Weak value

In quantum mechanics (and computation), a weak value is a quantity related to a shift of a measuring device's pointer when usually there is pre- and postselection. It should not be confused with a weak measurement, which is often defined in conjunction. The weak value was first defined by Yakir Aharonov, David Albert and Lev Vaidman, published in Physical Review Letters 1988, and is related to the two-state vector formalism. There is also a way to obtain weak values without postselection.Aharonov, Albert, Vaidman In quantum mechanics (and computation), a weak value is a quantity related to a shift of a measuring device's pointer when usually there is pre- and postselection. It should not be confused with a weak measurement, which is often defined in conjunction. The weak value was first defined by Yakir Aharonov, David Albert and Lev Vaidman, published in Physical Review Letters 1988, and is related to the two-state vector formalism. There is also a way to obtain weak values without postselection. There are many excellent review articles on weak values (see e.g. ) here we briefly cover the basics. We will denote the initial state of a system as | ψ i ⟩ {displaystyle |psi _{i} angle } , while the final state of the system is denoted as | ψ f ⟩ {displaystyle |psi _{f} angle } . 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 state the weak value of the observable A {displaystyle A} is defined as: A w = ⟨ ψ f | A | ψ i ⟩ ⟨ ψ f | ψ i ⟩ . {displaystyle A_{w}={frac {langle psi _{f}|A|psi _{i} angle }{langle psi _{f}|psi _{i} angle }}.} Notice that if | ψ f ⟩ = | ψ i ⟩ {displaystyle |psi _{f} angle =|psi _{i} angle } then the weak value is equal to the usual expected value in the initial state ⟨ ψ i | A | ψ i ⟩ {displaystyle langle psi _{i}|A|psi _{i} angle } or the final state ⟨ ψ f | A | ψ f ⟩ {displaystyle langle psi _{f}|A|psi _{f} angle } . In general the weak value quantity is a complex number. The weak value of the observable becomes large when the post-selected state, | ψ f ⟩ {displaystyle |psi _{f} angle } , approaches being orthogonal to the pre-selected state, | ψ i ⟩ {displaystyle |psi _{i} angle } , i.e. ⟨ ψ f | ψ i ⟩ ≪ 1 {displaystyle langle psi _{f}|psi _{i} angle ll 1} . If A w {displaystyle A_{w}} is larger than the largest eigenvalue of A {displaystyle A} or smaller than the smallest eigenvalue of A {displaystyle A} the weak value is said to be anomalous. As an example consider a spin 1/2 particle. Take A {displaystyle A} to be the Pauli Z operator A = σ z {displaystyle A=sigma _{z}} with eigenvalues ± 1 {displaystyle pm 1} . Using the initial state | ψ i ⟩ = 1 2 ( cos ⁡ α 2 + sin ⁡ α 2 cos ⁡ α 2 − sin ⁡ α 2 ) {displaystyle |psi _{i} angle ={frac {1}{sqrt {2}}}left({egin{array}{c}cos {frac {alpha }{2}}+sin {frac {alpha }{2}}\cos {frac {alpha }{2}}-sin {frac {alpha }{2}}end{array}} ight)}