Time and Quantum from Correlations Vlatko Vedral vlatko.vedral@gmail.com

Marginals Given some random variables and some marginal distributions, can we construct a global probability that reproduces the marginals? Sometimes the marginals are simply incompatible; e.g p(A,B) = ½,0,0, ½ while p(B,C)=1/3,0,0,2/3. (because p(B) is different in two cases)

Quantum This is known as quantum “contextualty”. Sometimes there is no overall probability distribution (even though there are no inconsistencies at the lower level), but there is an global quantum state that can account for the marginals. This is known as quantum “contextualty”.

Example - KCBS We have a pentagram. Each vertex can be coloured either red or blue. An edge is said to match if both of its vertices have the same colour. Otherwise, it's a mismatch. In a classical probabilistic model, the total number of mismatches has to be an even number, i.e. 0, 2 or 4. So, with a probability mixture over hidden variable assignments, the expectation value of the sum of mismatches over all of the 5 edges has to lie between 0 and 4. Can have a larger average with quantum states and quantum measurements.

Examples – Bell’s inequalities Given P(A1,B1), P(A1,B2), P(A2,B1), P(A2,B2), find P(A1,A2,B1,B2) such that the average of Bell operator is greater than 2. Cannot be done, but there is a quantum state and measurements that achieve 2root2. N.b. of course here we have an extra constraint of “locality”.

Quantum Principle The fact that there are marginals that are not incompatible, yet do not arise from an overall probability, forces us to use the quantum description. Could it be that there are marginals that cannot be explained quantumly, yet still exist in nature?

Pseudo-density matrix Pretend that different instances of time are like different Hilbert spaces and write a physical state that extends over multiple times. For instance, think of a qubit measured at two different times (with no evolution in-between).

Negative eigenvalue There is no physical state for two qubits of this type. Hence, this object is a pseudo-density (i.e. it has a negative eigen value.

Note… i.e. PDM is a partial transpose T of |00> + |11> maximally entangled Bell state.

Time from correlations We observe marginals that cannot be described by an overall density matrix. Forces us to acknowledge the existence of a time dimension.

Aside: an analogy with special relativity Start with time as fundamental and events as taking place in time. Then postulate: there are pairs of events for which neither one is before the other, nor are they simultaneous. This forces us to acknowledge that space has to have more than one dimension.

Space and time from PDMs? Given R, can we “distill” (reconstruct) space and time from it? Negativity of eigenvalues signals the presence of temporal correlations. However, temporal correlations can look like spatially separable as well as spatially entangled states.

3 time measurements Interesting possible states from 3 temporal pseudo densities: For All reduced states are physical, but the overall is not.

Dynamics out of R? Dynamics can be seen as a teleportation through time via the “entangled” pseudo-density matrix. Measure 12, and subsequently apply To get

Violation of monogamy In a pseudo density matrix we can have maximal entanglement between 1 and 2 and 1 and 3 at the same time. Monogamy: Pseudo-density:

Genovese experiments (INRiM)

Open Timelike curves problem       A’ A B

But… Could the state AA’B actually be a pseudo-state; This could save linearity at the price of introducing time-like correlations

Black-hole information loss? If black hole evaporates by emitting entangled pairs of particles, then after half of it has evaporated, it seems that the remaining black hole is now fully entangled to outside. Yet, if it continues to evaporate via the same (Hawking) mechanism, this seems to lead to a violation of the monogamy of entailment.

Final speculation Just after crossing the event horizon, time and space change their signature in the Schwarzschild metric. This looks like a transposition