The authors claim that the thought experiment shows that the following 3 apparently innocent and obvious assumptions about quantum measurement theory are mutually inconsistent.
- Quantum theory is universal, which means that agent - I translate it to conscious observer- can analyze second system, even a complex one including other agents, using quantum mechanics.
- The predictions made by different agents using quantum theory are not contradictory. This looks trivial but perhaps the point is in the meaning of "prediction".
- The outcome of quantum measurement is unique. This looks totally trivial but is not so in Many Worlds interpretation.
It is difficult to believe that authors could have made a computational error since the system is basically very simple and one essentially compares the outcomes of subsequent measurements for a pair of qubits with quantization axes rotated by 45 degrees with respect to those in the first measurements. I would seek the error is at the level of interpretation rather than computation. Authors assume that conscious entities are describable as extremely simple quantum systems (qubits) but simultaneously believe that they are classical entities with memories surviving in the further quantum measurements posed on them.
Scott Aaronson has a lot of fun with the assumption that conscious entities like humans are modelled as qubits.
The thought experiment
Alice and Bob measure their laboratories containing their friends FA and FB: the possible outcomes of measurements are specified. Reader can of course argue that measuring laboratories is not possible. Certainly it is not with recent technology but quantum theory does not deny this possibility. There are 4 measurements.
- FA measures a qubit - this is popularized as coin toss - and and codes the result to two spin states communicated to FB as spin states. These are non-orthogonal - this is essential. One can assume that tail correspons to spin UP in z-direction that is state |UP> and and head corresponds to - say - spin UP but in direction making angle of 45 degrees with z-direction. The spin up state in this direction is superposition proportional to |UP> - | DOWN > .
- FB measures the spin in z-direction for the state communicated to him by FA. The outcome is |UP> for tail but either |UP> or |DOWN> for head.
If FB observes |DOWN> he can conclude that FA got head. This is the crucial bit of information and assumed to be stored in memory of agent FB(whatever memory means!). Even more, FB is assumed to keep the memory in the sequel under the measurements applied to laboratory by Bob. It is also assumed that all observers have memory surviving futher measurements. This is an implicit assumption and is about consciousness rather than quantum mechanics.
Agents are assumed and to know their QM and be able to apply to it to deduce information about the measurement outcomes of others.
- Alice in turn measures the state of her lab containing FA, and coin. Now the state basis for coin (essentially qubit) is spanned by |OKA > = |tail P>- | head > and |FAILA > = |tail P>+ | head > .
- Bob does the same for his lab containing FB and spin. These states basis are rotated by 45 degrees with respect to those used by FA and FB. The state basis is spanned by |OKB > = |UP>- | DOWN > and |FAILB > = |UP> +| DOWN > .
- The 4 possible final states are of form |OKA > ⊗ |OKB > , |OKA > ⊗ |FAILB > , |FAILA > ⊗ |OKB > , and |FAILA > ⊗ |FAILB >
Authors claim that this is a paradox. If FA and FB, where just qubits, the authors would not speak of paradox. This kind of measurements have been done for ordinary spins and the predictions of QM have been verified.
There is no paradox if one just regards the systems as spins having no memory or if the memories are possible, they are affected in further measurements. Therefore the paradox must relate to the assumption that the outcomes of ealier measurements by agents FA and FB are stored in memory and that these memories are preserved under measurements by Alice and Bob. Since the agents in question have mind consisting of single qubit this assumption leads to a contradiction. There is no conflict between the 3 listed basic assumptions about QM. The paradox results from wrong assumptions about consciousness.
Suppose qubit minds are possible
What if one just for fun assumes that single bit minds are possible? The essential point is that coin⊗FA, spin⊗FB and FA⊗Alice and FB⊗ Bob represent different conscious entities than FA, FB, FA, Alice and Bob before the state function reduction taking place in measurement in question. WhenX and Y are entangled, it is X⊗Y, which is conscious whereas X and Y are unconscious! This means loss of the memory. The moment of state function reduction producing unentangled product state is moment of consciousness for both X and Y (even for spin!). Hence the information about earlier measurement outcome is destroyed.
For genuine conscious entities the situation is probably different. They can store information about previous measurements so that it is preserved in further quantum measurements involving enanglement and no paradoxes appear. For instance, in the many-sheeted space-time of TGD involving fractal hierarchy of p-adic length scales and scales coming as scales proportional to effective Planck constant the memory storage is possible and biology provides the actual realization. There is also hierarchy of ...selves-subselves-sub-sub-selves.... whre sub-selves of self correspond to its mental images and selves at lower and also higher levels of the hierarchy can store information preserved in the state function reductions.
The view provide by zero energy ontology (ZEO)
In TGD framework ZEO provides some general insights about the notion of memory.
- Zero energy states provide a generalization of the quantum states as pair of positive and negative energy states with vanishing total quantum numbers assignable to opposite boundaries of CD. Zero energy state can be regarded as superposition of deterministic classical time evolutions connecting initial and final states at boundaries of CD. The motivation for ZEO is that it resolves the basic paradox of quantum measurement theory since state function reduction replacing the superposition of deterministic time evolutions with a new superposition does not break the determinism of any time evolution.
- During the lifecycle of self identified as a sequence of what I call small state function reductions (analogs of weak measurements, see Wikipedia), the members of state pairs at the passive boundary of causal diamond (CD) - remain unaffected. One can talk about generalized Zeno effect. The state at passive boundary represents conserved memories. Note that one has hierarchy of CDs inside CDs so that the situation is rather complex.
- The states at the at the opposite -active - boundary of CD change. In each unitary evolution the active boundary of CD is de-localized in the moduli space of CDs and the small reduction involves localization in the moduli space of CDs, in particular time localization. The size of CD measured the temporal distance between its dips increases in statistical sense at least and this corresponds to clock time correlating strongly with subjective time defined by the sequence of reductions.
- In big state function reduction the role of passive and active states change and CD begins to increase in opposite direction: conscious entity dies and reincarnates as time reversed one. These reductions correspond to the state function reductions occurring in say particle physics experiments and involve drastic change of the quantum state. The memories represented by the state at passive boundary are destroyed and the outcome of the big state function reduction at active boundary represents the new memories.
- For minds with size of qubit, the memories would be indeed destroyed and new ones formed. For bigger minds it is quite possible that some sub-...-sub-selves in the hierarchy can preserve the memory and that it can be recalled in the subsequent re-incarnations in the original time direction. Sleep state could correspond at our level of consciousness to temporary death and re-incarnation at opposite time direction. We indeed remember something about yesterday, even previous year! Our mental images also die and re-incarnate and the interpretation would be as medemphychosis at the level of mental images.
For a summary of earlier postings see Latest progress in TGD.