1. By Joshua Barrow and Andrew Mogan
A Basic Overview of Quantum Key Distribution, and Recent Measurements of Orbital Angular Momentum Using Unitary Optical Transformations
By Joshua Barrow and Andrew Mogan

2. An Outline of This Presentation
Josh:
Present the basics of the quantum mechanics of qubits and entanglement
Simple entangled states
Mysterious properties when properly entangled and each are measured at the same time
How to use entangled states in quantum mechanically key encryption distribution
Discuss the built in security that arises from quantum measurement and entanglement
Presentation of the Orbital Angular Momentum basis for photons
Concept of quNits rather than simple qubits
Weak measurement thanks to mutually unbiased bases
Andrew:
Present the basics of current experimental work on the use of orbital angular momentum as a basis for quantum key distribution and computing
Introduce the orbital angular momentum state basis for photons
Compare and contrast some of the theoretical advantages of such a basis with classical computing and 2-level quantum systems
Explain some experiments on the subject
A brief overview of how they could be used for future QKD communications/computing technologies

3. Part I: Quantum Entanglement and Quantum Information

4. Overview of Quantum Spin States
For any spin particle, the most general wave function can be written in terms of the kets:
Ψ =𝑎 0 +𝑏| 1
where the |0 and | 1 kets are the “up” and “down” states of an electron (if we were to measure in the basis of the 𝑆 𝑧 operator), and we have
𝑎 ∗ 𝑎+ 𝑏 ∗ 𝑏=1
This creates a pure qubit, where any and all wavefunctions are a linear superposition of whatever basis states are available.
However, for our purposes, we refer to a qubit as the following:
| 𝜓 = ( |0 +| 1 )
There is nothing particularly remarkable about such a wave function, but it will become the basis of our proceeding discussion
We will use entanglement to eventually discuss the qubit as a fundamental unit of quantum information and computation in a 2-level system

5. The Bloch Sphere A way of visualizing any qubit

6. What Does Entanglement Look Like?
Mathematically, we can write any entangled wave function of two spin particles as
| 𝜑 = (| 01 − 10 = ( 0 𝐴 ⨂ 1 𝐵 − 1 𝐴 ⨂ 0 𝐵 )
where the subscripts A and B refer to the Hilbert spaces, states, and operators of Alice and Bob
We have used tensor products to combine Alice and Bob’s Hilbert spaces into a 4-dimensional tensor product space:
ℋ 𝐴𝐵 = ℋ 𝐴 ⨂ ℋ 𝐵
with associated operators, such as
𝑆 𝑧,𝐴;𝐵 = 𝑆 𝑧,𝐴 ⨂ 𝕀 𝐵
which allows Alice to measure the spin in the z-direction of her qubit, which doing nothing to Bob’s qubit (because they are separated, so of course she can’t)
Bob would have an associated operator which acts only on his Hilbert space

7. An Example
Let’s say that Alice and Bob are separated a great distance away from one another, and each of them shares an entangled qubit. They agree to measure the spin the z- direction at the same point in time. What will each of them measure?
𝑆 𝑧,𝐴;𝐵 | 𝜑 = 𝑆 𝑧,𝐴;𝐵 [ (| 01 − 10 ]= ( 𝑆 𝑧,𝐴 𝐴 ⨂ 𝕀 𝐵 𝐵 − 𝑆 𝑧,𝐴 𝐴 ⨂ 𝕀 𝐵 𝐵 )
= ℏ ( 0 𝐴 ⨂ 1 𝐵 𝐴 ⨂ 0 𝐵 )
𝑆 𝑧,𝐵;𝐴 | 𝜑 = 𝑆 𝑧,𝐵;𝐴 [ (| 01 − 10 ]= ( 𝕀 𝐴 𝐴 ⨂ 𝑆 𝑧,𝐵 𝐵 − 𝕀 𝐴 𝐴 ⨂ 𝑆 𝑧,𝐵 𝐵 )
= ℏ (− 0 𝐴 ⨂ 1 𝐵 − 1 𝐴 ⨂ 1 𝐵 )
meaning that, proportionally speaking, one will always measure a positive value and the other a negative one.
This implies that whenever Alice measures an “up” state, Bob will measure a “down” state, and vice versa

8. Quantum Key Distribution The Holy Grail of Future Communications
Alice and Bob want to communicate with each other quantum mechanically. Let’s say that they each have a set of N (a finite number) qubits at their disposal, each entangled with one another.
They decide to label qubits 1 through N, and then at a great distance at the same point in time, they measure each qubit consecutively with a randomly chosen Pauli spin operator from the set
{ 𝑆 𝑥 , 𝑆 𝑦 , 𝑆 𝑧 }
of course acting only on their individual Hilbert spaces/qubits
Once they have measured all of the qubits, they call one another (at the speed of light), and tell each other the order of their measurements (not what they measured)
Anytime that they disagree on what direction they measured, they delete the entry of their results from their logbook
Once they know which directions agreed, they in principle know the other’s results (and let’s say they only keep Alice’s, and so Bob just flips all of his results)
Thus, they have a quantum mechanically distributed encryption key! Encrypt away!

9. Part II: The Orbital Angular Momentum Basis for Photons

10. The Advent of QuNits
Similar to qubits (applying to 2-level systems), we can in principle manipulate 3-5 level systems
Consider spin of the photon: 1, and so has spinors of | −1 ,| 0 ,𝑜𝑟 | 1
Information density increases even though all basic mathematics remains unchanged
Can continue up to spin-2, a 5 level system with spinors of | −2 ,| −1 ,| 0 ,| 1 ,𝑜𝑟 | 2
This presupposes we must use the spin basis for all of our communications and computations
There are other alternatives: Orbital angular momentum is one!
In principle, we can write any state for a single photon as
| Ψ = 𝑙=−𝑚 𝑚 𝑎 𝑙 | 𝑙
where 𝑁=2𝑚+1 could approach infinity (though this is not practical)
Suddenly, we have the possibility of a N-level system for quantum informative purposes
However, experimentally characterizing such states is quite difficult

11. Mutually Unbiased Bases
A useful and empirically important property of the OAM basis is that it is mutually unbiased to the basis of angular position
In mathematical terms, if any two complete orthonormal systems {| 𝑒 𝑖 } and {| 𝑓 𝑗 } share the condition that | 𝑒 𝑖 𝑓 𝑗 | 2 = 1 𝑑 for some constant 𝑑𝜖ℕ
Here 𝑑=dim⁡[ℋ] where 𝑒 𝑖 ,{| 𝑓 𝑗 }𝜖ℋ
This implies that if a system is prepared in a state belonging to one of the bases, then all outcomes of the measurement with respect to the other basis will occur with equal probabilities
Said another way, their inner product always has the same magnitude
In terms of a Hilbert space of dimension d, we can say that for any unitary operators 𝑋 and 𝑍 , if we can find a phase factor 𝜔 such that 𝑋 𝑍 =𝜔 𝑍 𝑋 (such as 𝜔= 𝑒 2𝜋𝑖 𝑑 ), then the eigenbases of 𝑋 and 𝑍 are mutually unbiased
Examples of this for a 2-level system include:
| 0 and | 1 are a valid orthonormal basis
So are | 0 +| and | 0 −| , or even | 0 +𝑖| and | 0 −𝑖|

12. The “Weak” Projector
From here, we are ready to consider two CONS: the OAM basis and the angular position basis, given by {| 𝜃 𝑜 }
Because these two are mutually unbiased with respect to one another, it is useful to define the quantity
𝑐= 𝜃 𝑜 𝑙 𝜃 𝑜 Ψ ≅ 𝑒 𝑖𝑙 𝜃 𝑜 Ψ(𝜃)
and so
𝑐| Ψ =𝑐 𝑙=−𝑚 𝑚 𝑎 𝑙 | 𝑙 = 𝑙=−𝑚 𝑚 | 𝑙 𝜃 𝑜 𝑙 𝑙 Ψ 𝜃 𝑜 Ψ = 𝑙=−𝑚 𝑚 𝜋 𝑙 𝑤 | 𝑙
which represents the “weighted” or “weak projector” proportional to the amplitude 𝑎 𝑙
This is the OAM weak value, and is equal to the average result obtained by making a weak projection in the OAM basis using the traditional projector 𝜋 𝑙
If we follow this by a “strong measurement” in angular position, we develop scaled complex probability amplitudes 𝑐𝑎 𝑙 for any finite set of 𝑙 values, which can be renormalized the obtain the entire wavefunction!

13. Part III: Survey of Experimental Methods
Introduction. Simple quantum computers use spin states, more complex systems can use OAM states. Theoretically infinite state space. We’ll also briefly see one experiment that uses total angular momentum, but OAM generally suffices

14. Laguerre-Gaussian Modes
Laguerre-Gaussian modes possess well-defined OAM
Allen et al. showed how to convert Hermite-Gaussian to Laguerre-Gaussian
LG mode has an azimuthal dependence of 𝜃 𝑙 =𝑒 𝑖𝑙𝜃
Orbital angular momentum of 𝑙ℏ
The goal, then, is to measure OAM states without irreversibly disturbing the system
Hermite-Gaussian are more common, but Laguerre-Gaussian have well-defined OAM, which is what we want. The differences have to do with how the lasers are designed.

15. Mach-Zehnder Interferometer
Split beam, detect interference pattern. LIGO analogy!

16. Sorting OAM States
Dove prisms create phase differences equal to twice the relative angle. So here, the Dove prisms being at alpha/2 creates a phase difference of alpha = pi. Odd states are then out of phase by pi, even states are in phase. Even values of L get sorted into A1 and odd values into B1

17. Further Sorting
Cascaded interferometers. Output of the first stage acts as input for later stages. Gray boxes are interferometers from previous slide. First stage introduces a phase shift of pi and so sorts multiples of 2. Odd L are then put through a hologram to increase L by 1 and make them even. Second stage has alpha = pi/2, so it sorts into even and odd multiples of 2. Next hologram increases L by 2 for the odd states. Only about 10% efficient, despite theoretical 100% efficiency

18. Using SLMs to Sort OAM States
- Converts azimuthal position to lateral position
- Introduces a phase distortion that must be corrected for, hence the SLMs
Converts helical phase exp(ilphi) to transverse phase gradient. SLM1 generates the LG mode. SLM2 and SLM3 create desired phase profile for L1 and L2, respectively. Resulting beam is spatially separated on the CCD based on OAM state. Can successfully sort 11 OAM states [-5, 5]. 77% efficient

19. Limitation: Overlap of Spots
From left to right: input beam, phase corrected beam, modeled detector plane, observed detector plane. Top is L = 5, bottom is L = -1 and L = 2

20. Avoiding Overlap
The optical transformation converts the OAM modes to plane waves as they go through the first and second refractive phase elements. Cartesian to log-polar transformation. An SLM is used to create multiple copies of the unwrapped beam. A lens focuses the resulting wide beam into a spot, after the phase distortions are removed by the second SLM. Efficiency of 92% for 25 modes, [-12, 12]

21. Direct Measurement
“Measuring” a quantum system through sequential weak and strong measurements
Gives information on the state without irreversibly disturbing it
Can weakly measure OAM followed by a strong measurement of angular position to obtain complex probability amplitudes, along with the phase
From this, we can obtain all of the necessary information of the initial state with relatively high accuracy
If practical, could eliminate a huge obstacle in quantum computation

22. Direct Measurement
Setup is the same as the previous one up until QWP0. QWP0 is used in double pass with SLM4 to rotate the polarization of the OAM mode to be weakly projected. QWP1 and HWP1 are used to remove any ellipticity. A strong measurement of angular position is performed by a 10-mm slit placed in the Fourier plane of lens L3. Since the plane of the slit is conjugate to the plane where the OAM modes are spatially separated (SLM4), a measurement of the linear position by the slit is equivalent to a measurement of angular position. Result is complex probability amplitude that characterizes the state. Performed for L values of [-13,13]. Can “measure” not just L, but the initial state!

23. Applications and Limitations
High dimensionality of OAM basis increases tolerance to eavesdropping
Superdense coding: more information per bit
Most experimental setups are not yet space efficient
OAM state sorting is still not 100% efficient
Direct measurement seems promising, but further research is required

24. References