Unitary Time Evolution: Explaining Open Quantum Systems

In summary, unitary time evolution is used to explain the evolution of a closed quantum system, ensuring that the probability of all possible outcomes is always 1. However, for open quantum systems, the density matrix is used to describe the state of the system and environment, which may not have a unitary evolution when the environmental degrees of freedom are traced out. This is often described by the Copenhagen/Collapse Interpretation, which preserves the condition of Tr[ρ]=1, but is weaker than unitarity. This is because collapse is not invertible, unlike unitary time evolution which is described by U(t).
  • #1
jamie.j1989
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Hi, I am a bit confused about unitary time evolution, I understand that a closed quantum system can be explained by unitary time evolution which ensures that the probability of all possible outcomes is always 1. But for an open quantum system we can't in general explain it with a unitary time evolution. This is where I get confused, we explain the evolution of an open quantum systems with the density matrix ##\rho(t)##, which must always satisfy ##Tr[\rho(t)]=1##, so my question is, how is this not unitary time evolution? Thanks.
 
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  • #2
The density matrix may refer to the state of the system+environment which we expect to have a unitary evolution. But when we trace out the environmental degrees of freedom, we are left with a density matrix which describes what we should expect if we have don't have access to the environmental degrees the freedom. So ## \rho_{total} ## has a unitary evolution but ## \rho_{system}=Tr_{environment}(\rho_{total}) ## doesn't!
 
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Likes eloheim, vanhees71 and jamie.j1989
  • #3
More often than not open system measurements are described by the Copenhagen/Collapse Interpretation. Collapse is not invertible let alone unitary.
 
  • #4
Collapse preserves the condition ##\mbox{Tr}~ \rho=1##. This is a much weaker condition than nitarity, which says that ##\rho(t)=U(t)\rho(0)U(t)^*## with a unitary ##U(t)##.
 
  • #5
A. Neumaier said:
Collapse preserves the condition ##\mbox{Tr}~ \rho=1##. This is a much weaker condition than nitarity, which says that ##\rho(t)=U(t)\rho(0)U(t)^*## with a unitary ##U(t)##.

Indeed, ρ becomes the projection onto the collapsed eigenvector.
 

Related to Unitary Time Evolution: Explaining Open Quantum Systems

1. What is unitary time evolution in quantum mechanics?

Unitary time evolution is a concept in quantum mechanics that describes how a quantum system changes over time. It is represented by a mathematical operator known as the unitary operator, which acts on the state of the system to determine its evolution. This evolution is deterministic and reversible, meaning that the future state of the system can be determined from its current state, and the system can be returned to its original state by reversing the unitary operator.

2. How does unitary time evolution relate to open quantum systems?

Unitary time evolution is the fundamental mechanism that governs the behavior of all quantum systems, including open systems. In open quantum systems, the system is not isolated from its environment and is subject to external influences. These influences cause the system to undergo non-unitary evolution, meaning that its state cannot be fully determined by the unitary operator alone. Instead, a density matrix is used to describe the system's state, and the unitary operator is combined with other operators to account for the effects of the environment.

3. What are the challenges in explaining open quantum systems using unitary time evolution?

One of the main challenges in explaining open quantum systems using unitary time evolution is the complexity of the system-environment interactions. These interactions can be difficult to model and often involve a large number of degrees of freedom, making it challenging to fully determine the system's evolution. Additionally, the non-unitary evolution of open systems can lead to phenomena such as decoherence, which can be difficult to understand and predict using traditional unitary methods.

4. How is unitary time evolution used in practical applications?

Unitary time evolution is used in a wide range of practical applications, including quantum computing, quantum communication, and quantum simulation. In quantum computing, unitary operators are used to manipulate the state of qubits, the basic units of quantum information. In quantum communication, unitary operators are used to encode information onto quantum systems, allowing for secure communication. In quantum simulation, unitary time evolution is used to model and study the behavior of complex quantum systems that are difficult to simulate using classical computers.

5. What are some current research efforts in understanding open quantum systems using unitary time evolution?

Current research efforts in understanding open quantum systems using unitary time evolution focus on developing new theoretical and computational methods to model and study these systems. This includes developing new techniques for simulating large and complex open systems, as well as exploring new ways to incorporate the effects of the environment into the unitary evolution of the system. Additionally, there is ongoing research into the fundamental principles underlying open quantum systems and how they can be better understood and controlled using unitary methods.

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