Weak value

In

There are many excellent review articles on weak values (see e.g.[4]^[5][6][7] ) here we briefly cover the basics.

Definition

We will denote the initial state of a system as ${\displaystyle |\psi _{i}\rangle }$, while the final state of the system is denoted as ${\displaystyle |\psi _{f}\rangle }$. We will refer to the initial and final states of the system as the pre- and post-selected quantum mechanical states. With respect to these states, the weak value of the observable ${\displaystyle A}$ is defined as:

$${\displaystyle A_{w}={\frac {\langle \psi _{f}|A|\psi _{i}\rangle }{\langle \psi _{f}|\psi _{i}\rangle }}.}$$

Notice that if ${\displaystyle |\psi _{f}\rangle =|\psi _{i}\rangle }$ then the weak value is equal to the usual expected value in the initial state ${\displaystyle \langle \psi _{i}|A|\psi _{i}\rangle }$ or the final state ${\displaystyle \langle \psi _{f}|A|\psi _{f}\rangle }$. In general the weak value quantity is a complex number. The weak value of the observable becomes large when the post-selected state, ${\displaystyle |\psi _{f}\rangle }$, approaches being orthogonal to the pre-selected state, ${\displaystyle |\psi _{i}\rangle }$, i.e. ${\displaystyle |\langle \psi _{f}|\psi _{i}\rangle |\ll 1}$. If ${\displaystyle A_{w}}$ is larger than the largest eigenvalue of ${\displaystyle A}$ or smaller than the smallest eigenvalue of ${\displaystyle A}$ the weak value is said to be anomalous.

As an example consider a spin 1/2 particle.^[8] Take ${\displaystyle A}$ to be the Pauli Z operator ${\displaystyle A=\sigma _{z}}$ with eigenvalues ${\displaystyle \pm 1}$. Using the initial state

$${\displaystyle |\psi _{i}\rangle ={\frac {1}{\sqrt {2}}}{\begin{pmatrix}\cos {\frac {\alpha }{2}}+\sin {\frac {\alpha }{2}}\\\cos {\frac {\alpha }{2}}-\sin {\frac {\alpha }{2}}\end{pmatrix}}}$$

$${\displaystyle |\psi _{f}\rangle ={\frac {1}{\sqrt {2}}}{\begin{pmatrix}1\\1\end{pmatrix}}}$$

$${\displaystyle A_{w}=(\sigma _{z})_{w}=\tan {\frac {\alpha }{2}}.}$$

For ${\displaystyle |\alpha |>{\frac {\pi }{2}}}$ the weak value is anomalous.

Derivation

Here we follow the presentation given by Duck, Stevenson, and Sudarshan,^[8] (with some notational updates from Kofman et al.^[4] )which makes explicit when the approximations used to derive the weak value are valid.

Consider a quantum system that you want to measure by coupling an ancillary (also quantum) measuring device. The observable to be measured on the system is ${\displaystyle A}$. The system and ancilla are coupled via the Hamiltonian

$${\displaystyle H=\gamma A\otimes p,}$$

${\textstyle \gamma =\int _{t_{i}}^{t_{f}}g(t)dt\ll 1}$

${\displaystyle [q,p]=i}$

$${\displaystyle U=\exp[-i\gamma A\otimes p].}$$

Take the initial state of the ancilla to have a Gaussian distribution

$${\displaystyle |\Phi \rangle ={\frac {1}{(2\pi \sigma ^{2})^{1/4}}}\int dq'\exp[-q'^{2}/4\sigma ^{2}]|q'\rangle ,}$$

$${\displaystyle \Phi (q)=\langle q|\Phi \rangle ={\frac {1}{(2\pi \sigma ^{2})^{1/4}}}\exp[-q^{2}/4\sigma ^{2}].}$$

The initial state of the system is given by ${\displaystyle |\psi _{i}\rangle }$ above; the state ${\displaystyle |\Psi \rangle }$, jointly describing the initial state of the system and ancilla, is given then by:

$${\displaystyle |\Psi \rangle =|\psi _{i}\rangle \otimes |\Phi \rangle .}$$

Next the system and ancilla interact via the unitary ${\displaystyle U|\Psi \rangle }$. After this one performs a projective measurement of the projectors ${\displaystyle \{|\psi _{f}\rangle \langle \psi _{f}|,I-|\psi _{f}\rangle \langle \psi _{f}|\}}$ on the system. If we postselect (or condition) on getting the outcome ${\displaystyle |\psi _{f}\rangle \langle \psi _{f}|}$, then the (unnormalized) final state of the meter is

$${\displaystyle {\begin{aligned}|\Phi _{f}\rangle &=\langle \psi _{f}|U|\psi _{i}\rangle \otimes |\Phi \rangle \\&\approx \langle \psi _{f}|(I\otimes I-i\gamma A\otimes p)|\psi _{i}\rangle \otimes |\Phi \rangle \quad {\text{(I)}}\\&=\langle \psi _{f}|\psi _{i}\rangle (1-i\gamma A_{w}p)|\Phi \rangle \\&\approx \langle \psi _{f}|\psi _{i}\rangle \exp(-i\gamma A_{w}p)|\Phi \rangle .\quad {\text{(II)}}\end{aligned}}}$$

To arrive at this conclusion, we use the first order series expansion of ${\displaystyle U}$ on line (I), and we require that^[4][8]

$${\displaystyle {\begin{aligned}{\frac {|\gamma |}{\sigma }}\left|{\frac {\langle \psi _{f}|A^{n}|\psi _{i}\rangle }{\langle \psi _{f}|A|\psi _{i}\rangle }}\right|^{1/(n-1)}\ll 1,\quad (n=2,3,\dots )\end{aligned}}}$$

On line (II) we use the approximation that ${\displaystyle e^{-x}\approx 1-x}$ for small ${\displaystyle x}$. This final approximation is only valid when^[4][8]

$${\displaystyle |\gamma A_{w}|/\sigma \ll 1.}$$

As ${\displaystyle p}$ is the generator of translations, the ancilla's wavefunction is now given by

$${\displaystyle \Phi _{f}(q)=\Phi (q-\gamma A_{w}).}$$

This is the original wavefunction, shifted by an amount ${\displaystyle \gamma A_{w}}$. By Busch's theorem^[9] the system and meter wavefunctions are necessarily disturbed by the measurement. There is a certain sense in which the protocol that allows one to measure the weak value is minimally disturbing,^[10] but there is still disturbance.^[10]

Applications

Quantum metrology and tomography

At the end of the original weak value paper^[1] the authors suggested weak values could be used in quantum metrology:

This suggestion was followed by Hosten and Kwiat^[11] and later by Dixon et al.^[12] It appears to be an interesting line of research that could result in improved quantum sensing technology.

Additionally in 2011, weak measurements of many photons prepared in the same pure state, followed by strong measurements of a complementary variable, were used to perform quantum tomography (i.e. reconstruct the state in which the photons were prepared).^[13]

Quantum foundations

Weak values have been used to examine some of the paradoxes in the foundations of quantum theory. This relies to a large extent on whether weak values are deemed to be relevant to describe properties of quantum systems,^[14] a point which is not obvious since weak values are generally different from eigenvalues. For example, the research group of Aephraim M. Steinberg at the University of Toronto confirmed Hardy's paradox experimentally using joint weak measurement of the locations of entangled pairs of photons.^[15][16] (also see^[17])

Building on weak measurements,

Weak values have been implemented into quantum computing to get a giant speed up in time complexity. In a paper,[22] Arun Kumar Pati describes a new kind of quantum computer using weak value amplification and post-selection (WVAP), and implements search algorithm which (given a successful post selection) can find the target state in a single run with time complexity ${\displaystyle O(\log N)}$, beating out the well known Grover's algorithm.

Criticisms

Criticisms of weak values include philosophical and practical criticisms. Some noted researchers such as

Recently, it has been shown that the pre- and postselection of a quantum system recovers a completely hidden interference phenomenon in the measurement apparatus. Studying the interference pattern shows that what is interpreted as an amplification using the weak value is a pure phase effect and the weak value plays no role in its interpretation. This phase effect increases the degree of the entanglement which lies behind the effectiveness of the pre- and postselection in the parameter estimation.[23]

Further reading

• Zeeya Merali (April 2010). "Back From the Future". Discover. A series of quantum experiments shows that measurements performed in the future can influence the present.{{cite journal}}: CS1 maint: postscript (link)
• "Quantum physics first: Researchers observe single photons in two-slit interferometer experiment". phys.org. June 2, 2011. {{cite journal}}: Cite journal requires |journal= (help)
• Adrian Cho (5 August 2011). "Furtive Approach Rolls Back the Limits of Quantum Uncertainty". Science. 333 (6043): 690–693.
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