Greenberger–Horne–Zeilinger state

In physics, in the area of quantum information theory, a Greenberger–Horne–Zeilinger state (GHZ state) is a certain type of entangled quantum state that involves at least three subsystems (particle states, or qubits). It was first studied by Daniel Greenberger, Michael Horne and Anton Zeilinger in 1989.[1] Extremely non-classical properties of the state have been observed.

Generation of the 3-qubit GHZ state using quantum logic gates.

Definition

The GHZ state is an entangled quantum state of M > 2 subsystems. If each system has dimension ${\displaystyle d}$, i.e., the local Hilbert space is isomorphic to ${\displaystyle \mathbb {C} ^{d}}$, then the total Hilbert space of M partite system is ${\displaystyle {\mathcal {H}}_{tot}=(\mathbb {C} ^{d})^{\otimes M}}$. This GHZ state is also named as ${\displaystyle M}$-partite qudit GHZ state, it reads

${\displaystyle |\mathrm {GHZ} \rangle ={\frac {1}{\sqrt {d}}}\sum _{i=0}^{d-1}|i\rangle \otimes \cdots \otimes |i\rangle ={\frac {1}{\sqrt {d}}}(|0\rangle \otimes \cdots \otimes |0\rangle +\cdots +|d-1\rangle \otimes \cdots \otimes |d-1\rangle )}$.

In the case of each of the subsystems being two-dimensional, that is for qubits, it reads

${\displaystyle |\mathrm {GHZ} \rangle ={\frac {|0\rangle ^{\otimes M}+|1\rangle ^{\otimes M}}{\sqrt {2}}}.}$

In simple words, it is a quantum superposition of all subsystems being in state 0 with all of them being in state 1 (states 0 and 1 of a single subsystem are fully distinguishable). The GHZ state is a maximally entangled quantum state.

The simplest one is the 3-qubit GHZ state:

${\displaystyle |\mathrm {GHZ} \rangle ={\frac {|000\rangle +|111\rangle }{\sqrt {2}}}.}$

This state is non-biseparable[2] and is the representative of one of the two non-biseparable classes of 3-qubit states (the other being the W state), which cannot be transformed (not even probabilistically) into each other by local quantum operations.[3] Thus ${\displaystyle |\mathrm {GHZ} \rangle }$ and ${\displaystyle |W\rangle }$ represent two very different kinds of entanglement for three or more particles.[4] The W state is, in a certain sense "less entangled" than the GHZ state; however, that entanglement is, in a sense, more robust against single-particle measurements, in that, for an N-qubit W state, an entangled (N − 1)-qubit state remains after a single-particle measurement. By contrast, certain measurements on the GHZ state collapse it into a mixture or a pure state.

Properties

There is no standard measure of multi-partite entanglement because different, not mutually convertible, types of multi-partite entanglement exist. Nonetheless, many measures define the GHZ state to be maximally entangled state.

Another important property of the GHZ state is that when we trace over one of the three systems, we get

${\displaystyle \operatorname {Tr} _{3}\left(\left({\frac {|000\rangle +|111\rangle }{\sqrt {2}}}\right)\left({\frac {\langle 000|+\langle 111|}{\sqrt {2}}}\right)\right)={\frac {(|00\rangle \langle 00|+|11\rangle \langle 11|)}{2}},}$

which is an unentangled mixed state. It has certain two-particle (qubit) correlations, but these are of a classical nature.

On the other hand, if we were to measure one of the subsystems in such a way that the measurement distinguishes between the states 0 and 1, we will leave behind either ${\displaystyle |00\rangle }$ or ${\displaystyle |11\rangle }$, which are unentangled pure states. This is unlike the W state, which leaves bipartite entanglements even when we measure one of its subsystems.

The GHZ state leads to striking non-classical correlations (1989). Particles prepared in this state lead to a version of Bell's theorem, which shows the internal inconsistency of the notion of elements-of-reality introduced in the famous Einstein–Podolsky–Rosen article. The first laboratory observation of GHZ correlations was by the group of Anton Zeilinger (1998). Many more accurate observations followed. The correlations can be utilized in some quantum information tasks. These include multipartner quantum cryptography (1998) and communication complexity tasks (1997, 2004).

Pairwise entanglement

Although a naive measurement of the third particle of the GHZ state results in an unentangled pair, a more clever measurement, along an orthogonal direction, can leave behind a maximally entangled Bell state. This is illustrated below. The lesson to be drawn from this is that pairwise entanglement in the GHZ is more subtle than it naively appears: measurements along the privileged Z direction destroy pairwise entanglement, but other measurements (along different axes) do not.

The GHZ state can be written as

${\displaystyle |000\rangle +|111\rangle ={\big (}|00\rangle +|11\rangle {\big )}\otimes |L\rangle +{\big (}|00\rangle -|11\rangle {\big )}\otimes |R\rangle ,}$

where the third particle is written as a superposition in the X basis (as opposed to the Z basis) as ${\displaystyle |0\rangle =|L\rangle +|R\rangle }$ and ${\displaystyle |1\rangle =|L\rangle -|R\rangle }$.

A measurement of the GHZ state along the X basis for the third particle then yields either ${\displaystyle |00\rangle +|11\rangle }$, if ${\displaystyle |L\rangle }$ was measured, or ${\displaystyle |00\rangle -|11\rangle }$, if ${\displaystyle |R\rangle }$ was measured. In the later case, the phase can be rotated by applying a Z quantum gate to give ${\displaystyle |00\rangle +|11\rangle }$, while in the former case, no additional transformations are applied. In either case, the end result of the operations is a maximally entangled Bell state.

The point of this example is that it illustrates that the pairwise entanglement of the GHZ state is more subtle than it first appears: a judicious measurement along an orthogonal direction, along with the application of a quantum transform depending on the measurement outcome, can leave behind a maximally entangled state.

Applications

GHZ states are used in several protocols in quantum communication and cryptography, for example, in secret sharing[5] or in the Quantum Byzantine Agreement.

See also

• Quantum pseudo-telepathy uses a four-particle entangled state.
• Bell's theorem
• Bell state
• GHZ experiment
• Local hidden variable theory
• Quantum entanglement
• Qubit
• Measurement in quantum mechanics

References

1. Daniel M. Greenberger; Michael A. Horne; Anton Zeilinger (2007), Going beyond Bell's Theorem, arXiv:0712.0921, Bibcode:2007arXiv0712.0921G
2. A pure state ${\displaystyle |\psi \rangle }$ of ${\displaystyle N}$ parties is called biseparable, if one can find a partition of the parties in two disjoint subsets ${\displaystyle A}$ and ${\displaystyle B}$ with ${\displaystyle A\cup B=\{1,\dots ,N\}}$ such that ${\displaystyle |\psi \rangle =|\phi \rangle _{A}\otimes |\gamma \rangle _{B}}$, i.e. ${\displaystyle |\psi \rangle }$ is a product state with respect to the partition ${\displaystyle A|B}$.
3. W. Dür; G. Vidal & J. I. Cirac (2000). "Three qubits can be entangled in two inequivalent ways". Phys. Rev. A. 62 (6): 062314. arXiv:quant-ph/0005115. Bibcode:2000PhRvA..62f2314D. doi:10.1103/PhysRevA.62.062314.
4. Piotr Migdał; Javier Rodriguez-Laguna; Maciej Lewenstein (2013), "Entanglement classes of permutation-symmetric qudit states: Symmetric operations suffice", Physical Review A, 88 (1): 012335, arXiv:1305.1506, Bibcode:2013PhRvA..88a2335M, doi:10.1103/PhysRevA.88.012335
5. Mark Hillery; Vladimír Bužek; André Berthiaume (1998), "Quantum secret sharing", Physical Review A, 59 (3): 1829–1834, arXiv:quant-ph/9806063, Bibcode:1999PhRvA..59.1829H, doi:10.1103/PhysRevA.59.1829