Quantum mechanics says objective reality doesn't exist, that instead all we see are probabilities collapsing into one particular configuration... and all other possible realities might just exist together in a quantum multiverse. Here's the deadly experiment that could help you to test that very idea.
First, let's take a look at two major interpretations for the nature of quantum reality. The older and somewhat more preferred option is the Copenhagen interpretation, which was devised by legendary scientists like Niels Bohr and Werner Heisenberg in the 1920s. At its most basic, this interpretation says that all the subatomic particles that make up the universe can and should be thought of as wavefunctions, which are probabilistic representations of a particle's location and velocity at any given time. Measuring or observing these particles is what causes them to collapse into only one of all possible values, and that's how we get the universe that surrounds us.
The other idea was first put forward by Hugh Everett in 1957. He kept most of the Copenhagen interpretation but removed one crucial part: the wavefunction collapse. Without it, all probabilistic values for every subatomic particle would exist in superposition, all at once. In theory, this meant that was a very large and quite probably infinite number of universes in parallel existence.
The obvious question then is why we only seem to observe one universe, and why it looks for all the world as though wavefunction collapse is happening all the time. As Everett and those who followed him explained, the answer is another phenomenon called quantum decoherence. Basically, for all possible states of a particle to remain in superposition — to be coherent, in other words — their system needs to be isolated.
Being hit by even a single photon is enough to break that coherence, and what we see as wavefunction collapse is actually just one of the many realities that describe the possible states of the particle. Add it all together, and you get the quite possibly infinite universes of the many worlds interpretation.
There are certain theoretical advantages to this theory. The Copenhagen interpretation relies on the presence of an observer - not necessarily a sentient observer, just something capable of initiating wavefunction collapse — and a lot of the apparent paradoxes of quantum mechanics are eliminated if an observer is no longer necessary. For a start, it neatly solves the legendary problem of Erwin Schrödinger's cat, in which a cat is placed in quantum superposition inside a box so that is both dead and alive. The Many Worlds Interpretation has no issue with the cat being simultaneously dead and alive — it just spins the two results into different universes.
Despite these potential benefits, the Many Worlds Interpretation has always faced two seemingly insurmountable challenges. For one thing, there's no way to test it experimentally, which makes it unfalsifiable and arguably more a question for philosophy than for science. And, for a second thing, it's completely, utterly bonkers. It roars against every last shred of intuition we have about the world around us, violently disagreeing with everything we think must be true about the world. That doesn't mean it's wrong, of course, but that fact still does it no favors in the courts of popular and scientific opinion.
Actually, there is one way to prove the existence of a quantum multiverse, but if anything it only makes the bonkers problem even worse. For now — and probably for the indefinite future — it's just a thought experiment, but it's not totally out of the question that this test could one day be attempted. If successful, it would prove that the multiverse exists — but only to one person.
The twin notions of quantum suicide and immortality were first proposed by Hans Moravec in 1987 and independently a year later by Bruno Marchal, but the most work on the idea has been done by MIT's Max Tegmark. The most common version of this experiment goes like this — place the experimenter in a chamber with a life-terminating device, such as a high-powered rifle pointed at her head. Every ten seconds, the spin value of photons will be measured. Depending on the result — and there's a 50/50 chance of either measurement — the device will either fire and kill the experimenter or make an "all clear" noise that tells the experimenter she is safe.
What we've done here is tie the survival of the experimenter to a quantum state, meaning she now exists in a superposition of being both alive and dead. There's a 50% chance she survived the initial round, and she has the same chance for every subsequent repetition of the experiment. No matter how many times she repeats the experiment, half the time, she survives.
Of course, her overall survival chances are way less than 50%. The version of her that died in the initial experiment doesn't have a 50% chance of coming back to life in the next experiment. But each living version of the experimenter retains that chance at survival, even if the overall chance of survival keeps falling to 25%, then 12.5%, then 6.25%, and so on. Let's say that in one universe, an experimenter eventually emerges having survived 50 such tests in a row — something she has less than a one in quadrillion chance of surviving, which is way more than is needed to meet the 5-sigma level of certainty needed for an official discovery.
The experimenter can then distinguish between the Copenhagen and Many World interpretations — while there's less than a one in quadrillion chance of her being there in the former interpretation, there's a 100% chance in the latter, because some version of her must be around to observe this particular superposition, and all the other versions of her are dead. Thus, the Many Worlds Interpretation has been proven and this particular experimenter has tasted quantum immortality.
The only small hitch, of course, is that the interpretation has only been proven to the experimenter. No other observers will be choosing between 1 in a quadrillion and 100% chances — to them, the chances of the experimenter's survival will be equally unlikely no matter which interpretation one chooses. To be sure, the ridiculously low probability — 1 in a quadrillion is essentially impossible — might convince the experimenter's peers to accept the Many Worlds Interpretation as correct, but that still leaves countless more universes where the experimenter died. At best, Many Worlds is only going to be proven in a tiny sliver of all possible universes, because everywhere else the results just aren't unlikely enough.
For his part, Max Tegmark once said — probably with his tongue planted firmly in his cheek — that he might someday try the experiment, but only once he's old and crazy, and thus his death in most of the universe wouldn't be quite so hard for others bear. He also points out on his website that the experiment could potentially be expanded so that both you and a friend are either killed or spared in each round of the experiment, which at least would give you someone else to share your knowledge of the multiverse with.
I suppose one of sufficiently grandiose persuasion might imagine designing an experiment where everyone on Earth is placed in one colossal chamber, and so everyone is a participant in this exploration of quantum immortality. Of course, you'd better be damn sure the Many Worlds Interpretation is correct before risking the entire population of Earth on proving it once and for all, and besides — much as I'm a firm believer in the search for scientific truth, I'm not totally sure it's worth killing off the populations of countless Earths just so that one surviving world can have its answer. Although, ask me that same question in another universe, and I might just feel differently...
The Universes of Max Tegmark
The Interpretation of Quantum Mechanics: Many Worlds or Many Words? by Max Tegmark
Does the 'many-worlds' interpretation of quantum mechanics imply immortality? by James Higgo
Is there an evil, goateed version of you somewhere in the multiverse? by Dr. Dave Goldberg
The Many Minds Approach