# Can joint measurement be achieved in two labs far apart?

Consider the following scenario: Alice and Bob are in two labs far apart, and they each have one qubit. Can joint measurement (for bipartite projective measurement, they are measurements that cannot be written as the direct product of two measurements) be done between Alice and Bob? Is this kind of measurement reasonable even from a theoretical perspective (the axioms of measurement in quantum mechanics do not consider the factor of distance)?

There are some facts that might be helpful: I think joint measurement is actually be done by doing some joint operations and then followed by computational basis measurement, which is direct product form.

TL;DR No. Non-local gates and measurements between causally disconnected observers violate the no-signalling theorem and are therefore impossible.

## No-signalling theorem

The following protocol allows Bob to send one classical bit $$b$$ to Alice faster than light. Alice prepares two sets of qubits $$x_i$$ and $$y_i$$ in the $$|0\rangle$$ state. Bob prepares one set of qubits $$z_i$$ in the $$|0\rangle$$ state. If $$b=0$$, then Bob measures each pair $$x_iz_i$$ in the Bell basis. If $$b=1$$, then Bob measures each pair $$y_iz_i$$ in the Bell basis. Alice may now learn $$b$$ by performing quantum state tomography on qubits $$x_i$$ and $$y_i$$. If she finds $$x_i$$ to be in the maximally mixed state then $$b=0$$. If she finds $$y_i$$ to be in the maximally mixed state then $$b=1$$. This contradicts the no-signalling theorem.

## Channels

In general, Alice and Bob can perform non-local quantum gates and measurements if and only if they can exchange quantum information, for example via a quantum channel or quantum teleportation. Distance is not directly relevant. Instead, what matters is that the two systems are allowed to interact with each other or with other systems that mediate the interaction indirectly. Thus, distance is relevant insofar as many interactions tend to become weak at large distances.

In order to perform a non-local quantum gate or measurement on two qubits located in two distant laboratories we may send the qubit from the first laboratory to the second one via a quantum channel and then send the results back to the first laboratory. Alternatively, we could use intermediaries such as photon-encoded qubits. In any case, non-local operations require a causal connection between the laboratories which is realized by the use of a channel.

## Causal structure

In more abstract terms, quantum mechanics admits a causal structure$$^1$$, i.e. a partial order among events that specify whether one event may depend on another. However, the postulates of quantum mechanics do not prescribe any specific causal structure themselves. Instead, any analysis that depends on a causal structure needs to obtain it via other means rather than deriving it from quantum mechanics.

For example, in the quantum teleportation protocol we declare that the two qubits in Alice's possession are allowed to be jointly measured while neither of them is allowed to interact with Bob's qubit. These declarations imply a causal structure in which the protocol is analyzed.

## Distance

The role of distance - or more precisely its special relativistic generalization called spacetime interval - becomes more explicit in quantum field theory which incorporates special relativity and requires that operators associated with spacelike separated regions commute. This is analogous to how in ordinary quantum mechanics local unitaries and measurements take the form $$A\otimes I$$ which acts as identity on any non-local system.

$$^1$$ An elegant proof of this fact may be found in section $$6.3$$ on page $$309$$ of Picturing Quantum Processes by Bob Coecke and Aleks Kissinger.