**12.3 Decoherence-free subspace**

There is an interesting situation where it is possible to provided passive protection against errors — as opposed to the active error protection of the quantum error correction codes discussed earlier. The model assumes that all qubits in the register are affected by the same error at the same time. This is very different from an independent error model. Does such collective dephasing occur? It does in situations where the physical dimensions of the register are smaller than the shortest wavelength of the field.

Consider the following encoding:

The states |0〉* _{L}* and |1〉

*form an orthonormal basis. Note also that |0〉*

_{L}^{*}

*L*= |1〉

*.*

_{L}Decoherence can be modeled by an operation called *collective dephasing*. The operation transforms |1〉 into *e** ^{iθ}*|1〉 for both physical qubits at the same time and leaves |0〉 unchanged for both of them. This results in

If all operations are carried out on qubits encoded in the same way, collective dephasing only produces a global phase change, and hence does not affect measurements. Recall that in quantum mechanics only phase differences between qubits matter.

In February 2001, Wineland’s group^{18} reported an experiment in which they encoded a qubit into a decoherence free subspace of a pair of trapped ^{9}Be+ ions. They used encoding exactly like the one shown above. Then they measured the storage time under ambient conditions and under interaction with an engineered noisy environment and observed that the encoding increased the storage time by up to an order of magnitude.

### 13 Time-multiplexed interpretation of measurement

The macroscopic world is classical.

The microscopic world is quantum.

Every physicist agrees that despite its amazing weirdness quantum mechanics works brilliantly. The weirdness is shrugged off except by those who are incurably curious. If you want to calculate what experiments will reveal about subatomic particles, atoms, molecules and photons, then quantum mechanics will provide you the answer. It uses mathematical formulas and axioms that have withstood innumerable tests conducted with extraordinary rigor but were essentially pulled out of a hat by its creators in the early 20th century. It does not answer questions with certainties but with probabilities. Thus, the probability of measuring its observable properties is found by applying a mathematical operator to the wave function. The wave function contains all the information that we can know about the quantum system it describes. The rule for calculating probabilities was a bold intuitive guess by Max Born as was the wave equation by Schrödinger. Neither is supported by rigorous derivation. Quantum mechanics is not just a complex framework, but also an *adhoc* patchwork that lacks obvious physical interpretation or justification.^{19} At its core, quantum mechanics is probabilistic.

^{18}Kielpinski, *et al*(2001).

^{19}Ball (2017).