3.3.1
Schmidt projection
For completeness, we note that the Procrustean method is not the only method to be suggested that allows entanglement concentration. An alternative method is called Schmidt projection [70], where the protocol takes a ensemble of weakly entangled states and projects onto a maximally entangled state on a subspace of the initial entangled states. In the context of Gaussian entanglement concentration, this method has been employed by Duan et. al. [78, 79] for the two mode squeezed vacuum, with Alice and Bob initially sharingK copies of the entangled state|ζ(λ)i=
√
1−λ2P∞ n=0λ
n|n, niwhich can be written as
|Ψin(λ)i= K O i=1 |ζ(λ)iAi,Bi = (1−λ 2)K/2 ∞ X L=0 λL q d(LK)|LiAi,Bi. (3.40)
The |LiAi,Bi states are finite dimensional maximally entangled states |LiAi,Bi = 1 q d(LK) i1+i2+...+iK=L X i1,i2,...,iK |i1, i2, . . . iKiAi|i1, i2, . . . iKiBi, (3.41) withd(LK) = (L+K−1)!/(L!(K−1)!. Thus, this entanglement concentration protocol requires a non-demolition measurement of the total photon number in Alice’s modes and it yields the final shared state|Li with a probability PL(K) = (1−λ2)Kλ2Ld(LK). This non-Gaussian non-demolition measurement can be accomplished by using an array of noiseless beam splitters and photon counting [78, 79].
3.3.2
Non-Gaussian noise approach
Another alternative method of entanglement distillation for continuous-variable states lies in the introduction of a source of non-Gaussian noise which can then be corrected by Gaussian operations. Such schemes have recently been experimentally realised [80–82].
Chapter 4
Weak measurements and weak
values
The previous chapter ended with the desire to generalise the optical entanglement concentration protocol to include arbitrary ancilla systems. In attempting to do so, we are faced with two immediate problems: What measurement strategies can be employed to allow conditional entanglement concentration? And: What are the general constraints required to produced Gaussian preserving entanglement concen- tration? Both of these issues have a common resolution in the framework of weak measurements. Indeed, one can consider the previous entanglement concentration protocols as examples of weak measurements where the probe state is initially en- tangled. This realisation is surprisingly powerful as it yields a criterion for selecting different ancillary ingredients.
4.1
Weak Values
4.1.1
Definition of weak values
Any physical theory makes contact with empirical observations through the observ- able numbers it predicts. In quantum theories, there are three types of observable numbers: eigenvalues or measurement results, expectation values and weak values
[83]. The first set of numbers follow from the basic formulation of observables and measurements in quantum mechanics. Simply put, the results of a measurement of an observableA coincide the eigenvalues of its associated self-adjoint operator
ˆ
A=X
j
aj|ajihaj|. (4.1)
These numbers are observable in a single measurement on a quantum mechanical system. Next is the notion of an expectation value, i.e. the statistical average of an observable on a particular quantum state
hAi=hψ|Aˆ|ψi. (4.2)
Expectation values only emerge on a statistical level following measurements per- formed on an identically prepared ensemble. These numbers are also used to estab- lish a correspondence with classical theories [30].
The final set of observable numbers in quantum mechanics are a recent addition called weak values [83, 84]. A weak value, like an expectation value, is only a statistically observable number, but unlike an expectation value or eigenvalue, it can be complex. Weak values are only applicable to quantum systems which have been both pre and post-selected in particular quantum states. Thus, the weak value of the observable A on a system which is pre-selected in the state |Φ1i and post-
selected in|Φ2i is defined as
AW =
hΦ2|Aˆ|Φ1i hΦ2|Φ1i
, (4.3)
where it is assumed that |Φ1i and |Φ2i are non-orthogonal. From a physical point
of view, weak values are regarded as the possible values of the observable at inter- mediate times between the pre and post-selections [85].
4.1.2
Some properties of weak values
To gain a better appreciation for weak values it is worthwhile to consider how they are related to both eigenvalues and expectation values of a given observable. In the
first case, the weak value of ˆA coincides with an eigenvalue if either of {|Φ1i,|Φ2i}
coincide with an eigenstate of ˆA:
AW = haj|Aˆ|Φ1i haj|Φ1i =aj haj|Φ1i haj|Φ1i =aj, (4.4) AW = hΦ2|Aˆ|aki hΦ2|aki =ak hΦ2|aki hΦ2|aki =ak. (4.5)
The weak value becomes undefined if both the pre and post-selected states are dis- tinct eigenstates of ˆA(assuming that ˆAhas a completely non-degenerate eigenvalue spectrum). On the other hand, a weak value of ˆA coincides with an expectation value of ˆA if the pre and post-selected states are identical:
AW =
hψ|Aˆ|ψi
hψ|ψi =hψ|Aˆ|ψi=hAi. (4.6)
Furthermore, any expectation value of ˆA can be linearly decomposed into a sum of different weak values of ˆA [85, 86] since
hψ|Aˆ|ψi=hψ| X j |jihj| ! ˆ A|ψi=X j |hψ|ji|2hj|Aˆ|ψi hj|ψi = X j P(ψ|j)AW(j), (4.7)
where the complete basis used in the above does not coincide with the eigenbasis of ˆ
AandP(ψ|j) is the probability of obtaining|ψigiven|ji. This allows an alternative interpretation of expectation values as a probabilistic mixture of weak values [86]. In addition, it demonstrates two possibilities for the imaginary components of weak values, either=(AW(j)) = 0 for allj or they are mixed with some positive for some
j and others negative. This follows from
=hψ|Aˆ|ψi= 0⇒X
j
P(ψ|j)=(AW(j)) = 0. (4.8)
Thus, since not all of the P(ψ|j)s can be zero then either =(AW(j)) = 0 for all
j or they are mixed. So, in contrast to both eigenvalues and expectation values, weak values can assume complex numerical values whilst remaining observable at the statistical level. To understand the process by which this is possible, we now discuss the notion of weak measurements [83].