*How to know if your vampire friend without murdering them accidentally* is a video project made for PC3130: Quantum Mechanics II. Its aim was to teach a chosen topic on quantum mechanics to fellow students in the same class; as such, certain basic concepts are glossed over or unexplained in the video, although some are expounded in the write-up below.

Usually, to know if an object is present somewhere, one needs to interact with it in some way. Even if it were merely *looked* at, at least one particle of light – a “photon” – has to bounce off the object to reach the eye – a “detector”. The “measurement” here would be the answer to the question, “Is the object here?”

The referenced paper^{[1]} takes advantage of the quantised nature of light to get around this necessity. Essentially, if a single photon goes through a path containing the object, it either interacts with the object, or *not* interact with the object. If it does, and the process absorbs the photon (say, the object is a vampire and the light is absorbed by them, which kills them), no light will be detected at the end of the path.

Yet, even if the photon takes the path where it doesn't interact with the object, the outcome might be different depending on the object's presence! This difference in outcome allows one to know an object's presence without actually interacting with it, thereby performing an *interaction-free measurement*.

The symbols used in the video, and their corresponding bra–ket notation counterparts, will be laid out in this section. Some prior familiarity with the bra–ket notation is assumed.

FADE IN:

INT. 4-DIMENSIONAL HILBERT SPACE

Space is well-lit with a background that is subtly off-white. The time dimension not immediately visible but its presence definitely felt.

Photons – individual particles of light – are the primary agent of this phenomena. Of its many properties, only two are needed here: its **polarisation** and its **direction of travel**. The polarisation is an angle between 0° (vertically polarised) and 90° (horizontally polarised), while the direction of travel is one of two: rightwards or downwards. The restriction on the direction of travel is because of how the equipment is set up, as will be clearer later.

Vertically polarised photons are denoted

|0°, ⋅ ⟩

|90°, ⋅ ⟩

|00010203040506070809101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990°, ⋅ ⟩

A photon with an arbitrary polarisation can be written as

|00010203040506070809101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990°, ⋅ ⟩ = cos(00010203040506070809101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990°)|0°, ⋅ ⟩ + sin(00010203040506070809101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990°)|90°, ⋅ ⟩

Meanwhile, a photon travelling rightwards is

As the above photons have a polarisation of 45°, they are denoted

The Born rule states that the probability of finding a photon of state

where

Essentially, the above means

- There is no vertical component in the state of a horizontally polarised light
- The photon is normalised to unity, or
: “the length of projection of a photon's state onto itself is 1”. That is, the length of the original photon is 1 - The length of the vertical component of the state of a 45°-polarised photon is
. It is the same for the horizontal component, since it has equal parts horizontal and vertical components

More generally, using Equation

and

Finally, the probability is given by the *square* of the projected length:

If the photon is polarised by 45°, there is a 50% chance of finding it to be vertically polarised, and a 50% chance of finding it to be horizontally polarised. Visually, the square of the length of the final photon indicates how likely it is to find it in that state.

Dude These are *polarising* beam splitters.

A singular beam splitter slides in from below to the centre of the frame.

Dude They let vertically polarised light through, and reflects horizontally polarised light.

Beam splitter duplicates itself, positions themselves one on top of the other. Two photon appears from frame left, pauses, top one passes through, and the bottom reflects downwards, in tandem with narration. Top (Bottom) photon has graphic with arrow pointing up-down (left-right).

The beam splitter, here represented by *only if its polarisation is horizontal*:

In other words, it is only the horizontally-polarised component of the photon that gets reflected. Generally,

With a photon of polarisation 45°,

and hence the probabilities are

It is important to realise that the visuals represent the *state* of the photon rather than the photon itself – the photon cannot split into half! The above situation is therefore intepreted as follows:

“*If a single photon with polarisation 45° moving rightwards passes through a beam splitter, there will be*

*a 50% chance of finding it on the right path of the beam splitter with a vertical polarisation,**a 50% chance of finding it on the bottom path of the beam splitter with a horizontal polarisation,**and no chance of finding it anywhere or anyhow else.”*

Mirrors, represented with

Dude If we pass vertically polarised light through a Faraday rotator - rotating the polarisation by some angle theta - (MORE)

Both beam splitter disappears upwards as the camera follows the top photon to right frame, where a Faraday rotator is present. The arrow rotates clockwise a little as it passes through the rotator.

When light of a certain polarisation passes through a uniform magnetic field, its plane of polarisation shifts due to the Faraday effect. By controlling the strength of the magnetic field, the polarisation of a photon can be rotated to a certain angle: here depicted as

Dude (cont'd) - then through this inteferometer, the photon ends up at the bottom with polarisation theta.

Without first including Tommy, the experimental setup acts on the incoming vertically-polarised photon via the following steps:

- Start with a vertically polarised photon moving rightwards:
- Rotate its polarisation by an angle of 45°:
- Pass it through the beam-splitter:
- Reflect it with the mirror:
- Pass it through the beam-splitter
*again*: - Detect if a photon is present at the bottom (
*Is there a photon moving downwards?*): , where is the state of the photon after it goes through the experimental setup with the number of loops

The final state of the photon

where we have defined

Hence,

Meanwhile, the presence of Tommy means that

which is representative of the fact that the rightwards-moving photon is “absorbed” by Tommy, while the downwards-moving photon continues in its path. Therefore, with this setup,

So there is a 50% chance of detecting the photon at the bottom, and a 50% chance of Tommy absorbing the photon (and dying).

- Kwiat P., Weinfurter H., Herdzog T., Zeilinger A., & Kasevich M. (1995). Experimental Realization of Interaction‐free Measurements.
*Annals of the New York Academy of Sciences, 755*(1): 383-393. https://doi.org/10.1111/j.1749-6632.1995.tb38981.x