In quantum mechanics, every measurable quantity, i.e. everything that can be measured about a system, is called an observable.  If we measure an observable, the outcome of the measurement will be one of its eigenvalues, and the state of the system is the corresponding eigenstate of the observable.

Examples: 

• Assume that wee measure S_z, the z-component of the spin of an electron.  We will obtain one of the two eigenvalues of S_z, either h/2 or -h/2.  If  we measure h/2, then the spin state of the electron immediately after the measurement will be the eigenstate |+>.  If  we measure -h/2, then the spin state of the electron immediately after the measurement is the eigenstate |->.  If we repeat the same measurement before measuring something else, the second measurement will yield the same eigenvalue as the first measurement.
• Assume that we measure the linear polarization of a photon along the axes of the coordinate system shown below. 

  

  We will obtain the eigenvalues "h" (horizontal) or "v" (vertical).  If we measure "v", then the linear polarization state of the photon immediately after the measurement is the eigenstate |v>.  If we measure "h", then the linear polarization state of the photon immediately after the measurement is the eigenstate |h>.  If we repeat the same measurement before measuring something else, the second measurement will yield the same eigenvalue as the first measurement.

In quantum mechanics there are compatible and incompatible observables.  Compatible observables have common eigenstates.  If observable 1 and observable 2 are compatible, then measuring observable 2 after having measured observable 1 will not destroy any information we obtained from the first measurement.  Measuring observable 1 again after having measured observable 2 will with certainty yield the same eigenvalue as the first measurement of observable 1.

Example: 

• The energy and angular momentum of an electron in a hydrogen atom can be determined simultaneously.  For the hydrogen atom these are compatible observables.

Incompatible observables do not have common eigenstates.  Measuring observable 2 after having measured observable 1 will destroy the information we obtained from the first measurement.  For a subsequent second measurement of observable 1 we can only make probabilistic predictions.

Examples:

• The spin components S_z and S_x are incompatible observables.  A measurement of S_x will destroy the information we obtained from a prior measurement of S_z.
• The linear polarization components of a photon measured along the axes of the two coordinate systems shown below are incompatible observables.

    

  Measuring the linear polarization components along the primed coordinate system will destroy the information we obtained from a prior measurement of the linear polarization components along the unprimed coordinate system.

Today's experiment:

A Do-It-Yourself Quantum Eraser

(a)  In a double slit experiment, there are two paths a photon can take from a light source to a detector.  Let us denote these paths by |left> and |right>.  If we cannot know which path each individual photon is taking on its way from the source to the detector, then the arrival positions of many photons on the detector form an interference pattern.  If we can know which path each individual photon is taking, then no interference pattern forms at the detector.
If we do not observe an interference pattern, but two distinct distributions, then the eigenstate corresponding to one of those distributions is |left> and the eigenstate corresponding to the other distribution is |right>.
If we do observe an interference pattern, then we denote the eigenstate corresponding to this observation by 2^-1/2(|left> ± |right>).
We will first produce photons in such an eigenstate.

(b)  We now will entangle two different observables of a single photon.  We will set up a experiment so that a photon that takes the right path to the detector will have to go through a polarizer with its transmission axis aligned vertically, and a photon that takes the left path to the detector will have to go through a polarizer with its transmission axis aligned horizontally.  When the photon leaves the source we do not know which path it will take.
We therefore must denote its state by  2^-1/2(|left>|h> ± |right>|v>).
The two properties of the photon are entangled.  A priori, we do not know the outcome of either measurement, but the outcome of the "which path:" measurement is inextricably linked with the outcome of the linear polarization (h-v) measurement.
When a photon now arrives at the detector, we can know which path it took.  Detectors can be constructed that measure the arrival position and the polarization.  If we can have path information for each photon, no interference pattern is produced.  The photon polarization labels the path, and the interference pattern is destroyed.
This experimental setup produces a quantum eraser.

Note: It does not matter if we actually measure the polarization at the detector and therefore determine the path.  All that is needed to destroy the interference pattern is that the path has been labeled and that we can look at the label if we want to.  Interference patterns are produced if it is impossible to determine the path because no label exist, not because we choose to not look at the label.

(c)  We now will place another polarizer between the labeler and the detector, with its transmission axes oriented along h' or v'.  This second polarization measurement is incompatible with the first polarization measurement (the labeler).  With the second polarizer we destroy the information we obtained by using the first polarizer, i.e. we erase the label and therefore the path information.  We no longer know which path each photon has taken, and we expect the interference pattern to reappear.