# Qutrit

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A qutrit (or quantum trit) is a unit of quantum information that is realized by a 3-level quantum system, that may be in a superposition of three mutually orthogonal quantum states. [1]

## Contents

The qutrit is analogous to the classical radix-3 trit, just as the qubit, a quantum system described by a superposition of two orthogonal states, is analogous to the classical radix-2 bit.

There is ongoing work to develop quantum computers using qutrits and qubits with multiple states. [2]

## Representation

A qutrit has three orthonormal basis states or vectors, often denoted ${\displaystyle |0\rangle }$, ${\displaystyle |1\rangle }$, and ${\displaystyle |2\rangle }$ in Dirac or bra–ket notation. These are used to describe the qutrit as a superposition state vector in the form of a linear combination of the three orthonormal basis states:

${\displaystyle |\psi \rangle =\alpha |0\rangle +\beta |1\rangle +\gamma |2\rangle }$,

where the coefficients are complex probability amplitudes, such that the sum of their squares is unity (normalization):

${\displaystyle |\alpha |^{2}+|\beta |^{2}+|\gamma |^{2}=1\,}$

The qubit's orthonormal basis states ${\displaystyle \{|0\rangle ,|1\rangle \}}$ span the two-dimensional complex Hilbert space ${\displaystyle H_{2}}$, corresponding to spin-up and spin-down of a spin-1/2 particle. Qutrits require a Hilbert space of higher dimension, namely the three-dimensional ${\displaystyle H_{3}}$ spanned by the qutrit's basis ${\displaystyle \{|0\rangle ,|1\rangle ,|2\rangle \}}$, [3] which can be realized by a three-level quantum system. However, not all three-level quantum systems are qutrits. [4]

An n-qutrit register can represents 3n different states simultaneously, i.e., a superposition state vector in 3n-dimensional complex Hilbert space. [5]

The quantum logic gates operating on single qutrits are ${\displaystyle 3\times 3}$ unitary matrices and gates that act on registers of ${\displaystyle n}$ qutrits are ${\displaystyle 3^{n}\times 3^{n}}$ unitary matrices (the elements of the unitary groups U(3) and U(3n) respectively). [6]

Qutrits have several peculiar features when used for storing quantum information. For example, they are more robust to decoherence under certain environmental interactions. [7] In reality, manipulating qutrits directly might be tricky, and one way to do that is by using an entanglement with a qubit. [8]

## Related Research Articles

In physics, the no-cloning theorem states that it is impossible to create an independent and identical copy of an arbitrary unknown quantum state, a statement which has profound implications in the field of quantum computing among others. The theorem is an evolution of the 1970 no-go theorem authored by James Park, in which he demonstrates that a non-disturbing measurement scheme which is both simple and perfect cannot exist. The aforementioned theorems do not preclude the state of one system becoming entangled with the state of another as cloning specifically refers to the creation of a separable state with identical factors. For example, one might use the controlled NOT gate and the Walsh–Hadamard gate to entangle two qubits without violating the no-cloning theorem as no well-defined state may be defined in terms of a subsystem of an entangled state. The no-cloning theorem concerns only pure states whereas the generalized statement regarding mixed states is known as the no-broadcast theorem.

Quantum computing is the exploitation of collective properties of quantum states, such as superposition and entanglement, to perform computation. The devices that perform quantum computations are known as quantum computers. They are believed to be able to solve certain computational problems, such as integer factorization, substantially faster than classical computers. The study of quantum computing is a subfield of quantum information science. Expansion is expected in the next few years as the field shifts toward real-world use in pharmaceutical, data security and other applications.

Quantum teleportation is a technique for transferring quantum information from a sender at one location to a receiver some distance away. While teleportation is commonly portrayed in science fiction as a means to transfer physical objects from one location to the next, quantum teleportation only transfers quantum information. Moreover, the sender may not know the location of the recipient, and does not know which particular quantum state will be transferred.

In quantum computing, a qubit or quantum bit is the basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state quantum-mechanical system, one of the simplest quantum systems displaying the peculiarity of quantum mechanics. Examples include the spin of the electron in which the two levels can be taken as spin up and spin down; or the polarization of a single photon in which the two states can be taken to be the vertical polarization and the horizontal polarization. In a classical system, a bit would have to be in one state or the other. However, quantum mechanics allows the qubit to be in a coherent superposition of both states simultaneously, a property that is fundamental to quantum mechanics and quantum computing.

Quantum entanglement is a physical phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of the others, including when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between classical and quantum physics: entanglement is a primary feature of quantum mechanics lacking in classical mechanics.

In quantum mechanics, wave function collapse occurs when a wave function—initially in a superposition of several eigenstates—reduces to a single eigenstate due to interaction with the external world. This interaction is called an "observation". It is the essence of a measurement in quantum mechanics which connects the wave function with classical observables like position and momentum. Collapse is one of two processes by which quantum systems evolve in time; the other is the continuous evolution via the Schrödinger equation. Collapse is a black box for a thermodynamically irreversible interaction with a classical environment. Calculations of quantum decoherence show that when a quantum system interacts with the environment, the superpositions apparently reduce to mixtures of classical alternatives. Significantly, the combined wave function of the system and environment continue to obey the Schrödinger equation. More importantly, this is not enough to explain wave function collapse, as decoherence does not reduce it to a single eigenstate.

Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wave function is used to explain various quantum effects. As long as there exists a definite phase relation between different states, the system is said to be coherent. A definite phase relationship is necessary to perform quantum computing on quantum information encoded in quantum states. Coherence is preserved under the laws of quantum physics.

In quantum information theory, a quantum circuit is a model for quantum computation in which a computation is a sequence of quantum gates, which are reversible transformations on a quantum mechanical analog of an n-bit register. This analogous structure is referred to as an n-qubit register. The graphical depiction of quantum circuit elements is described using a variant of the Penrose graphical notation.

In quantum computing and specifically the quantum circuit model of computation, a quantum logic gate is a basic quantum circuit operating on a small number of qubits. They are the building blocks of quantum circuits, like classical logic gates are for conventional digital circuits.

Quantum error correction (QEC) is used in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. Quantum error correction is essential if one is to achieve fault-tolerant quantum computation that can deal not only with noise on stored quantum information, but also with faulty quantum gates, faulty quantum preparation, and faulty measurements.

The Bell states or EPR pairs are specific quantum states of two qubits that represent the simplest examples of quantum entanglement; conceptually, they fall under the study of quantum information science. The Bell states are a form of entangled and normalized basis vectors. This normalization implies that the overall probability of the particle being in one of the mentioned states is 1: . Entanglement is a basis-independent result of superposition. Due to this superposition, measurement of the qubit will collapse it into one of its basis states with a given probability. Because of the entanglement, measurement of one qubit will assign one of two possible values to the other qubit instantly, where the value assigned depends on which Bell state the two qubits are in. Bell states can be generalized to represent specific quantum states of multi-qubit systems, such as the GHZ state for 3 or more subsystems.

A trapped ion quantum computer is one proposed approach to a large-scale quantum computer. Ions, or charged atomic particles, can be confined and suspended in free space using electromagnetic fields. Qubits are stored in stable electronic states of each ion, and quantum information can be transferred through the collective quantized motion of the ions in a shared trap. Lasers are applied to induce coupling between the qubit states or coupling between the internal qubit states and the external motional states.

In quantum computing, a quantum register is a system comprising multiple qubits. It is the quantum analog of the classical processor register. Quantum computers perform calculations by manipulating qubits within a quantum register.

In physics, in the area of quantum information theory, a Greenberger–Horne–Zeilinger state is a certain type of entangled quantum state that involves at least three subsystems. It was first studied by Daniel Greenberger, Michael Horne and Anton Zeilinger in 1989. Extremely non-classical properties of the state have been observed.

Time-bin encoding is a technique used in quantum information science to encode a qubit of information on a photon. Quantum information science makes use of qubits as a basic resource similar to bits in classical computing. Qubits are any two-level quantum mechanical system; there are many different physical implementations of qubits, one of which is time-bin encoding.

A decoherence-free subspace (DFS) is a subspace of a quantum system's Hilbert space that is invariant to non-unitary dynamics. Alternatively stated, they are a small section of the system Hilbert space where the system is decoupled from the environment and thus its evolution is completely unitary. DFSs can also be characterized as a special class of quantum error correcting codes. In this representation they are passive error-preventing codes since these subspaces are encoded with information that (possibly) won't require any active stabilization methods. These subspaces prevent destructive environmental interactions by isolating quantum information. As such, they are an important subject in quantum computing, where (coherent) control of quantum systems is the desired goal. Decoherence creates problems in this regard by causing loss of coherence between the quantum states of a system and therefore the decay of their interference terms, thus leading to loss of information from the (open) quantum system to the surrounding environment. Since quantum computers cannot be isolated from their environment and information can be lost, the study of DFSs is important for the implementation of quantum computers into the real world.

Circuit quantum electrodynamics provides a means of studying the fundamental interaction between light and matter. As in the field of cavity quantum electrodynamics, a single photon within a single mode cavity coherently couples to a quantum object (atom). In contrast to cavity QED, the photon is stored in a one-dimensional on-chip resonator and the quantum object is no natural atom but an artificial one. These artificial atoms usually are mesoscopic devices which exhibit an atom-like energy spectrum. The field of circuit QED is a prominent example for quantum information processing and a promising candidate for future quantum computation.

In quantum mechanics, the cat state, named after Schrödinger's cat, is a quantum state that is composed of two diametrically opposed conditions at the same time, such as the possibilities that a cat be alive and dead at the same time.

Linear Optical Quantum Computing or Linear Optics Quantum Computation (LOQC) is a paradigm of quantum computation, allowing universal quantum computation. LOQC uses photons as information carriers, mainly uses linear optical elements, or optical instruments to process quantum information, and uses photon detectors and quantum memories to detect and store quantum information.

The no-hiding theorem states that if information is lost from a system via decoherence, then it moves to the subspace of the environment and it cannot remain in the correlation between the system and the environment. This is a fundamental consequence of the linearity and unitarity of quantum mechanics. Thus, information is never lost. This has implications in black hole information paradox and in fact any process that tends to lose information completely. The no-hiding theorem is robust to imperfection in the physical process that seemingly destroys the original information.

## References

1. Nisbet-Jones, Peter B. R.; Dilley, Jerome; Holleczek, Annemarie; Barter, Oliver; Kuhn, Axel (2013). "Photonic qubits, qutrits and ququads accurately prepared and delivered on demand". New Journal of Physics. 15 (5): 053007. arXiv:. Bibcode:2013NJPh...15e3007N. doi:10.1088/1367-2630/15/5/053007. ISSN   1367-2630.
2. "Qudits: The Real Future of Quantum Computing?". IEEE Spectrum . Retrieved 2021-05-24.
3. Byrd, Mark (1998). "Differential geometry on SU(3) with applications to three state systems". Journal of Mathematical Physics. 39 (11): 6125–6136. arXiv:. doi:10.1063/1.532618. ISSN   0022-2488.
4. "Quantum systems: three-level vs qutrit". Physics Stack Exchange. Retrieved 2018-07-25.
5. Caves, Carlton M.; Milburn, Gerard J. (2000). "Qutrit entanglement". Optics Communications. 179 (1–6): 439–446. arXiv:. doi:10.1016/s0030-4018(99)00693-8. ISSN   0030-4018.
6. Colin P. Williams (2011). Explorations in Quantum Computing. Springer. pp. 22–23. ISBN   978-1-84628-887-6.
7. Melikidze, A.; Dobrovitski, V. V.; De Raedt, H. A.; Katsnelson, M. I.; Harmon, B. N. (2004). "Parity effects in spin decoherence". Physical Review B. 70 (1): 014435. arXiv:. Bibcode:2004PhRvB..70a4435M. doi:10.1103/PhysRevB.70.014435.
8. B. P. Lanyon,1 T. J. Weinhold, N. K. Langford, J. L. O'Brien, K. J. Resch, A. Gilchrist, and A. G. White, Manipulating Biphotonic Qutrits, Phys. Rev. Lett. 100, 060504 (2008) (link)