When humans go to Mars, will they be able to transmit HD/4k video back to earth?

The transmission will never be live. The transmission delay at any speed and quality from Mars to Earth is between 3 and 21 minutes depending on the distance between the planets as they move around their orbits.

A recording can be transmitted at lower bit rates to save bandwidth and ensure quality. There will be transmissions, as there are from the Mars rovers. Whether they will use HD/4k is debatable. Maybe that will be used occasionally for more detail

It would be very difficult, and wasteful. HD/4k video needs 20 Mbps. The current data rates back to Earth are about 32,000 bits per second. So it would take about 10 minutes 25 seconds to transmit just 1 second of HD video.

Now you might imagine improving the technology by a factor of 625 times. That would just barely be enough. But there are so many other really useful things to do with that extra bandwidth, and HD video is mostly just entertainment. Maybe a few minutes a day might be allowed, but I don't see more than that for a long time.

Depends entirely on the amount of electrical power they are willing to use. The higher the frame rate and the higher the resolution of each frame, the greater the bandwidth needed - Shannon's Law. It's perfectly possible at the higher end of the ultra high frequency UHF band (300 - 3,000 MHz) or the super high frequency band (3,000 - 30,000 MHz). At such frequencies high gain antennas do not have to be gigantic, so there is that going for it.

Good high gain antennas on Earth will be needed as well, you could get away with three so that at least one is pointed at Mars all the time, but six would be better. Once Mars is near or behind the Sun, there is no direct communication.

The greater the bandwidth, the more power needed at the transmitter.

DISTANCE make it difficult.

4K video requires a LOT of information, and so it requires a high speed data link. THAT requires a Very High Frequency carrier, and THAT is hard to transmit over such long distrances

With current technology, live video would not be possible at that resolution. The bandwidth needed would take an ungodly amount of power to broadcast. That's why they don't use very high definition pictures to broadcast live from Mars.

A recorded video, yes, it could be done. It would be a very large file, and transmitted at low bandwidth, so it could take a very long time to send it, but it could be done.

they can send it off-line, at a lower data rate, any time they like.

sending it live will require significant transmit power, but is not impossible. please estimate the transmit power required to do so. all the information is publicly available.

It's a good question... and, it depends on a couple of factors - mostly, compression of the video for transmission. The New Horizons probe just recently finished sending all it's Pluto-encounter data back to Earth; and, that's because of the limited bit rate it could afford - so, it stored it's data onboard, then transmitted for more than a year after it's encounter with Pluto all the data it collected in a few short days.

Mars will be similar, in that we won't really have a direct need for 'live' video; instead, we'll be seeing video clips of the astronauts minutes to hours later, since there is no immediate communications possible. With current technology, my guess would be they'd video some action (posting the flags of the explorers on the Martian surface, say), and while transmitting in low-speed, low-res video, they'd later transmit the high-def video when there was less traffic (say, when the astronauts are sleeping, for example...)

Unless we really up the bandwidth, or unless compression of video data quadruples (or more), my guess is we won't have immediate, hi-res video capability...

It's just a matter of signal-to-noise ratio and bandwidth. There are no fundamental limitations. A high-quality, high-bandwidth data stream can be achieved using enough transmitter power, large enough antennas at either end, and possibly a high-frequency transmission carrier, perhaps even at optical wavelengths.