Is that possible?

Submarines do indeed use sonar as aircraft use radar, to see where they are going, or see around themselves.

Undersea cables don't count for submarines, or communication under the sea, because they transmit data from continent to continent; AFAIK they are not designed with little substations that transmit & receive from undersea vessels.

But if the issue is submarine vessels communicating with the rest of the world, which is how I interpret the OP, then that typically means electromagnetic radiation. And there are severe limitations on that, because sea water is conductive, which means it attenuates the strength of an electromagnetic field traveling through it. Specifically, the absorption is inversely proportional to the square root of frequency, through a phenomenon known as skin depth.

That means that the lower the frequency, the farther it goes before the signal strength is too low to be usable. Here is a quantitative example of how this works. It's a long story, but at the end you will understand the limitations, so I hope it's worth it.

But first, no you aren't going to get useful communications 4 km under the sea. That's not even a normal submarine, that's some kind of bathyscape, and you should be able to use fiber optics strung in parallel with your bathyscape to mother-ship tether.

At 4 km under the sea, the attenuation of even the very low frequency signals discussed below would be attenuated to levels well below the ability of any radio to receive. I worked out the attenuation at 4 km under the sea, at 10 kHz, and it's over 10,000 dB! There isn't enough power in the world to overcome that much attenuation.

Here's the real world.

In the bad old days of the Cold War, submarines and their sea-launched ballistic missiles provided one-third of the Triad of Mutually Assured Destruction (MAD) between the USA and the USSR. The other partners were manned bombers and intercontinental ballistic missiles.

The advantage of submarines was that no one knew where they were, and the deeper they sailed, the harder it was to find them. If you needed to tell them to launch missiles, you wanted to be able to do that without the submarines having to be too near the surface.

Submarines were contacted via very low frequency (vlf) 14 kHz to 30 kHz radio links. Consider that the bottom of the AM broadcast band in the USA is 530 kHz, and in Europe, 150 kHz. So you are looking at an order of magnitude lower frequency spectrum being used, so that it could penetrate deeper into the ocean.

In the late 1950s and 1960s, the USA's main such submarine was the Polaris class. That submarine had a vlf radio designated the AN/BRR-3, a fully operational specimen of which I happen to own. I have measured its noise figure, and it is 1 dB. Noise figure is a measure of how perfect a receiver is. A perfect receiver (unrealizable) would have a noise figure of 0 dB. So this receiver is within 1 dB of perfection, and there would be no point in trying to make it marginally better, because 1 dB of improvement wouldn't mean much in increased depth.

The sensitivity of that receiver was classified at the time, because if you knew the sensitivity, then you could figure out fairly accurately the maximum distance below the surface the subs were traveling, because skin depth is strictly a function of sea water conductivity, which is easy to measure. The only other variable involved is antenna gain, and that can be bracketed fairly easily if you know what kind of antennas are being used.

Now the sensitivity and the noise figure are two separate entities, but they are not independent of each other. The sensitivity of a radio is defined by how much above the noise floor the signal must be in order to get a certain quality of reception. The noise floor is given by:

noise floor (Watts) = kTBF,

where,

k = Boltzmann's constant, which is the ideal gas law constant divided by Avogadro's number,

T = radio receiver front end (first mixer) temperature, in units of Kelvin (absolute temperature),

B = bandwidth, in units of cycles per second, or Hertz (Hz), and

F = noise figure, which is a ratio without units.

Looking at that equation, for a given noise figure (which as we saw above was already optimized), all you can change are the bandwidth and temperature. Boltzmann's constant is just that, a fundamental physical constant.

The AN/BRR-3 operated at ambient (room) temperature, and there were three selectable bandwidths. I'm not home right now, so I can't check precise bandwidths, but the top bandwidth was under 200 Hz.

That is way too low for voice communications, which requires at least two kilohertz (and that sounds somewhat nasal). The AN/BRR-3 baseband output (what would normally be audio for a broadcast band radio such as AM or FM or shortwave) was instead data fed to a teletype machine. The teletype would spit out the instructions for launching Armageddon.

So we have come almost to the end, and to the point of the story. In order to maximize the depth at which the submarines could communicate, they were forced to use very small bandwidths that were unsuitable for voice communications. If you remember getting your internet connection over a telephone modem, you should be able to appreciate that limiting bandwidth to less than a tenth of the voice bandwidth is a huge limitation for anything other than very specific and limited instruction sets, such as "Launch missiles at target X immediately."

You may be wondering how I came to own one of these receivers, and why I know so much about it, and am willing to share the info.

It's no longer classified because newer receivers use cryo-cooled front-ends, which greatly increases their sensitivity and renders the AN/BRR-3 and radios like it obsolete. I bought mine at a surplus electronics outlet for $600, and that included the classified instruction manual...

So now you have the rest of the story, as the late, great Paul Harvey would have said.