In
quantum physics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
, the "monogamy" of
quantum entanglement
Quantum entanglement is the phenomenon that occurs when a group of particles are generated, interact, or share spatial proximity in a way such that the quantum state of each particle of the group cannot be described independently of the state of ...
refers to the fundamental property that it cannot be freely shared between arbitrarily many parties.
In order for two qubits ''A'' and ''B'' to be
maximally entangled, they must not be entangled with any third qubit ''C'' whatsoever. Even if ''A'' and ''B'' are not maximally entangled, the degree of entanglement between them constrains the degree to which either can be entangled with ''C''. In full generality, for
qubits
, monogamy is characterized by the Coffman-Kundu-Wootters (CKW) inequality, which states that
:
where
is the
density matrix
In quantum mechanics, a density matrix (or density operator) is a matrix that describes the quantum state of a physical system. It allows for the calculation of the probabilities of the outcomes of any measurement performed upon this system, using ...
of the substate consisting of qubits
and
and
is the "tangle," a quantification of bipartite entanglement equal to the square of the
concurrence
In Western jurisprudence, concurrence (also contemporaneity or simultaneity) is the apparent need to prove the simultaneous occurrence of both ("guilty action") and ("guilty mind"), to constitute a crime; except in crimes of strict liability ...
.
Monogamy, which is closely related to the
no-cloning property,
is purely a feature of quantum correlations, and has no classical analogue. Supposing that two classical random variables ''X'' and ''Y'' are correlated, we can copy, or "clone," ''X'' to create arbitrarily many random variables that all share precisely the same correlation with ''Y''. If we let ''X'' and ''Y'' be entangled quantum states instead, then ''X'' cannot be cloned, and this sort of "polygamous" outcome is impossible.
The monogamy of entanglement has broad implications for applications of
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
ranging from
black hole physics to
quantum cryptography
Quantum cryptography is the science of exploiting quantum mechanical properties to perform cryptographic tasks. The best known example of quantum cryptography is quantum key distribution which offers an information-theoretically secure solution ...
, where it plays a pivotal role in the security of
quantum key distribution
Quantum key distribution (QKD) is a secure communication method which implements a cryptographic protocol involving components of quantum mechanics. It enables two parties to produce a shared random secret key known only to them, which can then be ...
.
Proof
The monogamy of bipartite entanglement was established for tripartite systems in terms of concurrence by Coffman, Kundu, and
Wootters in 2000.
In 2006, Osborne and Verstraete extended this result to the multipartite case, proving the CKW inequality.
Example
For the sake of illustration, consider the three-qubit state
consisting of qubits ''A'', ''B'', and ''C''. Suppose that ''A'' and ''B'' form a (maximally entangled)
EPR pair
The Bell states or EPR pairs are specific quantum states of two qubits that represent the simplest (and maximal) examples of quantum entanglement; conceptually, they fall under the study of quantum information science. The Bell states are a form o ...
. We will show that:
:
for some valid quantum state
. By the definition of entanglement, this implies that ''C'' must be completely disentangled from ''A'' and ''B''.
When measured in the standard basis, ''A'' and ''B'' collapse to the states
and
with probability
each. It follows that:
:
for some
such that
. We can rewrite the states of ''A'' and ''B'' in terms of diagonal basis vectors
and
:
:
:
Being maximally entangled, ''A'' and ''B'' collapse to one of the two states
or
when measured in the diagonal basis. The probability of observing outcomes
or
is zero. Therefore, according to the equation above, it must be the case that
and
. It follows immediately that
and
. We can rewrite our expression for
accordingly:
:
:
:
This shows that the original state can be written as a product of a pure state in ''AB'' and a pure state in ''C'', which means that the EPR state in qubits ''A'' and ''B'' is not entangled with the qubit ''C''.
References
{{Quantum information
Quantum information science