1. Quantum Information with Continuous Variables
Blaine Heffron

2. Continuous Variables
We learned about qubits and qudits, in which the measurables can take on one of N possible values (discrete variables)
Continuous variables are measurables with an infinite spectrum of possible values
Experimentally photon beams are of interest as a possible candidate for continuous variable quantum computing
Unitary transformations and entanglement can be achieved at room temperature using phase shifters, beam splitters, and optical squeezing
Instead of using the number operator as the measurable (discrete), the real and imaginary amplitudes of the electric field are used (called “quadratures” of the field)

3. Quantum Optics Basics
Beams of light can be described via the Quantum Harmonic Oscillator
Operators X and P obey the commutation relation

4. Quadratures of the Electromagnetic Field
Electric Field of polarized beam of light, polarization 𝞳
Operators for the quadratures are the dimensionless position and momentum operators
Electric field written in terms of quadratures (with phase of 0):

5. Uncertainty Relationship, Coherent States
“Coherent” state is one in which the uncertainty relation is minimum
These states happen to be eigenstates of the annihilation operator (for H.O.)
Which are obtained via a displacement of the vacuum state

6. Coherent States cont’d
Expansion of coherent state in terms of Fock states (eigenstates of Hamiltonian, or number operator)
Yields a probability distribution for the number of photons detected that obeys poisson statistics
Standard deviation of number of photons detected scales with the square root of the average number detected

7. Quadratures with Phase
In general, the operators can be parameterized by a phase angle 𝚯
These can be obtained via a rotation of the 0 degree phase operators:

8. Detection of Quadratures
In general, detectors measure photocurrent i, proportional to the number of photons detected
To detect quadratures, one can use homodyne detection, where the photon beam is split using a 50:50 beam splitter, and it is combined with an intense local oscillator that is a coherent state |α> yielding a large number of average photons (|α| large). This allows a classical approximation for the coherent state, and the outgoing two beams can be written as

9. Homodyne detection The outgoing currents are then
These are then subtracted and measured
Which is the x quadrature at a phase 𝚯, where
𝚯 is the relative phase between oscillator and signal,
Image source: cryptography-and-network-security-protocols-and-technologies/optical- communication-with-weak-coherent-light-fields

10. Unitary Transformations
2x2 Unitary matrices can be implemented via beam splitters and phase shifters
Ideal phase free beam splitter:
sin(𝜃) reflectivity, cos(𝜃) transmittance
With phase shifts:
Because of this, any N x N matrix can be constructed via a sequence of beam splitters and phase shifters

11. Entanglement
Entanglement requires optical squeezing and beam splitters
Optical squeezing can be performed by combining the signal wave with a pump wave at twice the frequency of the signal wave in a nonlinear crystal
In the interaction picture, this interaction is described by the Hamiltonian
𝞳 is a constant that absorbs the amplitude of the coherent pump state (treated as a classical wave) |α_pump|

12. Optical squeezing
Given the squeezing operation, the quadratures evolve according to
Which increases the uncertainty in x while decreasing the uncertainty in p
Source:

13. Two-mode Squeezing
Squeezing two modes is done with the same type of setup, a nonlinear crystal combined with a pump frequency, but now two squeezed modes are then combined at a 50:50 beam splitter. This is shown by the hamiltonian
With solutions for the quadratures given by
where r is the dimensionless interaction time

14. Entanglement
Bipartite entanglement can be produced with two-mode optical squeezing and beam splitters, which can then be performed pairwise on any number of states to produce n-partite entanglement.

15. An example: Quantum Key Distribution
Problem: Alice wants to securely send a key (string of 0’s or 1’s) to Bob without it being intercepted
Solution: A laser beam is split into two beams, one with a large number of photons (10^6 photons/pulse) and the other with a small (<1 photons/pulse) number of photons. Alice then applies a random phase shift chosen from (0,90,180,270) degrees to the large pulse. Bob then receives the signal and applies a random (0, 90) degree phase shift to the signal. He then uses homodyne detection on the signal, where the local oscillator used in detection is the large pulse sent by alice. The phase shift is denoted

16. QKD cont’d
Depending on the relative phase shift, Bob will have the following probability distribution for measuring the quadrature X:
Image source: [3]

17. QKD cont’d
If the detected amplitude is above a threshold ( in this case it would be 0), Bob assigns a 1 to that pulse, and if it is less than threshold, he assigns a 0. After a number of pulses have been sent and detected in this method, Bob calls Alice and tells her the phase shifts he made to the signal for each pulse. Alice then calculates what the relative phase was, and if it was 90 or 270 degrees, she tells him to throw out that signal, if not she tells him to keep that corresponding bit. Alice can then figure out each value because 0 always corresponds to a 180 or 270 degree shift by Alice, and 1 always corresponds to a 0 or 90 degree shift by Alice. Because of the overlap between the two accepted cases in their probability distribution, error correction schemes must be implemented.

18. Thanks for listening

19. References
[1] U. L. Andersen, G. Leuchs, C. Silberhorn. Continuous-variable information processing.Laser & Photon.
Rev. 4, No. 3, , 2010.
[2] S. L. Braunstein and P. van Loock. Quantum information with continuous variables. Reviews of
Modern Physics, Vol. 77: , April 2005.
[3] T. Hirano, T. Konishi, and R. Namiki. arXiv:quant- ph/
[4] Z. Wang, S. Ma, and F. Fan. Quantum fresental transformation responsible for squeezing mechanism
and squeezed state of a degenerate parametric amplifier. International Journal of Theoretical Physics,
May 2015, Vol. 54 5: