Extracting Large Amounts of Information with Quantum Entanglement

Quantum entanglement is well-known in quantum information science as a resource for quantum teleportation and quantum computers. In this article, we will explain the potential for this entanglement to also be used as a resource for precision measurement across various fields of physics. Currently, this technology is primarily used in quantum optical experiments, but it is expected that its applications will expand to diverse condensed matter systems, including semiconductors, particle and nuclear physics experiments, and even space observation technologies.

The goal of many physics experiments is to uncover unknown interaction processes. For example, there are many situations where it is important to accurately estimate the magnitude of parameters such as the very small coupling constants of interactions. Here, we would like to use a simple example to explain the usefulness of quantum entanglement in parameter estimation.

Consider a device where the inside cannot be seen from the outside, as shown in Figure 1. Suppose there is a magnetic field inside this box oriented perpendicular to the front of the box.

If we put one neutron with its spin in the upward state into this box, its spin will rotate by a very small angle of 2θ. (In a normal case, θ takes a value from 0 to π/2.) This θ is an unknown quantity, and we estimate its magnitude by measuring the spin of the neutron that emerges from the box. By knowing the value of θ, we can estimate the strength of the magnetic field inside the box.

In the conventional method, as shown in Figure 2, N neutron spins were all prepared in the same state, passed through the box individually, measured separately, and then θ was estimated by taking a statistical average.

In this case, as is well known, the estimation error of θ decreases in inverse proportion to the square root of N. However, by using quantum entanglement, it becomes possible to reduce the estimation error to 1/N even when using the same N neutron spins. This means that when N is very large, θ can be estimated much more precisely than with the conventional method.

To explain the mechanism of this high-precision parameter estimation, let's first rotate the box by 90 degrees, as shown in Figure 3. Then, the direction of the magnetic field becomes perpendicular to the ground.

Let's try putting a neutron with its spin in the upward state into this rotated box. The spin direction will not change, and it will emerge from the box in the same state. However, the state of the spin that emerges has a phase factor of exp(iθ) applied compared to the state before it was put in.

Such a phase factor applied to the entire state vector is not a physical observable, so no matter what measurement is performed, information about θ cannot be obtained. Therefore, in the situation shown in Figure 3, estimation appears to have become more difficult than in Figure 1.

However, a hint for increasing the estimation accuracy of θ can be obtained when N upward-state neutrons are put into the box, as shown in Figure 4.

At this time, the state of the N spins emerging from the box has a phase factor of exp(iNθ) applied to the whole. In other words, the effect of a small θ has been amplified by a factor of N.

If N is made sufficiently large, Nθ can reach about 1, and there is a possibility of utilizing this amplification effect. However, even in a case like Figure 4, the amplification appears in the phase factor applied to the entire state vector, which is not an observable. Therefore, this Nθ cannot yet be directly observed. Thus, to effectively utilize such an amplification effect, we use quantum entanglement.

As shown in Figure 5, consider a quantum entangled state of N neutron spins.

This is the sum of the state where all N are upward and the state where all N are downward, and it is named the GHZ state. It is also called a cat state. When N neutrons in this state are passed through the box one by one, the final state obtained has a phase factor of exp(iNθ) applied to the upward part and exp(−iNθ) to the downward part. At this time, a relative phase of exp(i2Nθ) appears between the two states, which is an observable quantity. And, due to the 2N-fold amplification effect, it becomes possible to estimate θ with an error of approximately Δθ = π/(2N). In the conventional method, the estimation error could only be reduced to the reciprocal of the square root of N, but by using quantum entanglement, the error can be reduced at a rate of 1/N in this way.

Of course, this discussion assumes that we can successfully control cat states and suppress decoherence (the collapse of quantum states due to interaction with the environment). At present, such control is not technically easy. However, if technology in quantum computer development advances in the future and precise control of quantum entanglement becomes possible, such high-precision quantum parameter estimation will likely become a reality.

Furthermore, by using quantum entanglement, another type of quantum estimation becomes possible. As shown in Figure 6, we prepare an entangled state of spin A and spin B, and pass spin A through a box that performs an operation dependent on an unknown infinitesimal parameter g. As a result, information about g is written into spin A.

In this case, if we perform a quantum measurement across both A and B, the estimation error of g generally becomes smaller than in the case where quantum entanglement is not used. This is because information about g is written not only into spin A alone, but also into the quantum correlation existing between A and B. A general theory of such quantum estimation can be rigorously constructed using the framework of quantum Fisher information. Such quantum entanglement is also expected to be a very useful resource in future experiments and observations in various fields of basic science.