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Projective measurement

by Yee Wei Law - Sunday, 4 June 2023, 12:31 PM
 

The discussion here follows from the discussion of the positive operator-valued measure (POVM).

The POVM cannot be used to determine the post-measurement state because the post-measurement state may not be pure [WN17, p. 13]. Instead, a Kraus operator representation of the POVM, known as projective measurement, is necessary to specify the post-measurement state.

Projective measurements are repeatable, and the outcome observed as a result of projective measurement is deterministic [MM12, Sec. 2.7], as explained below.

This repeatability implies many important measurements in quantum mechanics are not projective measurements; for instance, if we use a silvered screen to measure the position of a photon, we destroy the photon in the process, and obviously this measurement cannot be repeated.

Projective measurements are often discussed in terms of an observable, which is defined as a Hermitian operator acting on the state space of a system.

Every observable has a spectral decomposition [Mey00, p. 517]:

where are the eigenvalues of , and are so-called spectral/orthogonal projectors onto the null space of . More precisely, 1️⃣ are Hermitian and idempotent, 2️⃣ , and 3️⃣ are mutually orthogonal.

Clearly, are valid POVM elements.

A (non-unique) Kraus operator representation or Kraus decomposition of is defined as [WN17, Definition 1.5.2]:

where .

  • If we define , then .
  • If we define , where is any unitary matrix, then .
  • Thus, there is no unique Kraus representation/decomposition.

With Kraus operators, we can now define projective measurement:

Definition 1: Projective measurement [WN17, Definition 1.5.4; MM12, p. 153]

A projective measurement is given by a set of orthogonal projectors  such that . By default, the Kraus operator is chosen to be .

The probability of observing measurement outcome given initial state is

The post-measurement state is

For pure state , the post-measurement state is

Most authors equate the term “von Neumann measurement” to “projective measurement”, but some consider the former to be a special case of the latter [KLM07, p.50].

Since the default Kraus operators are the same as the orthogonal projectors, most authors skip discussion of Kraus operators altogether 🤷‍♂️.

Measuring the observable is equivalent to performing a projective measurement with respect to the decomposition , where the measurement outcome corresponds to eigenvalue .

Example 1 [KLM07, Example 3.4.1]

Consider the Pauli observable , i.e., the Pauli-Z operator/matrix:

which does not change an input of , but flips to (which is equivalent to with a phase change).

The Pauli-Z operator acts as a NOT operator in the Hadamard basis.

has eigenpairs and , where the eigenvectors and are also called eigenstates [MM12, p. 338].

Thus, has spectral decomposition:

with orthogonal projectors

Interpreting the above, a projective measurement of is a measurement in the standard basis with eigenvalue corresponding to final state and eigenvalue corresponding to final state .

Example 2 [WN17, Example 1.5.3]

Given a two-qubit state , suppose we want to measure the parity of the two qubits in the standard basis.

One method is to measure in the (-dimensional) standard basis, obtain two classical bits, and take their parity.

In this case, the probability of obtaining outcome “even” is

and the post-measurement state is

Another method is to measure the parity using projective measurement which directly projects onto the relevant subspaces, without measuring the qubits individually. Define projectors:

which we can quickly verify to be orthogonal and conformant with the completeness relation:

In this case, the probability of getting outcome “even” is

which is the same as before; and the post-measurement state is

The preceding two methods produce different outcomes on the EPR state , or in density-matrix representation,

By measurement in the standard basis, the probability of getting outcome “even” is

while the post-measurement state is

By projective measurement, is the same as before, but the post-measurement state is different:

Thus, projective measurement with does not change the EPR state.

This is one of the main advantages of using projective measurement as opposed to basis measurement: the former enables simple computation (e.g., parity) on multi-qubit states without fully “destroying” the state, as basis measurement does.

In Example 2, the observation that the projective measurement preserves the quantum state is not a coincidence. In fact, building on Definition 1, applying measurement to , i.e., applying to twice gives us [MM12, p. 153]:

where .

References

[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.
[MM12] D. C. Marinescu and G. M. Marinescu, Classical and Quantum Information, Elsevier, 2012. https://doi.org/10.1016/C2009-0-64195-7.
[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.
[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/.