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A Basic Overview of Quantum Key
Distribution, and Recent Measurements of
Orbital Angular Momentum Using Unitary
Optical Transformations
BY JOSHUA BARROW AND ANDREW MOGAN
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
Part I:
Quantum
Entanglement and
Quantum Information
Overview of Quantum Spin
States
1
For any spin- particle, the most general wave function can be written in terms of the
2
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 𝑆𝑧 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:
1
|𝜓 =
(|0 + |1 )
2
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
The Bloch Sphere
A way of visualizing any qubit
What Does Entanglement
Look Like?
1
Mathematically, we can write any entangled wave function of two spin-2
particles as
1
1
|𝜑 =
(|01 − 10 =
( 0 𝐴⨂ 1 𝐵 − 1 𝐴⨂ 0 𝐵 )
2
2
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:
ℋ𝐴𝐵 = ℋ𝐴 ⨂ℋ𝐵
with associated operators, such as
𝑆𝑧,𝐴;𝐵 = 𝑆𝑧,𝐴 ⨂𝕀𝐵
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
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 zdirection at the same point in time. What will each of them measure?
1
1
𝑆𝑧,𝐴;𝐵 |𝜑 = 𝑆𝑧,𝐴;𝐵 [ (|01 − 10 ] =
(𝑆𝑧,𝐴 0 𝐴 ⨂𝕀𝐵 1 𝐵 − 𝑆𝑧,𝐴 1 𝐴 ⨂𝕀𝐵 0 𝐵 )
2
2
ℏ
=
( 0 𝐴⨂ 1 𝐵 + 1 𝐴⨂ 0 𝐵 )
2 2
1
1
𝑆𝑧,𝐵;𝐴 |𝜑 = 𝑆𝑧,𝐵;𝐴 [ (|01 − 10 ] =
(𝕀𝐴 0 𝐴 ⨂𝑆𝑧,𝐵 1 𝐵 − 𝕀𝐴 1 𝐴 ⨂𝑆𝑧,𝐵 0 𝐵 )
2
2
ℏ
=
(− 0 𝐴 ⨂ 1 𝐵 − 1 𝐴 ⨂ 1 𝐵 )
2 2
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
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
{𝑆𝑥 , 𝑆𝑦 , 𝑆𝑧 }
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!
Part II:
The Orbital Angular
Momentum Basis for
Photons
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
𝑚
|Ψ =
𝑎𝑙 |𝑙
𝑙=−𝑚
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
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 {|𝑒𝑖 } and
1
{|𝑓𝑗 } share the condition that | 𝑒𝑖 𝑓𝑗 |2 = 𝑑 for some constant 𝑑𝜖ℕ
◦ Here 𝑑 = dim[ℋ] where
𝑒𝑖 , {|𝑓𝑗 }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 𝑋𝑍 = 𝜔𝑍𝑋 (such as 𝜔 = 𝑒
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 +|1
2
and
|0 −|1
2
, or even
|0 +𝑖|1
2
and
|0 −𝑖|1
2
2𝜋𝑖
𝑑
),
The “Weak” Projector
From here, we are ready to consider two CONS: the OAM basis and the
angular position basis, given by {|𝜃𝑜 }
◦ Because these two are mutually unbiased with respect to one another, it is useful
to define the quantity
𝜃𝑜 𝑙
𝑒 𝑖𝑙𝜃𝑜
𝑐=
≅
𝜃𝑜 Ψ
Ψ(𝜃)
and so
𝑚
𝑚
𝑚
𝜃𝑜 𝑙 𝑙 Ψ
𝑐|Ψ = 𝑐
𝑎𝑙 |𝑙 =
|𝑙
=
𝜋𝑙 𝑤 |𝑙
𝜃𝑜 Ψ
𝑙=−𝑚
𝑙=−𝑚
𝑙=−𝑚
which represents the “weighted” or “weak projector” proportional to the
amplitude 𝑎𝑙
◦ 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 𝜋𝑙
◦ If we follow this by a “strong measurement” in angular position, we develop
scaled complex probability amplitudes 𝑐𝑎𝑙 for any finite set of 𝑙 values, which can
be renormalized the obtain the entire wavefunction!
Part III:
Survey of
Experimental
Methods
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
Mach-Zehnder Interferometer
Sorting OAM States
Further Sorting
Using SLMs to Sort OAM
States
- Converts azimuthal position to lateral position
- Introduces a phase distortion that must be corrected for, hence the
SLMs
Limitation: Overlap of Spots
Avoiding Overlap
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
Direct Measurement
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
References