For Q
Preface
A classical bit is a binary unit of information. It has two distinct states. We can label these as "0"
or as "1".
It could be off...
or on.
Amplitude includes the range between 1 and 0. We can assign a probability of the bit being in either of the two states. This arrow represents the bit being 0 with probability 100%.
By Laura De Decker
The notation used is called Bra-Ket or Dirac notation. This is a zero-ket and...
1 0 this is a one-ket. The Greek letter Ψ, pronounced psi, represents a wave function. This can be written as Ψ=1|0> and Ψ=1|1>. There is only one amplitude if there is only one eigenstate as in these equations. Normally the 1 representing 100% is left out when there is only one state.
The zero-ket can also be expressed as a vector with 100% probability in the top position.
0 1 The one-ket can also be expressed as a vector with 100% probability in the bottom position.
This colour can also represent the zero state or ground state.
And this colour can represent the one state or excited state.
A quantum bit can hold more information. Think of it as having colour instead of just greyscale.
Qubits have more possible states. We think of it in terms of X (red), Y (green), and Z (blue).
If you have 256 values in each axis you end up with a cube with 16 777 216 colours.
One of the challenges of quantum computing is to make use of the extra information available in a qubit. The cube from the previous page is transformed into my colour version of the Bloch sphere.
Like its classical counterpart, qubits can have uncertain states. The certainty is the distance from the centre. The state of a qubit has two amplitudes, one for "0" and the other for "1". The farther from the centre the more certain we are about the amplitude.
Here, colour planes meet at where red is 115, green is 255, and blue is 15. It can be written as a tuple: RGB(115,255,15).
Polar coordinate version showing where Ψ is the outer point of the intersecting disks.
Ψ can be represented as a vector in a Bloch sphere. If you measure a qubit with respect to the 0-1 axis, you observe the qubit in either state |0> or |1>. The choice of measurement affects the state.
This octahedron represents six states with its vertices. Many operations only require these six states.
This shows 2x2 matrix multiplication. You can multiply operators together to calculate the equivalent single operator. The density matrix of one qubit is a 2x2 matrix.
Above are circuits showing the unitary nature of Pauli operations using the colour associated with Bloch sphere position. When you apply a Pauli gate twice you go back to where you started. In quantum mechanics the bit flip is called an X operator. Sometimes I use a colour version of the Bloch sphere that fits inside an RGB cube. It doesn't contain all of the bright hues of the RGB colours but is less distorted.
A rotation of the Bloch Sphere represents a change in phase. If the qubit is in the |0> state, the state remains as the point at the top however the vector direction changes. Ψ = eiφ |0>
A quantum coin can be flipped in two incompatible bases, for example the top and bottom or the front and back points of the Bloch sphere.
This superposition of red and green dots on a blue background optically blend to create This superposition of red and green dots on a blue background optically blends to create a colour spectrum. colour spectrum.
Until we measure a qubit, it can exist in a superposition of two distinct states, like |0> and |1>. For example, the |+> = 1/√̅ 2|0> + 1/√̅ 2|1>.
+ +
This equation shows matrix multiplication.
If the X (red) direction represents one eigenstate and the y (blue) another, the superposition represents the diagonal (magenta).
From amplitudes to probabilities: If the side of the square is the amplitude of a certain state, then the area is the probability to observe that state.
An operator acting on a state is represented by a matrix multiplying a vector. Visualization of the inner product between the top row of a 2x2 matrix and a 2-dimensional vector. The four Hadamard states are all real numbers represented by the disk containing the x and z axes whereas the other complex numbers contain an imaginary component.
Visualization of the inner product between the bottom row of a 2x2 matrix and a 2-dimensional vector.
Visualization representing matrix multiplication of Identity-operator on |0>. X-operator acting on |0>.
Bloch colours distinguish Y-operator acting on |0> from the previous image. The Z matrix rotates the state around the Bloch sphere Z-axis 180 degrees. Z|+>=|->
H|0>=|+> H|+>=|0>
H|1>=|-> H|->=|1>
The Hadamard operation on a zero-ket can be thought of as rotating around a pole at 45 degrees from the z-axis. The Hadamard operator along with the Pauli gates are unitary operations that have special properties such as when you square the function you get the identity operation.
The Schrӧdinger picture has cats representing potential states in superposition and the basis is static.
I like to think of the Heisenberg picture as a rotating arrow or vector.
The Interaction (Dirac) picture combines the two previous pictures.
The Many-Worlds theory envisions that for all possible outcomes, for example, in making a quantum measurement, there is a world where such an outcome exists such that reality branches off into multiple realities. These images represent possible worlds.
α β A state vector has amplitudes whose squares add up to 1.
In Bra-Ket Notation: This is a zero-bra and...
this is a one-bra.
This 1x2 row matrix can be the top row of a 2x2 density matrix and means the same thing as the 0-bra.
This 2x1 column matrix can be the left column of a 2x2 density matrix and means the same thing as the 0-ket.
This expression, conventionally written the second way, represents the inner product.
|ᴪ>
ϴ
ɸ
This image represents the inner products of this particular qubit state.
<0|ᴪ> = cos(ϴ) <+|ᴪ> = cos(ɸ)
This state is composed of 50% theta (Blue) and 50% phi (red and green).
Multiplying in the other order calculates the outer product.
So far we have discussed the pure states, that make up the outer shell of the Bloch Sphere.
In the cat theme this is a tensor product of density matrices showing outer products.
A tensor results from multipling several density matrices together. This image represents the Kronecker delta which states that δij = 0 if i≠j and 1 if i=j. Think of these matrices as tables where the rows are i and the columns are j. Only the diagonal entries where the row and column are the same get used. Above think of the second and third planes as containing a matrix for each of the previous image matrices. The following page is the tensor product ZZZ from this page.
A pure state of two qubits written as a density matrix. In general this is entangled.
The non-diagonal terms of each quadrant of the matrix cancel out for the reduced density matrix representing the partial trace of one of the entangled qubits in the two qubit system.
This is the reduced density matrix.
The state of each of two entangled qubits exists in the interior of the Bloch Sphere.
The state is not pure because of entanglement.
A state is not pure if it is missing information.
In the case of entangled qubits, the missing information is in the other qubit.
This is a discrete quantum walk model that shows how the potential entangled states expand from time zero at the top using a Hadamard coin with initial state +↓ using the (↑,↓) basis. There is entanglement between coin and position. This shows the interference pattern in the middle between positive and negative values of the up and down spins compared with the highly ordered patterns at the sides.
+|↑> -|↑> +|↓> -|↓>
Sine and cosine waves corresponding to the Bloch sphere.
Shows that looking at the Bloch Sphere from the top, motion is counter-clockwise and clockwise when you imagine viewing from the bottom, relating to next image.
This image represents half a Bell State or EPR (Einstein–Podolsky–Rosen) pair and on the next page the other half of the pair. When you combine these the charge represented by colour cancel each other out.
Together they become grey. If you were to take the partial trace of the maximally entangled Bell State you would end up with the centre point of the Bloch sphere which in my colour model is middle grey.
This is a squeezed state.
Nuclear magnetic resonance lab rule.
When a photon interacts with a 50/50 optical beamsplitter it either goes through or bounces off. Metaphorically the stairs in this picture either go up or down. If you see both, a flat plane, or a flower you are like a photon in superposition and have yet to go through the beamsplitter.
Stay tuned after these messages
Double-slit experiment: Without an observer the particle behaves like a wave as if it went through both slits interfering with itself.
Double-slit probability distribution. Artistic interpretation of interference pattern. Quantum mechanics is fascinating because in actuality once you try to deconstruct the interference pattern from each slit, you no longer see an interference pattern!
Situation Normal All Fuzzed Up Tie! 1:1
It's impossible to know what belongs to one qubit as opposed to the other since by definition they are entangled. When you try to solve it algebraically there is a paradox. If you set α0β1=0 then α1β0=0 cannot add to 1 and it doesn’t fit the identity requirement.
Non-locality: Bell state correlations (entanglement) has no causal explanation since it occurs beyond the reach of known physical causal relationships. It occurs at distances greater than what light can travel between particles.
Bell States (Bloch sphere colours). Let this represent the Phi+ Bell state or EPR pair. |Φ+>= 1/√̅ 2(|0>A|0>B+|1>A|1>B)
This represents the Phi- Bell state.
This represents the Psi+ Bell State.
|Φ->=1/√̅ 2(|0>A|0>B-|1>A|1>B)
|Ψ+>= 1/√̅ 2(|0>A|1>B+|1>A|0>B)
This represents the Psi- Bell State. |Ψ->=1/√̅ 2(|0>A|1>B-|1>A|0>B)
Classical Limit Probability Game with Bell States. When the players share a Bell State they can win more frequently than if they didn't share an entangled state.
Bell states are useful for practical quantum applications such as cryptography, due to their stronger than classical correlations also known as entanglement.
For more context on the rules of the game refer to Gisin (2012).
Logarithmic RGB cube showing values for 24-bit colour. Notice that each bit represents a different amount.
Visualization of Big Bell Test (2016) data collected from participants around the world and used to test Bell's theorem. In this representation it looks unordered or random.
Data from a pseudo-random source. Notice this data seems to have an underlining pattern, it appears somewhat ordered.
Visualization of binary data from 50/50 Optical Beam-splitter. This data was the least ordered in all representations. I would say it is unordered or random.
Uncertainty principle: The more you know about the position, the less you know about the momentum.
Visualization representing a Harmonic oscillator in phase space. The x axis represents position and the y axis is named p for momentum.
IDENTITY REQ"D
Polar colouring of phase space.
Pointed curve (best case) The uncertainty principle is Δx Δp ≥ ħ/2.
The small coloured circles represent two opposite coherent states in phase space.
Image inspired by the Quantum Scissors amplification device. The quantum noise gets amplified too.
A coherent state with quantum and classical noise.
Series of images inspired by the State Comparison Amplifier, based on coherent state comparison, followed by photon subtraction (Eleftheriadou 2015).
Coherent state comparison: two beams interact at beamsplitter, resulting in a mixed output state. One output beam is post-selected if no photons are recorded in the other output arm.
The post-selected output beam from the first beamsplitter goes through a highly transmissive beamsplitter.
Photon subtraction: a detector firing at the beamsplitter's reflected output confirms the post-selection of the final output beam.
An XOR operation on two bits, as used in the Vernam Cipher. The point of Quantum Key Distribution is to share a secret to encode and decode messages used in the Vernam Cipher.
Vernam visualization: encrypted message
Vernam visualization: secret key
Vernam Visualization: message
Quantum Error Correction codes are like double bagging your candy. You get a chance to adjust the situation before bits are on the floor. So you can have your bits and eat them too!
Grover 24-bit unit.
Randomly coloured waves representing Grover algorithm.
Visualization of GHZ (Greenberger-Horne-Zeilinger) Game: GHZ state with three-qubit entanglement. Alice's ones are red; Bob's ones are blue; and Cleo's ones are green.
Visualization representing BB84 Quantum Key Distribution by Bennett and Brassard.
GHZ truth table visualized as a modified Venn Diagram. The strategy is any colour composed of zero or two primary colours wins.
Taking out the single primaries from the possible scenarios shows the winning combinations. Each quadrant represents all possible plays for Alice and Bob if Cleo plays a game: from top to bottom left to right: all zeros, same, different, and all ones.
|0>A|0>B=1/√̅ 2(|Φ+>AB+|Φ->AB) |0>A|1>B=1/√̅ 2(|Ψ+>AB+|Ψ->AB) |1>A|0>B=1/√̅ 2(|Ψ+>AB-|Ψ->AB) |1>A|1>B=1/√̅ 2(|Φ+>AB-|Φ->AB)
|χ>A|Φ+>BC=1/√̅ 2(α|0>A+β|1>A) (|0>B|0>C+|1>B|1>C) = 1/2|Φ+>AB(α|0>C+β|1>C) +1/2|Φ->AB(α|0>C-β|1>C) +1/2|Ψ+>AB(β|0>C+α|1>C) +1/2|Ψ->AB(-β|0>C+α|1>C) Teleportation of Alice's secret qubit to Cleo. Bob and Cleo share a Bell State. Bob receives Alice's qubit making a three-party state. The Bell States are substituted into the three-party state to rewrite it as a four-term superposition.
=(α|Φ α|Φ+>AB|0>C+α|Φ->AB|0>C +α|Ψ+>AB|1>C+α|Ψ->AB|1>C +β|Ψ+>AB|0>C-β|Ψ->AB|0>C +β|Φ+>AB|1>C-β|Φ->AB|1>C)1/√̅ 2 =(α|Φ α|Φ+>AB|0>C+α|Φ->AB|0>C +α|Ψ+>AB|1>C+α|Ψ->AB|1>C +β|Ψ+>AB|0>C-β|Ψ->AB|0>C +β|Φ+>AB|1>C-β|Φ->AB|1>C)1/√̅ 2 = 1/2|Φ+>AB(α|0>C+β|1>C) +1/2|Φ->AB(α|0>C-β|1>C) +1/2|Ψ->AB(-β|0>C+α|1>C) +1/2|Ψ+>AB(β|0>C +α|1>C)
|χ>A
|φ>BC |χ>C
From top to bottom: initial 3-Party State; 4-term superposition; 3-party superposition; and 3-party superposition (re-arranged). In math calculations the equation is organized by state whereas below the second and fourth rows shown as columns show the physical rearrangment in teleportation for the wires in the teleportation circuit.
Teleportation circuit. With measurement, the three-party state becomes one of the Bell state pairs. Bob tells Cleo. Cleo can make adjustments to obtain Alice's secrert qubit |χ>.
Rainer Müller and Franziska Greinert, “Playing with a Quantum Computer, arXiv: 2108.06271v1 [physics. ed-ph] (2021).” J. Kempe, "Quantum random walks – an introductory overview, arXiv:quant-ph/0303081v1 (2003)." 4–5. Nicolas Gisin, Quantum Chance, trans. Stephen Lyle (Switzerland: Springer International Publishing, 2014), 7-20. The Big Bell Test (2016), http://thebigbelltest. icfo.eu/. Electra Eleftheriadou, “Quantum Optical State Comparison Amplification (2015).” University of Strathclyde, Glasgow: Student Thesis. Accessed October 25, 2021. https://pureportal.strath.ac.uk/en/ studentTheses/quantum-optical-state-comparison-amplification. Laura De Decker, “Quantum Catwalk, Journal of Mathematics and the Arts 12, no. 4 (2018).” 244-247. References
Dr Electra Eleftheriadou is an Educational Consultant and Coach with a focus on advancing equity and inclusion. She has a PhD in Theoretical Physics and her career path includes Science Outreach, Physics Education and supporting institutional transformation. Stefan A. Rose is a large-format photographer, video artist, and poet who received BFA and BSc degrees from Mount Allison University. He was 2010 Artist-In-Residence for City of Kitchener and documentary photographer/videographer for New Adventures in Sound Art (NAISA), and new music concerts. His video collaborations with Penderecki String Quartet were presented in concert internationally; he was sound artist for an experimental video with his wife Laura De Decker and Herménégilde Chiasson that received a Special Mention at FICFA 2022 (Festival international du cinéma francophone en Acadie) and was long-listed for Aesthetica Art Prize 2023.
Biographies
Laura De Decker is a graduate of the Art and Art History program of University of Toronto and Sheridan College and MFA program in Visual Arts, University of Victoria where she started to write computer programs to have control over the colours in her artworks. She was a panelist at Banff New Media Institute, at ISEA2014 in Dubai, and at OCADU. She was artist-in-residence for CAFKA (Contemporary Art Forum for Kitchener + Area)/Christie Digital and for Institute for Quantum Computing, University of Waterloo. She has been the recipient of numerous Canada Council and Ontario Arts Council grants. She published an article in the Journal of Mathematics and the Arts called Quantum Catwalk about her 50-foot by 7-foot laser-cut vinyl installation at Perimeter Institute for Theoretical Physics in Waterloo. De Decker is currently working on two collaborative projects relating to art and physics. She lives in Dorchester, NB.
Artist's Biography
Acknowledgement I would like to thank Electra and Stefan for your support and friendship in doing this project. I also would also like to thank Quasar for the patience, enthusiasm, and insight you provided during this project.
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