3.1.

Optical Elements

Elements in Virtual Lab are based on idealized versions of optical devices that can be used on an optical table in a laboratory. We provide visual icons as symbolic representations of these elements along with logical operators; see Fig. 3. Many elements have additional physical properties that can be modified in a tooltip that opens after a right-click; see Fig. 4.

Fig. 3

Elements available in Virtual Lab grouped by type and described from left to right. (a) Photon generators: a single-photon source, a Bell-pair generator, the three-photon GHZ state, and the three-photon W state. (b) Measurements: a detector, a photo-sensitive bomb, a rock, a neutral-density filter, a linear polarizer, and a nondemolition detector. (c) Nonabsorptive optical elements: a non-polarizing beamsplitter, a polarizing beamsplitter, a mirror, a corner cube, an optical circulator, a waveplate, a Faraday rotator, a sugar solution, a vacuum jar, and a glass slab. (d) Input and output: a switch, a randomized switch, an output variable, a correlator, and a goal. (e) Classical logic gates: AND, NAND, OR, NOR, and XOR; arithmetic operations: SUM, MIN, and MAX. (f) Quantum gates^41,42 for a single qubit: identity, Pauli-$X$, Pauli-$Y$, Pauli-$Z$, Hadamard, phase shifts, and a square root of NOT. For two qubits: controlled-NOT (CNOT) and controlled-Z. (g) A text comment field.

Fig. 4

Configurable parameters for a nonpolarizing beamsplitter: rotation, reflectance, and reflection phase from one of its sides.

The Bell pair generator is based on an abstracted version of the spontaneous parametric down-conversion in a beta barium oxide crystal,⁴³ hence the gem icon. We do not display the incident laser beam and focus only on the generated state. The detector is represented by a venus flytrap to add playfulness with a recognizable image of the piranha plant from the Mario game series. Similarly, we represent a fully-absorbing element as a sentient rock. A photo-sensitive bomb icon takes its inspiration from another popular game, Minesweeper. We have found that this approach makes Virtual Lab a more welcoming introduction to quantum physics and adds recognizability – to the point that we named the company after this plant. Each classical logic operation is depicted by a Venn diagram, while a quantum gate is given as a letter on its matrix visualization; see Sec. 3.4.

In optics, phase delays are typically modified by a slight change of optical path, which is challenging to present with a grid-based system. Therefore, we delay the phase with a glass slab (becauseit has a higher refraction index than air) and a vacuum chamber (because it has a lower refraction index than air). This representation serves as a further introduction to optical phenomena. Similarly, in our mirrors, both sides are reflective. Linear polarizers and phase plates are represented in the board plane, rather than perpendicularly, for visual clarity. At the same time, we made sure that all elements are expressed with physically-correct operators.

3.2.

Photon Amplitude Visualization

We represent a photon as a wavepacket – oscillations within a Gaussian envelope; see Fig. 5. We draw an electric field, according to its polarization, in a pseudo-3D way. We show the phase by shifting the oscillation. The amplitude is shown as opacity, rather than more traditionally as the height of the electric field amplitude. We wanted to ensure that small amplitudes are visible and use an intuitive, visual language that is suitable for a general audience. In our visualizations of vectors and matrices, we use hue to indicate phase. We refrained from the same approach for photons approach because it would be confusing to use color where it could be confused with the wavelength.

Fig. 5

A single-photon amplitude is represented as oscillations within a Gaussian envelope. Note that horizontal polarization $|H⟩$ is parallel to the board, and vertical $|V⟩$ is perpendicular to the board.

3.3.

Quantum State Visualization

Users can check the exact quantum state at a given simulation step in the typical form of kets.⁴⁴ We provide a few improvements for readability and interactivity to create a clear and minimalistic visual data representation;^45,46 see Fig. 6. We employ color-coding to match the formula with its description.^47–49 Users can choose between various representations of complex numbers – polar, polar using full rotations ($\tau =2\pi$), Cartesian, and a color circle (amplitude as radius and phase as hue). Users can change between these representations according to their preferences or what is most suitable for a given phenomenon.

Fig. 6

A Bell state $|{\mathrm{\Phi }}^{+}⟩$ represented as a ket vector, with various choices of bases and complex number formatting. In this notation, $|64\to H⟩$ is a single-photon superposition component located at position (6,4), traveling right, with horizontal polarization; $|64\to H44\leftarrow H⟩$ is a two-photon state.

Similarly, we provide a way to dynamically change between bases – horizontal ($|H⟩$, $|V⟩$), diagonal ($|D⟩$, $|A⟩$), and circular ($|L⟩$, $|R⟩$). This feature can be used to set a basis that works best for a given phenomenon or to toggle between bases to explore the same phenomena in a few different ways. For example, the Faraday rotator, when expressed in $|H⟩$ and $|V⟩$, acts as a rotation. But for the circular basis of $|L⟩$ and $|R⟩$, it is just a phase change between polarizations. Likewise, for a mirror in the horizontal basis, due to changing the coordinate system, horizontal polarization gets a sign flip, which might look unremarkable. Yet, in the circular basis, we see that this sign flip swaps left with right circular polarization.

We distribute an extended version of this feature as an open-source library BraKetVue.⁵⁰ This public version supports arbitrary discrete states (spin, qubits, n-energy states, and user-defined), sequences of states, and light and dark modes; see Fig. 7. Because BraKetVue is a JavaScript library, it can be used on websites, in interactive materials, and on web-based slides or be embedded in Jupyter Notebook. One of the examples is an explorable explanation of quantum logic gates for a single qubit.⁴²

Fig. 7

A different example of the BraKetVue state visualizer, a single-qubit state propagating through a quantum circuit.

3.4.

Operator Visualization

One typical way to visualize quantum operators involves showing them in a braket notation, or as any other matrix – displaying them as a table with numerical values. This approach is insufficient for a clear presentation.

In data visualization, arrays can be visualized with heatmaps – two-dimensional tile plots, with tile color representing the numerical value. For an interactive example, see Clustergrammer.⁵¹ However, typical plotting packages (such as Python Matplotlib or R ggplot2) do not support complex numbers.

We present a quantum operator matrix as an array of colored disks, with basis state labels and with further interaction options for label rearrangement and selection of basis; see Figs. 8 and 9. We follow a similar visualization scheme to a recursive pattern of Qubism,^52,53 applied to operators (matrices) instead of wavefunctions (complex vectors). For a complex amplitude $z=r\exp (i\varphi )$, we represent its length $r$ as a disk radius and its phase $\varphi$ as hue. The color scheme follows a domain coloring technique.⁵⁴ Instead of the common choice to map $r$ to brightness or opacity, we use disc radius for a few reasons. First, it is easier to read regardless of screen settings, even for low amplitudes. Second, the disk area $\pi r^2$ is proportional to another vital property – probability. We show the amplitude and phase values on mouseover, with three applications in mind: as a chart legend, for access to the exact values, and to provide accessibility for people with color blindness (regardless of its type).

Fig. 8

BraKetVue visualizations of matrices representing quantum operators for a nonpolarizing beamsplitter. The interactive legend show $r$ and $\varphi$ of highlighted matrix entry. (a) The horizontal/vertical basis, direction-polarization tensor ordering. (b) The circular basis, polarization-direction tensor ordering.

Fig. 9

BraKetVue visualizations of matrices beyond Virtual Lab. (a) The controlled-NOT (CNOT) quantum gate. Note that, with the input column highlighted, we get a truth table; in this case, we see that $|10⟩\mapsto |11⟩$. (b) The Pauli-Z operator for a spin-1/2 system.

3.5.

Visualizing Entangled Amplitudes

To visualize entanglement, we invent a new method because we did not find any procedural way of visualizing arbitrary entangled states. Visually, we represent entanglement by coordinated blinking of the entangled particles; see Fig. 10. For two particles in state ${|\mathrm{\Psi }⟩}_{12}$, we use the following procedure:

Fig. 10

Three snapshots of the amplitude blinking visualizing entanglement in the singlet state $|{\mathrm{\Psi }}^{-}⟩$. Note that, within an image, the states are orthogonal to each other.

In other words, we sample the state of the first particle $|{\psi }_1⟩$ if the other particle is randomly measured. Then we get the corresponding state of the second particle $|{\psi }_2⟩$. We get two states, $|{\psi }_1⟩$ and $|{\psi }_2⟩$, which display separately the same way. For an easy example, take Bell state $|{\mathrm{\Psi }}^{-}⟩=(|01⟩-|10⟩)/\sqrt{2}$, also known as the singlet state. In this state, two particles are always orthogonal. We check that

We can easily generalize this operation for $n$ particles. First, we generate $n-1$ random vectors $|{\chi }_2⟩,\dots |{\chi }_n⟩$. Taking a random unit vector for $d$-dimensional space is simple and does not require parameterization. We generate $2d$ random numbers with Gaussian distribution⁵⁵ (for the real and imaginary parts of each coordinate) and then normalize the vector. We do not need to generate all random vectors (i.e., around 1000 per photon) – it suffices to generate a subset for which photon probabilities are nonzero. Then, we follow an iterative procedure:

An alternative approach that we considered is the Schmidt decomposition of an entangled state:

Both approaches (ours and the Schmidt decomposition) can be applied simply to the visualization of any quantum state, including qubits, electrons, and other systems. In general, it is impossible to present entangled quantum states as probabilistic representations. Consequently, this coordinated blinking should be understood as a quantum correlation rather than a statistical mixture.

3.6.

Entanglement Between Particles

Although the blinking of the waves is an indicator of entanglement, it might be hard to see the degree of entanglement between parts of the system. Because we did not find an existing way to represent the degree of entanglement between particles, we created our own. Working with pure states simplifies the task, as all measures of correlations are related to entanglement.

A quantum physics novice might expect to see values of entanglement between each particle pair, that is, for three particles A, B, and C, one would expect to have some measures of entanglement for A–B, B–C, and C–A. However, it is not possible. The only meaningful measure is the entanglement of a particle with the rest of the system: A–BC, B–AC, and C–AB. We use Rényi-2 entanglement entropy, due to its computational simplicity.

Next, we turn this measure into a digestible visualization; see Fig. 11. We place particles in a circle and represent them as dots. We draw a string connecting them and set the respective entanglement entropy (of a particle with the rest of the system) as the force constant. If all strings are equally stiff, their connecting points land in the middle. If only two particles are entangled with each other, then the connecting point lands between them. Next, we turn the strings into amoeba-blobs (force constants become connection widths), so we can clearly see which particles are entangled with each other. The same approach can be employed to visualize entanglement in any other system of quantum particles.

Fig. 11

An entanglement graph for three particles, a bipartite system of a photon and an entangled Bell pair. (a) The spring model, with an entanglement measure serving as the spring coefficients. The connecting point is the equilibrium. (b) The same graph visualized as a blob connecting entangled particles – each connection width is a function of the entanglement measure.

3.7.

Multiverse Tree

We represent the evolution of quantum states as a tree rather than a sequence of steps; see Fig. 12. Each measurement event splits evolution into as many branches as there are outcomes with nonzero probability. The multiverse tree serves as a summary of all possible outcomes and as a way to explore the experiment. The user can click to select each timestep of each branch.

Fig. 12

A multiverse tree with a highlighted path to the state that we currently explore. Each absorption event is represented by an icon of the absorbing element, including polarizers and detectors. As the tree can be infinite for setup with loops, we show the percentage of all simulated paths, weighted by their probabilities.

There are several interpretations of quantum mechanics, including the indeterministic Copenhagen interpretation and the deterministic Everettian many-worlds interpretation.⁵⁶ Even among physicists, the choice of interpretation is a matter of philosophical belief rather than a scientific hypothesis.^57,58 The multiverse tree works the same regardless of the interpretation, just with a difference in what the nodes mean – separate probabilistic branching events or components of the global wavefunction.

3.8.

Classical Randomness, Table of Measurements, and Correlators

In addition to quantum states, Virtual Lab supports classical states based on the initial setup and detection events. They are transferred through wires, with logic operations, and used for modifying elements on the board, measuring output correlations, and logging the results for arbitrary analysis; see Fig. 13. The data are downloadable as a comma-separated values (CSV) file. For a full example, see the Bell inequality violation in Fig. 17.

Fig. 13

Inputs connected through logic gates (AND and OR) to an element and an output variable. A gray wire carries 0 (false) and a highlighted 1 (true). An element connected to a value has two sets of adjustable parameters.

Input variables can be deterministic (so they work as switches) or randomized (so on each run, they are randomly set to on/off). Both have their application in setups – especially in quantum key distribution protocols and other correlation-based experiments. Because elements are affected by detection events, it is possible to create conditional setups, e.g., quantum teleportation.

All wires carry their signals instantly, without the speed-of-light delay, to simplify the interface.

Correlators measure correlation between $\pm 1$-valued variables $s_1,s_2,\dots s_n$, conditional on the control variables $c_1,c_2,\dots c_m$. For a practical example, see the Bell inequality in Fig. 17.

3.9.

Quantum Game

The sandbox mode offers a way for the user to create arbitrary setups. However, the sandbox mode by design lacks any objectives and can be intimidating as it provides all functionalities at once. It is entirely up to the user to decide how to interact with Virtual Lab.

Based on the earlier success of Quantum Game with Photons, we created a similar experience; see Fig. 14. Rather than releasing a separate product, we made Quantum Game a part of Virtual Lab. Although our focus was to provide an enjoyable experience for beginners and advanced users alike, we wanted it to serve as a step-by-step tutorial on both quantum physics and the Virtual Lab interface. We aimed to make it clear that Quantum Game is an advanced simulation of quantum mechanics, rather than a visual gimmick loosely based on actual physics.

Fig. 14

A screen from Quantum Game after running a simulation shows level goals and their status.

For each level, the gameplay consists of creating a setup, running it many times, and collecting the results – a typical pattern for programming games,⁵⁹ e.g., SpaceChem and Shenzhen I/O by Zachtronics. The probabilistic nature of quantum mechanics makes it challenging to design clear goals. We opted for setting detection percentage thresholds (e.g., a given detector needs to absorb a photon with the probability $\ge 25\%$). Although we show a finite number of experimental runs (e.g., 20), the win condition is based on probabilities in the infinite limit.

4.1.

State Representation

The state of a single photon is characterized by four components: position (along the $x$ and $y$ axes), direction, and polarization. A photon has two polarization states. Because we use grid mechanics, light travels along four cardinal directions, rather than at an arbitrary angle. Grid size is a parameter that can be adjusted according to needs, and we use the default of $13\times 10$, which we find suitable for most applications. It gives rise to around 1000 dimensions per single particle.

Virtual Lab simulation supports primarily distinguishable particles. However, photons are indistinguishable bosons, but they can be made distinguishable by assigning them a wavelength. A nonentangled product state is represented as a tensor product of single-particle states:

This approach is equivalent to the typical description of bosons with creation and annihilation operators for states with a fixed number of photons.^53,63 However, from a numerical point of view, it is easier to work with vectors than polynomials. One significant drawback of our choice is the restriction to Fock states – the current framework does not support coherent or squeezed states.

4.2.

Unitary Operations

In each step, we perform three operations: free propagation, measurement, and the action of unitary operation. The propagation step $T$ involves shifting the photon’s position according to its direction:

We use a limited number of two-photon interactions. This choice is motivated by optical setups in which photon-photon interactions are challenging to implement. We apply the following operation:

4.3.

Measurement

We allow measurements at any step, including destructive measurements (absorbing a photon) and nondemolition measurements⁶⁴ (not absorbing a photon). In each measurement phase and for each photon, there are three exclusive outcomes:

Note that, after a measurement event, the state remains a state with a fixed number of photons.

All measurements happen within the elements on the board. Destructive measurements happen when a photon hits a fully absorptive element (e.g., a rock, a detector, the side of certain elements), a partially absorptive element (e.g., a neutral density filter), or a polarization-dependent element (e.g., linear polarizer) or falls out of the board. Nondemolition measurements happen when a photon passes through a nondemolition detector or when none of the other detection events occur.

Destructive measurements are represented with a non-normalized (bra) vector, $\sqrt{w_i}⟨{\varphi }_i|$. For example, for a single direction $|\to ⟩$, these vectors for polarization are

After a measurement of the $m$’th photon, the remaining state is

To deal with the general case of destructive and nondemolition measurements, we need to use a general scheme of positive operator-valued measures (POVMs). A general POVM requires a set of positive semidefinite matrices $M_i$ that sum to identity.⁴¹ The resulting non-normalized state for three photons is

4.4.

Performance

To create a smooth experience, we need to have fast response times. As a rule of thumb, up to a 100 ms response time feels like a system reacts instantaneously, while up to a 1 s response time interupts the user’s flow of thoughts.^65–67 This sets the upper bound for the simulation time of up to three photons. For comparison, 60 frames per second required for a smooth video game experience⁶⁸ corresponds to 17 ms. None of the off-the-shelf libraries (NumPy, SciPy, TensorFlow, and PyTorch) were able to support this kind of operation with these constraints, in addition to working in a browser.

Note that, even for a three-photons state with a typical (dense) representation, ${10}^9$ complex numbers are required. A single unitary operator would require ${10}^{18}$. Yet, in typical experimental cases, we use only a few nonzero amplitudes – prompting us to use sparse vectors. This task goes far beyond typical interactive simulators of quantum circuits. For eight qubits, it is $2^8\approx {10}^3$ dimensions, which can be easily simulated with dense vector and matrix operations.

However, even with sparse vectors for three particles, an identity matrix would still have ${10}^9$ nonzero elements (i.e., the ones on the diagonal). Hence, we need to apply operations to take advantage of the structure of the tensor, for speedup of operators acting only on a subset of dimensions.

The initial approaches are in an open-source TypeScript library Quantum Tensors. To improve performance even further, we developed a high-performance sparse array simulation in Rust, a low-level programming language. The typical timescale of simulating experiments is $30\hspace{0.17em}\mathrm{to}\hspace{0.17em}150\hspace{0.17em}\mu \mathrm{s}$. To use the engine in the browser, we compiled Rust code to WebAssembly.⁶⁹ The execution time measured in the browser is only $1.5\times$ slower than that in native code.

5.1.

One Photon, Classical

Interference is a wave phenomenon in which waves add by their amplitudes and amplify or cancel each other out depending on their relative shift. This behavior might be counterintuitive to people accustomed to geometric optics, in which light adds by its intensity.

In optics, interferometers are applied for sensitive measurements of distance. The famous experiment of Michelson and Morley⁷³ in 1887 showed that light velocity is independent of the frame of reference, thus setting the footing for the special relativity theory. For an accessible introduction to special relativity, we suggest Unusually Special Relativity by Dragan.⁷⁴ For an interactive version of the experiment, see Ref. 75.

The Mach–Zehnder interferometer⁷⁶ is applied in precise measurements of tiny changes of the refraction index and for a wide range of quantum experiments. In the interactive version, the user can modify the relative delay between paths and see the result in real-time; see Fig. 15.

Fig. 15

Changing relative optical paths in the Mach–Zehnder interferometer. We use the high-intensity visualization mode.

The Sagnac interferometer,^77,78 originally created to detect the luminiferous aether (as Michelson–Morley), is now used in ring laser gyroscopes and fiber optics gyroscopes.

Another set of phenomena involves the polarization of light. Understanding polarization is important for learning the nature of light and its application in numerous fields including photography, LCD displays, and telecommunications. The three polarizer paradox⁷⁹ is an experiment in which two perpendicular polarizers block light, but inserting a third allows some of the light to pass. This represents a classical version of the quantum Zeno experiment.

Optical activity is a phenomenon in which chiral molecules (e.g., D-glucose, a naturally occurring enantiomer) rotate the polarization of light. Although in quantum optical laboratories this effect is not being applied to rotate polarization, it is being utilized to measure the concentration of sugar. The Faraday effect is a magneto-optic effect in which the polarization of light rotates in the presence of a magnetic field in a transparent material. Although this rotation may seem to be the same as the one caused by optical activity, it has different symmetries, that is, the rotation is opposite depending on if the light propagates in the same or opposite direction of the magnetic field. The effect is used in virtually all other optical elements that are not symmetric in time, such as optical circulators and optical diodes; see Ref. 80.

5.2.

One Photon, Quantum

There is a common misunderstanding that at least two entangled particles are required to get distinctively quantum phenomena. Even some parts of quantum computing are possible without entanglement.^81,82 First and foremost, in quantum mechanics, light is absorbed in discrete portions (or quanta), with energy $\hslash \omega$ each. Although at classical intensities of light we can measure the power of a light beam (or the energy of a pulse), for low intensities, the detection phenomenon has a discrete nature.

Even the detection of a single particle serves as a way to explore the fundamentals of quantum mechanics, such as the shot noise (randomness of detection events), Born’s rule (the probability of measurement is ${|\psi |}^2$), and the nature of measurement. Shot noise can be seen on photographic film and digital sensors. Consumer smartphone cameras can be used to extract pure quantum randomness.⁸³ We can simulate more advanced detection schemes, such as an optical implementation of optimal nonorthogonal state discrimination;⁸⁴ see Ref. 85.

The Elitzur-Vaidman bomb tester^86,87 is a combination of interference and the nature of measurement. In the classical physics world, it is a tautology that a photo-sensitive bomb cannot be detected with light without setting it off. In the quantum world, there is room for “interaction-free measurement” by observing the changes in the interference pattern. A bomb can be placed in one arm of the Mach–Zehnder interferometer. Some detector clicks testify that there is something blocking one path and therefore breaking the interference; see Fig. 16.⁸⁸

Fig. 16

(a) The Elitzur–Vaidman bomb tester with a detection event that would not be possible if the bomb were not present. (b) Deutsch–Jozsa algorithm for a function $\{\mathrm{0,1}\}\to \{\mathrm{0,1}\}$ with its values $f(0)$ and $f(1)$ implemented as phase shifts. A single detection event tells us whether the functiom is constant.

Quantum measurement affects the measured quantum systems, even when no absorption is involved. It is impossible to measure two noncommuting observables without affecting them. Discovering where a particle is located destroys the interference pattern, which is called quantum erasure;⁸⁹ see Ref. 90. Within Virtual Lab it is possible not only to place a nondemolition measurement on one path but also to define its efficiency, thus showing a continuous transition between fully breaking quantum interference and not touching the system at all; see Ref. 91.

The Quantum Zeno effect⁹² is another phenomenon in which a nonmeasurement affects the result. The state evolution is inhibited completely by the limit of infinitely frequent projective measurements.

Due to the fundamental rules of quantum mechanics, it is impossible to copy an arbitrary quantum state, also known as the “no-cloning theorem.”⁹³ This property is a direct consequence of the linearity of quantum mechanics and is necessary for forbidding superluminal communication. This is a substantial challenge for quantum computing that provides an opportunity to use quantum states for cryptography. The BB84 quantum key distribution protocol⁹⁴ sends photons with random bits encoded quantum in different quantum bases – any eavesdropping would be both imperfect (no-cloning) and discoverable (as measurement affects the system); see Ref. 95.

The Deutsch-Jozsa algorithm^96–98 is an algorithm showing exponential speedup of a quantum algorithm over a classical one. It can be shown even within a single qubit that one needs a single measurement rather than two; see Fig. 16.

Photon polarization serves as a carrier of quantum information. This dimension alone encodes a qubit. In Virtual Lab, on top of optical operations, we provide abstract quantum gates. Last but not least, it is possible to encode two classical qubits within a single photon, using polarization for one dimension and direction for the other.⁹⁹

5.3.

Two Photons

It takes two particles to show one of the most striking quantum properties – entanglement.

Bell pairs are a set of four quantum states forming a basis for two fully-entangled qubits. In Virtual Lab, we provide an idealized nonlinear crystal (based on spontaneous parametric down-conversion), which generates Bell pairs. Bell pair detection, necessary for some protocols (such as quantum teleportation), can be performed with a CNOT gate that we provide. Experimental setups within Virtual Lab can be applied to transform Bell states, explore correlation patterns, and create cryptography setups such as the Ekert quantum key distribution protocol;¹⁰⁰ see Ref. 101.

The Bell inequality¹⁰² is a correlation that should hold for any probabilistic correlations between subsystems. It was created to turn a philosophical debate on hidden variables into an experimentally testable hypothesis. Quantum mechanics violate it with a Bell pair and a set of measurements. In Virtual Lab, there is a correlator between results, which is set so that it can measure the Bell inequality in the Clauser–Horne–Shimony–Holt (CHSH) form.¹⁰³ It can be used in two main ways – as an exact result and as a quantity based on the number of detections (involving shot noise); see Fig. 17.¹⁰⁴

Fig. 17

CHSH Bell inequality violation simulated with Virtual Lab. We use the detection visualization mode.

5.4.

Three Photons

In quantum teleportation,¹⁰⁵ a quantum state is transferred to another particle, regardless of how far away it is. It requires three particles: one to be teleported and an entangled pair, with one particle being close to the source and the other in the target location. See Fig. 1 and the interactive version.¹⁰⁶

In Virtual Lab, we simulate quantum teleportation with a photon source (in which photon polarization is set to an arbitrary state), with the nonlinear crystal generating a Bell pair. The Bell state measurement is implemented with a CNOT Gate, and two classical bits are sent to the remote location.

For two qubits, there is only one type of entanglement, that is, each entangled state can be turned into another by applying only local operations and classical communication (LOCC). For three and more qubits, this is no longer the case.¹⁰⁷ There are two distinct states that cannot be changed locally – the W and Greenberger–Horne–Zeilinger (GHZ) states,¹⁰⁸ which can be used to test nonlocality as¹⁰⁹