“When two systems enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems
separate again, then they can no longer be described in the same way as before. I would not call that one but rather the characteristic trait of quantum mechanics,the one that enforces its entire departure from classic lines of thought. By the interaction the two systems have become entangled”
[E. Schroedinger, 1935]
Being a physicist or a mathematician and living in the normal world is not a simple task. This is not only about physical aspect, conventions, or appearances.
This is also about the weird sensation that you prove before answering the question “What do you do in your life?” Generally speaking, saying you studied maths or physics leads to the answer “Wow! You’re very smart“, that in general is how the conversation ends. However there are more brave people who go further and deeper in the discussion asking you what is your research on.
Sometimes it happened to me that I had to explain to my – non-physicist – friends what entanglement is. This is how usually this thing goes:
– I’m studying a type of correlation between quantum systems called entanglement.
– Sounds interesting! Can you tell me more? –
– Well, basically you let two very small systems interact and it can happen that they end up being correlated in such a way that even if they separate again you can’t describe them separately anymore.–
… and here comes the awkward part…
– Oooh! So it’s like they fell in love!–
Since I’m not a very romantic person, the first time I heard this reaction my face looked like this:
But in time I learned not only to take things less seriously but also that sometimes concepts that you would never relate, explain each other more easily (especially to people that are completely unfamiliar with one of the two subjects).
The first time I came in contact with the concept of “monogamy of entanglement” it was reading B. Terhal paper celebrating C. H. Bennett’s legacy on quantum information theory [B. Terhal, Is entanglement monogamous? IBM J. Res. Dev. 48, 71 (2004)]. Monogamy of entanglement: it makes sense, no? If entanglement can be compered to love, then we should also question its monogamy. So let me explain you what monogamy of entanglement is.
Suppose that you have three systems. Let me address them with A, B and C, for simplicity. Furthermore, suppose A and B are maximally entangled. It turns out that each of these two systems, say A specifically, can’t share any entanglement with C. If we want to maintain the romantic vision of entanglement as love, it makes perfect sense to call this property monogamy of entanglement: it can immediately become the story of a woman, Alice, and a man, Bob, that are so in love with each other that for them is impossible to find room for anyone else.
Now we can move this concept further and ask more questions. For instance, Alice and Bob may be not so keen on spending all their time always together and with no one else and maybe they would like to meet other people. This is how they meet Charlie. Anyway, being so in love with each other, Alice and Bob cannot dedicate too much time, energies and attentions to someone else, let’s say Charlie.
This is how we arrive to a more interesting concept of monogamy of entanglement. In 2000 it was proved that if two qubits (elementary quantum systems) are entangled, the entanglement that they can share with a third part is bounded [V. Coffman, J. Kundu, and W. K. Wootters, Distributed entanglement Phys. Rev. A 61, 052306 (2000)]. In other words the entanglement that A shares with B summed with the one that A shares with C can’t exceed the entanglement that A shares with B and C seen as a whole system. The work of Coffman, Kundu and Wootters caught what seemed to be an essential feature of quantum entanglement and since then many other works followed, trying to investigate deeper this property in many directions. In particular, from the rough description that I gave you there are at least two possible directions that are not so difficult to see:
- Study what happens enlarging the number of subsystems;
- Study what happens enlarging the dimension of the single subsystems.
But maybe the experts among you, readers, have noticed that I move forward a crucial information: what measure did they use to quantify the entanglement? Indeed, there’re a lot of mathematical functions that can be addressed as entanglement measures and each of them is able to catch different aspects of this quantum feature. In their paper, Coffman, Kundu and Wootters proved the inequality shown in the previous picture (that from now on I’ll call CKW inequality) using as measure the (squared) concurrence [S. Hill and W. K. Wootters Entanglement of a pair of quantum bits Phys. Rev. Lett. 78, 1997; W. K. Wootters, Entanglement of formation of an arbitrary state of two qubits Phys. Rev. Lett. 80, 1998]. This means that there is a third possible direction in which one can try to investigate further:
3. Study what happens changing the employed entanglement measure.
Here it comes one of the main problems regarding CKW inequality. Indeed, it can be shown that its validity is not universal, but rather depends on the specific choice of the measure. In fact, useful entanglement monotones, such as the entanglement of formation or the distillable entanglement do not directly obey the inequality.
Anyway monogamy seems to be a key feature of entanglement as hinted, for instance, apart for the nice parallelism with love, by the shareability problem that states that it is not possible to create a symmetric extension of an entangled state of two parties, for infinitely many parties.
So we are led to raise the question: Should any valid entanglement measure be monogamous in a CKW-like sense? In particular, given an entanglement measure E, should it satisfy an inequality like
with f a suitable function that doesn’t make everything trivial? Clearly we would like to have f as general as possible meaning that we would like it to be dimension independent, for instance.
The first step in order to answer our question is to define what are the properties that we would like the entanglement measure to satisfy in order to be considered valid. Here I will skim through some technicalities, but let me just say that we would like our entanglement measure to be additive, in the sense that if you bring two copies of the same quantum state the total entanglement you get is twice that of a single copy, and to respect the geometry of quantum states, in the sense of assigning a high value to states which stay far away from the subset of unentangled states no matter their dimension (one such example is the so-called fully antisymmetric state in arbitrary dimension, which has a constant distance from unentangled states). This last requirement can be considered a kind of faithfulness of the measure. (Now you see where I’m going, don’t you?) But before going further and before you start thinking that I’m a cheater, let me tell you that these properties have not been chosen in such a way that there is no possible E satisfying them. Indeed, important measures of entanglement such as the regularization of the relative entropy of entanglement and the entanglement cost satisfy them.
As maybe you’re expecting, the interesting thing in this discussion is that it can be proved that a suitable f for E satisfying the properties that we required, including the non-dimensional dependence (or faithfulness), can’t exist (what a surprise!). Indeed, what can be shown is that enlarging enough the dimension of the system we can always find a state for which the entanglement of part A with B and C taken together is effectively the same as the entanglement that A shares with B and C individually.
This leads us to the starting title and directly to the end of the story. The moral is that even if monogamy and faithfulness are properties that we would like entanglement (maybe also love!) to be endowed with, they’re not compatible to each other at all (is it also true with love?), at least within the weird quantum world.
[based on a real story:
C. Lancien, S. Di Martino, M. Huber, M. Piani, G. Adesso, A. Winter
Ehm… If you’re wondering, the answer is yes: there is a closing scene after the ending titles.
There is a way to escape the monogamy vs faithfulness problem: you can decide to give up the universality of f regarding the dimension of the system, thus preserving both monogamy and faithfulness.
Would you rather care for monogamy, faithfulness or dimensions?