, and Axiom 1 reduces to the requirement that ; in this case, the quantum operation is trace-preserving.

  • If does not completely describe the operation, then there exists such that , and is so-called non-trace-preserving.
  • Every physical quantum operation on satisfies the requirement that .

Justification of Axiom 2: Let , then

where is a normalisation factor.

By Bayes’ rule,

Justification of Axiom 3: A quantum operation maps a density operator to a density operator — by itself or as part of a larger system — and density operators are always positive, so must be a completely positive map.

Example 1 [Pre18, p. 19]

This example is meant to show that the transpose operation is not a completely positive map and hence not a quantum operation.

We first observe that is positive if is positive because the quadratic form

However, the transpose operation is not completely positive due to the following.

Suppose system contains entangled subsystems and , and has state , ignoring the normalisation constant.

The density matrix of is

Above, we applied the identity .

Now consider the composite map , where is the transpose operation, which transforms to .

Applying the composite map to , we get

Applying the composite map to the above, we get back

Therefore, if we represent the map with a square matrix, the matrix is involutory (square = identity), and it is trivial to show that the eigenvalues of an involutory matrix are (see below), implying the composite map is not positive.

Note: For involutory , .

In other words, the transpose operation is not a completely positive map, and hence not a quantum operation.

Axioms 1-3 lead to the following important theorem.

Theorem 1 [NC10, Theorem 8.1]

The map satisfies Axioms 1-3 if and only if

(1)

for some set of operators that map the input Hilbert space to the output Hilbert space, and .

In Theorem 1,

  • The statement is equivalent to the statement is positive-semidefinite.
  • are so-called Kraus operators or operation elements [Pre18, Sec. 3.2.1], and Eq. (1) is called an operator-sum representation of [MM12, Sec. 2.14].
    • Operator-sum representations are not unique because of the unitary freedom in these representations [NC10, Theorem 8.2]; see Example 2.
  • If , then form a POVM [WN17, Definition 1.5.2], and we have a nonunique Kraus decomposition of .
    • A Kraus decomposition always exists for POVM by simply setting , the positive square root of (Python function scipy.lingalg.sqrtm, MATLAB function sqrtm).
    • If is a projector, then and we can set .
Example 2 [MM12, p. 176]

Consider the Kraus operators or operation elements: and , where .

An equivalent operator-sum representation can be provided by and , because

The freedom in the operator-sum representation is especially useful for studying quantum error correction.

Theorem 2 [NC10, Theorem 8.3]

Any quantum operation on a system of -dimensional Hilbert space can be generated by an operator-sum representation containing at most elements:

where .

Watch lectures on Kraus representations by Artur Ekert, inventor of the E91 QKD protocol:

References

[HXK20] Z. Hu, R. Xia, and S. Kais, A quantum algorithm for evolving open quantum dynamics on quantum computing devices, Sci Rep 10 (2020), 3301. https://doi.org/10.1038/s41598-020-60321-x.
[MM12] D. C. Marinescu and G. M. Marinescu, Classical and Quantum Information, Elsevier, 2012. https://doi.org/10.1016/C2009-0-64195-7.
[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.
[Pre18] J. Preskill, Lecture Notes for Ph219/CS219: Quantum Information Chapter 3, 2018. Available at http://theory.caltech.edu/~preskill/ph219/chap2_15.pdf.
[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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