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We humans have mirror neurons in our brains. Makes this test a pretty trivial one. Just show to one person some negative and positive things done to other person, record activity in that part of the brain.

I'm pretty sure the response will be uneven when testing different pairs of humans as all sort of biases would manifest.

Unfortunately this sort of experiment has been done. I read about it as a teenager. It may have been carried out by Josef Mengele I choose not to remember.

In essence relatives were paired up and one was tortured. It was noted whether the relative would be willing to take the pain instead of letting their relative suffer. E.g. would a mother agree to take torture to save her child from torture.

I prefer not to remember any more about this or dig up its history. The records are out there if you are interested.

EDIT

You may also wish to investigate the Milgram Experiment which is less harrowing.

It involved subjects giving what they believed to be electric shocks to another subject (who was actually an actor). There were no electric shocks involved but the actor pretended to be in increasing pain.
https://www.simplypsychology.org/milgram.html

Fortunately, even with "primitive" (relatively speaking), technology empathy is fairly easy to measure even without understanding your subjects' languages.

To complete this experiment you'll first need to locate the main cognitive center of your species of choice. Look for a dense network of stuff that is electromagnetically active. It'll likely be found encased in some sort of dense protective shell/box. Use the clustering of sense organs in one place to inform your search, the main cognitive center will usually be found within the general vicinity of this clustering.

Once you're confident you've found what we're looking for you're going to need a large number of individuals from your chosen species-oh, it looks like you've already got that sorted. Excellent.

Next, designate pairs of individuals. One will be the "experiencer" and the other will be the (prospective) "empathizer".

Now find a way to measure the electromagnetic activity of your species' cognitive center. I realize that's vague but there's not much to say besides suggesting a bunch of trial and error. Depending on the anatomy of the species in question you may be able to simply place electromagnetic sensors on the surface of the individual encasing its cognitive center, or you may need to implant sensors directly onto the latter by means of surgery.

Anyways, once you've figured that out outfit each of the members of a pair of creatures with your electromagnetic measuring system.

Now here comes the fun, uh-the necessary part. Inflict damage of some kind (be creative) onto the designated "experiencer" while the "empathizer" is in proximity. Depending on what sort of senses your creatures have and where their sense organs are located you may need to orient your empathizer in a specific direction relative to the location of the experiencer for proper experimental efficacy.

Now compare the measured cognitive response of the experiencer and the potential empathizer. Are they the same? If, so, you've detected empathy!

Of course, this could simply be a fluke, any sound experiment needs to be repeated to weed out anomalous readings. That's where the previously mentioned large number of individuals comes in. Repeat the experiment many times to comfirm your results. Remember to vary what damage you inflict onto the experiencer in each instance. Oh what fun!

For an Extremely Low Tech Solution, Drop the Capitives on Island(s)

Solitary creatures will scatter. Some, like male bears, will completely isolate themselves. Some, like female bears and their cubs, will keep together with their young until the young are old enough, while others, like lions, will remain in pairs demonstrating a band of increasing - but very low - empathy.

Empathic creatures will form groups. Dogs will form packs where even the weak are cared for. Monkeys will develop specialized labor. These things are impossible - absent language - without empathy to cover the gap.

In your aliens + humans case, if you drop them on an island with limited resources, you might get one of the following behaviors. A group of sociopaths would likely scatter to "do it themselves". A group of highly empathetic will form a village of some sort. The level at which the weakest in the group, or rebels against authority are cared for gives you an idea of how empathetic the society is.

The aliens may even quantify empathy: 100% - $(number of isolated captives) over (total number captives)$

Crude/low tech version: Show them a video of someone getting kicked in the groin really hard, with audio of a sickening "crunch". Observe resulting wince/flinch/shudder/protective covering of area.

Edit in response to comments

Empathy is about understanding or sharing a feeling or reaction. Without necessarily understanding what a specific feeling is or reaction means, the aliens can invoke various actions and see that there is a reaction, and an approximate magnitude thereof.

Once they find a suitable action/reaction they can then display it to the humans - they have no idea whether they are causing pain, laughter, sadness, or erotic stimulation, and quite frankly, they don't care. The important thing is observing whether the display generates a sympathetic reaction in the second subject.

For best results, this will be repeated with a variety of action/reaction pairs

Tl/Dr: It's always difficult to write short answers about tricky words like "empathy." However, if we use some philosophy to pin it down to a more abstract concept based on relating to others, we can then expand the concept to how empathy affects societies and groups, and then look for those group patterns. We should see patterns in humans along well recognized scales of 5, 15, 40, 150, and 1500 individuals, and that would be a strong indicator of something we would have to call empathy.

It's actually surprisingly difficult to identify empathy unless you already have an intuitive sense of what it is and understand the being. Its too easy for individuals to fake empathy if you don't have a good enough connection. For example, it can be really difficult for us to determine if the empathy of politicians is genuine or an illusion. The cynics would argue it's always an illusion, but it's hard to tell.

You and I "feel" empathy because we're already bound together in how we view the world. To scientifically measure empathy from an alien perspective, we're going to have to be a bit more specific about it.

Arne Naess is a 20th century philosopher who came up with a concept called the "Ecological Self." He was looking at different definitions of self ("My body is my self" "my mind is my self" "my body and mind is my self", etc), and found great problems with all of them. The one he settled on "The Ecological Self is that with which the self relates to."

In his essays, he gave a story of a scientists who was looking through a microscope into a petri dish. A fly buzzing around the room landed in the dish. The dish itself contained a rather strong acid, rapidly deteriorating the fly's wings so that even if it escaped, it would not survive. But it takes time to be dissolved by acid. The scientist could do nothing but watch as the fly painfully disassociated into tiny molecules and integrated into the liquid of the dish. In those moments, Naess argued the scientist's Ecological Self extended to the fly. He had some sense of relating to what the fly was going through. Of course the scientist had never been dissolved in acid, but he had been splashed with it and burned by it, so there was something with which he related to the fly.

Naess then went on to argue that much of altruism could be explained by selfishness from such a wide concept of self. He argued that Mother Theresa was the most selfish person ever. However, her Self was so wide and all encompassing that acting in her self-interest meant supporting the countless people whom she helped during her life. Her Ecological Self encompassed more people than most of us can even imagine.

I use this philosophical example because it provides a larger more abstract structure the aliens can look for. They can look for evidence of this Ecological Self. If they find that the Ecological Self of any individual extends beyond their trivial body, then they have found empathy.

Thus, the best way to scientifically search for empathy is to put people together into groups and observe whether they exhibit these sorts of behaviors. This, of course, will require trying to categorize human behaviors into simple egocentric behaviors and wider behaviors founded in the Ecological Self.

Measuring this is excruciatingly difficult, which is why we don't have any scientific tests for politicians. However, one fascinating possibility stems from the Integrated Information Theory of Consciousness (IIT). IIT quantifies the gestalt effects which come from bringing processing units together. It studies how much information is contained in the individual units (neurons in the usual IIT case, but test-subjects in this alien case), versus how much information is contained in groups of them. If there's no integration, no larger selves to be had, then a collection contains no more information than the individual units. Put a bunch of CDs together, and they contain little more information than the stamped contents of the individual CDs themselves. Really the only information stored beyond that is a few bits of information stored in the order you stack the CDs. However, put humans together, and they tend to form groups and societies which contain far more information in their structure than in the individual bodies themselves!

A key to using this theory here is that information has a tendency to decay over time unless something is preserving it. If there's a lot of information stored in the collection, it will tend to decay unless it's providing some value to the individual units.

So with this, we can divide up our sample of half a million people into small lots and see if they form structures with measurable information stored in their interactions. You can start with large or small scale samples, but I'd probably start with small because I have a limited population of test subjects to work with.

As it turns out, our aliens will notice really interesting results occurring at some regular intervals. They'll find similarities between groups of 3 4 and 5. Likewise, they'll find patterns in the 5-15 region which are different from the 3-5 region. Another region is 15-40. Then 40-150. 150-500 is a bit fuzzy, then there's 500-1500. 1500 on takes on a very different nature. These numbers have been found by anthropologists in virtually all cultures. If I may gloss them:

• 3-5 people is your close knit group. These are the people you will let you see at your worst, or at your most vulnerable. These are the people you rely upon when everything else has gone wrong. In most militaries, 3-5 people form a fireteam, lead by a corporal.


• 5-15 is your wider friend circle. These are the people you rely on directly on a day to day basis. In hunter-gather societies, these are the sizes of hunting parties in many parts of the world. In most militaries, 12 people form a squad, lead by a corporal or a sergeant.


• 15-40 people is a tricky group to give a single word for. This is smaller than a tribe, but is a close knit group with a lot of tribal elements to it. The hunting party of 15 people will come back to the great group of 40. Here you will have true "leaders" in the sens of them making decisions which everyone else has to follow. In most militaries, 30-50 people will form a platoon, lead by a lieutenant, which is the first officer we've seen in this hierarchy of military organizations.


• 4-150 is a tribe. This is the size where we see real tribal allegiance. Many companies target 150 as the size of a department or center. The upper end of this is a number called "Dunbar's number," as the size of a tribe, though he actually gave it as a range rather than a single number. IDEO, made famous by documentaries in the '90s, would build a new building when they hit 150 people rather than go larger within a building. 80-250 people is a company in the military, lead by a captain or a major.


• 150-500 is a tough region that's been nicknamed the "megatribe." In anthropology, we find this when tribes form alliances. These are the extended tribe. In the military, these are battalions, lead by a Lt. colonel.


• 500-1500 is a fun region to explore. It is currently believed that 1500 is the maximum number of faces we can pair to identities. If you have a society larger than that, you become forced to have people who you do not identify by their identity. You have to identify them by classes or roles, such as "serf" or "plumber" or "teacher." Larger than that, you see written down hierarchies dominate, as our brains can't handle the size of the structures without writing.

I point all of these out because they seem to be very universal, so the aliens would almost certainly notice those structures. They just seem to form when humans are involved, so any reasonable scientific experimentation will eventually find them. From an IIT perspective, this would be sufficient to argue that there's something somewhat-empathic going on. If you stressed one human in a way specific to its individual self, you could watch how the group responds to reach out and help that individual.

It would be very hard to explain the patterns we see by simple rules, as opposed to ants which have very clear hard-coded social structures built into their DNA to handle the 4 thousand to 4 million ant populations. Ours would be more fluid and adapting. The groups always seem to form, though the actual social structures they create vary depending on the circumstances. This could be tested easily by dividing out groups and seeing how they interact before bringing them back into the fold (three cheers for nondestructive testing!) To turn this one around on you, you might ask yourself whether you would consider ants to be empathic in any way at all. There's something to their structure that suggests there's more than meets the eye!

Now that would be the scientific way. The other more intuitive way would be to introduce yourself into the system. You, as an alien, try to extend your Ecological Self to include the humans. Then you see whether this self is returned or not. This, of course, may introduce empathy into the humans where it did not exist before. Depending on your goals, this may be a good thing or a bad thing. Clarke's Rendezvous with Rama series explored this.

Of course, if your alien species has no empathy, this will not work. Then the question will be "what do they think they're looking for?" Is this a cold calculating study before planning an invasion? If so, then the pure scientific approach is best. On the other hand, if they're looking for something they lost, they might try integrating themselves into this human society. Perhaps they can learn empathy from us!

Scientists have performed a similar test on rats to test for altruism, which is closely related to empathy, and the same under some definitions.

The experiment involved two rats, one of which was in a small cage in the middle of the enclosure while the other was free to roam. Approximately 70% of rats chose to free the trapped one, even when it meant sharing the supply of treats in the room.

The aliens could easily set up the same or similar experiments with pairs or small groups of the human captives to observe the results.

Another method could simply be observation - place the humans on a planet somewhere with a little food and supplies and see how they interact and perform when setting up their own small civilisation.

How are we so sure the aliens even comprehend empathy?
Would the aliens even have a sense of "self"?

They could be insectoid .. spider, ant or bee-like megamonsters who only have some kind of hive-mind but each creature cannot feel anything in particular at all but just acts on insectoid insticts. Actually this is a quite popular type of alien in sci fi.

I think it is a better question how we humans can test the aliens for these qualities. Because at least humans know what empathy is.

Aliens want to test humanity for empathy.

They stage a crash landing of one of their spacecraft near a small town in New Mexico, to see if the local humans will offer assistance to the crew.

Humanity leans about the names "Roswell" and "Area 51".

Aliens learn that humanity doesn't really have much in the way of empathy and have left us well alone ever since.

The aliens could set up a scenario similar to this:

Create a huge pool of water.
Place one group of humans on one side of the water.
Place another group of humans on the other side.
Now the aliens would arrange the living conditions on both sides of the water in a way that one side is very hostile and dangerous, a place you wouldnt want your children to grow up in, and the other one to be a flourishing society.
Finally the aliens would put some boats on the "bad" side.

As a result of this setup humans from the group in "bad" living conditions would try to cross the body of water on their boats. However the aliens would have prepared the boats in a way that they would not be fit to make the trip and sink en-route.

Now the aliens could observe the reactions of the other group. Different strategies might surface:
Some might do it like the rats mentioned in another example and try to save the drowning ones.
Others might choose to ignore it.
Some might build up barbed walls on their shores to keep even the good swimmers from entering their land.
And few might even search for ways to make the "bad" land more liveable.

It seems like a pretty reliable empathy test. Luckily noone would do such a thing.

perhaps in the same way we tested for empathy in elephants and dolphins? expose some members of a group of humans to certain stimuli such as pain and view how they react. if the other humans appear to be attempting to comfort the other members of their group it would be a good indicator of empathetic tendencies existing in humans

1. Isolate them, but need all their needs


2. After a period of time give them a companion (either of the same species, or something that's non-threatening to them that they can socialize with)


3. Threaten or treat their companion badly and observe their reaction

If empathy is: the ability to understand the feelings of others,

then we need a test that can't be passed by someone who doesn't understand the feelings of others.

Here is a short list of things that can be learned through training, repetition, conditioning, etc. that are obvious contenders for empathy but actually don't fit the bill.

1. Saving people that are in danger.




2. Comforting someone who is visibly crying / screaming /
   hyperventilating.




3. Performing really any action in response to an obvious stimuli.

These do not necessarily indicate any understanding of another's emotions. Such responses can be learned or programmed. For example a robot can be programmed to respond to a sound frequency that approximates crying, with ice cream and words of affirmation. Does the robot understand what sadness is and how the other person feels, or what its actions are doing? If this is the only code in the robot, then no it wouldn't understand emotion. Action and response.

In the same way a dog can be trained that when a person wearing a mask (so emotions cant be seen) stamps their foot, then the dog should sit on their lap and lick them.
What is a way to make this happen with out any emotional understanding? Just consistently rewarded the dog with a treat afterwards.

I hope I am beginning to show that actions that might seem like they indicate empathy, don't always have to.

So since we are an advanced alien race, I presume that we want to know if humans are capable of empathy with a high scientific degree of certainty. I propose that we test human's ability to conceive the emotions of others.

Empathy: the ability to understand and share the feelings of another.

This is the emphasis of my answer (and in my opinion the core component of empathy).

A test to prove this can take any form, but there are some key points we should hit.

1. We should establish a baseline. How does the subject being tested react to certain situations, and how do they feel about things. We want to make sure the subject is not just saying how they would react in every situation.



2. 
   

   We give the subject a hypothetical test using the same situations as step 1, but with regard to different people. We want to make sure to use examples of people that are known to react differently to the subject. Provide appropriate background information for each person the subject is empathizing with, provide a situation and ask the subject how the person would react. How do they feel. What are some of their likely future actions. For example:

   
   
   

   "Person A goes to law school and dedicates her life to it for 4 years excelling at it the whole time. She is then expelled for failing one class. She has never failed a class before. How does she feel? What are some likely future actions?".

   
   



3. We see if the subject can identify the emotions of others with some degree of accuracy (maybe 50% of answers are right, or something like that).

Now obviously there are some caveats. For example not all humans experience empathy. We have plenty of psychopaths and they are markedly known for an absence of empathy. We may also want to weigh different parts of the test differently. For example when predicting the actions of others in response to an emotion is difficult, so we may be satisfied with any plausible action.

However the key point of this test is that the subject should have to mentally walk through what others would do, to clearly rule out other explanations of seemingly-empathetic behaviors.