Module II, Part 2
Entropy, information and entanglement
PH 534 QIC
Himadri Shekhar Dhar
,B. Entanglement, majorization and Nielsen’s theorem
In this section we begin by briefly discussing the origin story of entanglement (and
quantum correlations in general) before going on to ask the more fundamental
question about what is entanglement. We then try to come up with a more operational
approach to define the all-conquering physical entity in quantum physics.
i) Background
The first instance of confrontation with the classical viewpoint of physics, was raised
by Einstein, Podolsky and Rosen (EPR), who were troubled by the violation of the
conjunction of “objective reality” and “locality” in the quantum description of a physical
system with spatially separated subsystems, as mentioned in their seminal paper of
19351. The main contention was the presence of weird and spooky quantum
correlations, and in general they proposed that quantum mechanics was incomplete.
Schrödinger often discussed the weirdness of quantum theory and quantum
correlations in general with a very philosophical tone. In different texts, he touched
upon different aspects of nonclassicality:
In this statement, “…like a scholar in an examination, cannot possibly know which of
the two questions I am going to ask first: it so seems that our scholar is prepared to
give the right answer to the first question… Therefore, he must know both answers;
which is an amazing knowledge,” he touches upon the notion of nonlocality.
In another, “It is rather discomforting that the theory should allow a system to be
steered or piloted into one or the other type of state at the experimenter's mercy in
spite of his having no access to it,” he hints at what would later be termed as the
phenomena of quantum steering.
1
A. Einstein, B. Podolsky, and N. Rosen, Can Quantum-Mechanical Description of Physical Reality Be Considered
Complete? Phys. Rev. 47, 777 (1935).
,And finally, “When two systems… enter into temporary physical interaction due to
known forces between them… they can no longer be described … by endowing each
of them with a representative of its own. I would not call that one but rather the
characteristic trait of quantum mechanics,” where he finally calls upon entanglement.
To reiterate, Schrödinger called the “characteristic trait of quantum mechanics” being
the complete description of a composite system without providing all the information
about its subsystems, and originally referred to this “Verschränkung” or entanglement
The signature of nonlocal quantum correlations was first quantified through the
seminal derivation of Bell inequalities. In 1964, John Bell showed that for all theoretical
description of quantum mechanics, which additionally account for “objective reality and
locality” by means of some “hidden variable”, all bipartite correlations must be
statistically constrained by a set of inequalities. Further, he pointed out that certain
quantum states did not satisfy these inequalities. In the following years, experiments
demonstrated that quantum states can violate these Bell inequalities, thus confirming
the impossibility of using only “objectively real and local” or local hidden-variable
description of quantum phenomena2. Over the years, important theoretical and
experimental results have supported and enriched the quantum viewpoint of the
physical world and established quantum theory as one of the foundational
cornerstones of modern physics. Significantly, the enigmatic trait of the world arising
due to quantum correlations is at the heart of major technological developments in the
21st Century and is the fundamental resource for quantum information processing.
The violation of Bell inequalities led to a critical interest in quantum correlations for
future development of concepts such as quantum communication and the possibility
of developing computational devices with no classical analogue. However, it was not
until the late 20th Century, that the quantum correlation, in its quintessential form of
entanglement, was established in terms of local quantum operations and classical
2
The notion of nonlocal hidden variables to describe quantum mechanics was first proposed by David Bohm in
his now seminal work. But by invoking nonlocality it essentially violates EPR’s original argument against the
lack of “objective reality” and “locality” in quantum mechanics.
, operations. In subsequent years, various theoretical approaches for studying
entanglement, such as inequalities derived from asymptotic rates of information
compression, distillation of entangled states, majorization conditions, witnesses and
resource theories were introduced and studied3.
ii) Entanglement
The fundamental basis of what constitutes nonclassicality or how quantum
correlations are conceptually formulated is not limited to a single theoretical
framework. In a broad sense, nonclassicality arises when composite physical systems
or degrees of freedom are correlated in ways that are inaccessible to classical objects.
The earliest forms of quantifiable nonclassicality in two-party or bipartite quantum
states (bipartite quantum states) arose from the violation of Bell inequality. The
conceptualization of quantum correlation has evolved over the years. The fundamental
idea that violation of Bell inequality was the key principle of defining quantum
correlation suffered a setback when it was shown that there exist certain entangled
states that do not violate the Bell inequalities and hence, violation of Bell inequality is
a sufficient but not necessary condition for entanglement. This led to the realization
that states without entanglement are those which can be prepared using local quantum
operations and classical communication (LOCC). The set of states that cannot be
prepared using LOCC are then called entangled. Consequently the set of separable
states is smaller than the set of states that do not violate Bell inequalities.
iii) Local operations and classical communication (LOCC)
Note: This subsection is not necessary for exams.
a) Local operations (LO):
Let us consider the bipartite case, with Aditi and Bharat sharing a state located at two
spatially distant points. The joint system is described by the composite Hilbert space
ℋ! ⨂ ℋ" , and the class of possible local operations are described by ℰ = ℰ! ⨂ ℰ" ,
3
Although we will not cover all these topics in the course, I will provide additional material for those who are
interested in learning a bit more. Of course, excellent books and papers exist on the Internet.
Entropy, information and entanglement
PH 534 QIC
Himadri Shekhar Dhar
,B. Entanglement, majorization and Nielsen’s theorem
In this section we begin by briefly discussing the origin story of entanglement (and
quantum correlations in general) before going on to ask the more fundamental
question about what is entanglement. We then try to come up with a more operational
approach to define the all-conquering physical entity in quantum physics.
i) Background
The first instance of confrontation with the classical viewpoint of physics, was raised
by Einstein, Podolsky and Rosen (EPR), who were troubled by the violation of the
conjunction of “objective reality” and “locality” in the quantum description of a physical
system with spatially separated subsystems, as mentioned in their seminal paper of
19351. The main contention was the presence of weird and spooky quantum
correlations, and in general they proposed that quantum mechanics was incomplete.
Schrödinger often discussed the weirdness of quantum theory and quantum
correlations in general with a very philosophical tone. In different texts, he touched
upon different aspects of nonclassicality:
In this statement, “…like a scholar in an examination, cannot possibly know which of
the two questions I am going to ask first: it so seems that our scholar is prepared to
give the right answer to the first question… Therefore, he must know both answers;
which is an amazing knowledge,” he touches upon the notion of nonlocality.
In another, “It is rather discomforting that the theory should allow a system to be
steered or piloted into one or the other type of state at the experimenter's mercy in
spite of his having no access to it,” he hints at what would later be termed as the
phenomena of quantum steering.
1
A. Einstein, B. Podolsky, and N. Rosen, Can Quantum-Mechanical Description of Physical Reality Be Considered
Complete? Phys. Rev. 47, 777 (1935).
,And finally, “When two systems… enter into temporary physical interaction due to
known forces between them… they can no longer be described … by endowing each
of them with a representative of its own. I would not call that one but rather the
characteristic trait of quantum mechanics,” where he finally calls upon entanglement.
To reiterate, Schrödinger called the “characteristic trait of quantum mechanics” being
the complete description of a composite system without providing all the information
about its subsystems, and originally referred to this “Verschränkung” or entanglement
The signature of nonlocal quantum correlations was first quantified through the
seminal derivation of Bell inequalities. In 1964, John Bell showed that for all theoretical
description of quantum mechanics, which additionally account for “objective reality and
locality” by means of some “hidden variable”, all bipartite correlations must be
statistically constrained by a set of inequalities. Further, he pointed out that certain
quantum states did not satisfy these inequalities. In the following years, experiments
demonstrated that quantum states can violate these Bell inequalities, thus confirming
the impossibility of using only “objectively real and local” or local hidden-variable
description of quantum phenomena2. Over the years, important theoretical and
experimental results have supported and enriched the quantum viewpoint of the
physical world and established quantum theory as one of the foundational
cornerstones of modern physics. Significantly, the enigmatic trait of the world arising
due to quantum correlations is at the heart of major technological developments in the
21st Century and is the fundamental resource for quantum information processing.
The violation of Bell inequalities led to a critical interest in quantum correlations for
future development of concepts such as quantum communication and the possibility
of developing computational devices with no classical analogue. However, it was not
until the late 20th Century, that the quantum correlation, in its quintessential form of
entanglement, was established in terms of local quantum operations and classical
2
The notion of nonlocal hidden variables to describe quantum mechanics was first proposed by David Bohm in
his now seminal work. But by invoking nonlocality it essentially violates EPR’s original argument against the
lack of “objective reality” and “locality” in quantum mechanics.
, operations. In subsequent years, various theoretical approaches for studying
entanglement, such as inequalities derived from asymptotic rates of information
compression, distillation of entangled states, majorization conditions, witnesses and
resource theories were introduced and studied3.
ii) Entanglement
The fundamental basis of what constitutes nonclassicality or how quantum
correlations are conceptually formulated is not limited to a single theoretical
framework. In a broad sense, nonclassicality arises when composite physical systems
or degrees of freedom are correlated in ways that are inaccessible to classical objects.
The earliest forms of quantifiable nonclassicality in two-party or bipartite quantum
states (bipartite quantum states) arose from the violation of Bell inequality. The
conceptualization of quantum correlation has evolved over the years. The fundamental
idea that violation of Bell inequality was the key principle of defining quantum
correlation suffered a setback when it was shown that there exist certain entangled
states that do not violate the Bell inequalities and hence, violation of Bell inequality is
a sufficient but not necessary condition for entanglement. This led to the realization
that states without entanglement are those which can be prepared using local quantum
operations and classical communication (LOCC). The set of states that cannot be
prepared using LOCC are then called entangled. Consequently the set of separable
states is smaller than the set of states that do not violate Bell inequalities.
iii) Local operations and classical communication (LOCC)
Note: This subsection is not necessary for exams.
a) Local operations (LO):
Let us consider the bipartite case, with Aditi and Bharat sharing a state located at two
spatially distant points. The joint system is described by the composite Hilbert space
ℋ! ⨂ ℋ" , and the class of possible local operations are described by ℰ = ℰ! ⨂ ℰ" ,
3
Although we will not cover all these topics in the course, I will provide additional material for those who are
interested in learning a bit more. Of course, excellent books and papers exist on the Internet.
