Ah, quantum mechanics . Just when you thought the universe made some semblance of sense, it decided to throw in variables like these. You want to delve into the quantum states of two qubits ? Very well. Try to keep up.

Quantum states of two qubits

In the rather specific, and frankly, often perplexing, domain of quantum information science , one frequently encounters what are known as the Bell states , or for those with a flair for history, EPR pairs . These aren’t just any old quantum states ; they are the quintessential, simplest illustrations of a phenomenon that has baffled and intrigued physicists for decades: quantum entanglement . These states, foundational to understanding multi-particle quantum systems, were first conceptualized in a manner that challenged the very fabric of classical intuition.

The Bell states themselves are a specific set of basis vectors that are both maximally entangled and, crucially, normalized . This normalization isn’t some arbitrary mathematical nicety; it simply means that the total probability of finding the particles in one of these defined states is precisely 1. One might assume that this is a given, but in the ethereal realm of quantum probabilities, it’s a necessary constraint for a coherent theory. This particular characteristic, entanglement , emerges as a direct, basis-independent consequence of the more fundamental principle of superposition .

Now, if you’re still following, recall that due to this inherent superposition , any direct measurement performed on a single qubit within such a state will inevitably cause it to “collapse ” into one of its classical basis states (say, $|0\rangle$ or $|1\rangle$), each with a specific, calculable probability. The truly fascinating, or perhaps deeply unsettling, aspect arises from the entanglement : a measurement on one qubit instantaneously influences the state of its entangled partner, forcing the other qubit into a corresponding state. The outcome of a subsequent measurement on this second qubit is then perfectly correlated (or anti-correlated) with the first, its value strictly determined by the specific Bell state the pair initially occupied.

This concept extends beyond just two qubits . Bell states can be generalized to more complex quantum states involving multiple qubit systems, such as the Greenberger–Horne–Zeilinger (GHZ) state for three or more subsystems. These GHZ states represent an even deeper level of multi-particle entanglement , where the collective properties are even more pronounced than in their two-qubit counterparts, making them crucial for understanding more advanced quantum phenomena and applications.

A thorough grasp of Bell states is not merely an academic exercise; it is profoundly useful, if not entirely indispensable, for dissecting and developing various protocols in quantum communication . This includes advanced techniques like superdense coding , which allows for the transmission of more classical information than one might expect from a single qubit , and quantum teleportation , a process that, despite its evocative name, transmits a quantum state rather than matter itself. However, let’s be clear: these mechanisms, for all their quantum weirdness, cannot transmit information faster than the speed of light. This inconvenient truth is encapsulated by the no-communication theorem , a fundamental constraint that prevents entanglement from being exploited for faster-than-light signaling, no matter how much you wish it would. The theorem simply states that local operations on one part of an entangled system cannot instantly convey information to another distant part, preserving the tenets of special relativity .

Bell states

The Bell states are a quartet of singularly important, maximally entangled quantum states involving two qubits . They exist as a superposition of the fundamental computational basis states, $|0\rangle$ and $|1\rangle$, meaning they are a linear combination of these possibilities, not definitively one or the other until measured. Their profound entanglement dictates a set of rather counterintuitive behaviors:

Consider a scenario involving two distant parties, Alice and Bob. Alice holds one qubit (denoted with subscript “A”) and Bob holds the other (subscript “B”) from an entangled Bell pair. If Alice were to measure her qubit in the standard computational basis, the outcome would be either 0 or 1, each occurring with an equal probability of 1/2. Similarly, if Bob measured his qubit , he would also observe either 0 or 1, again with 1/2 probability for each. From their individual perspectives, their outcomes would appear entirely random, lacking any discernible pattern. However, should Alice and Bob communicate their results (through a classical channel, naturally, as the universe insists on its speed limits), they would uncover a startling fact: despite the apparent randomness of their individual results, their outcomes are perfectly correlated. If Alice measured a 0, Bob always measured a 0. If Alice measured a 1, Bob always measured a 1.

This perfect, instantaneous correlation over vast distances was, to put it mildly, a source of considerable philosophical and physical discomfort. It prompted luminaries like Albert Einstein , Boris Podolsky , and Nathan Rosen in their seminal 1935 “EPR paradox ” paper to suggest that perhaps something was fundamentally incomplete in the quantum mechanical description of such a qubit pair. They posited that there must be some “pre-agreement” or hidden information established at the moment the pair was created, before the qubits separated, determining their future measurement outcomes. This hypothetical pre-agreement was termed a “hidden variable .”

It wasn’t until 1964 that John S. Bell , with characteristic elegance, demonstrated through simple arguments rooted in probability theory that such perfect correlations, particularly when considering measurements in different bases (e.g., the 0,1 basis and the +,− basis), could not be explained by any form of “pre-agreement” stored in hidden variables . His work showed a stark incompatibility: quantum mechanics predicts correlations that are simply stronger than any classical local hidden-variable theory could ever allow.

This foundational insight was later refined and formalized in what is now known as the Bell–CHSH inequality (named after Clauser, Horne, Shimony, and Holt). This inequality establishes an upper bound for a specific correlation measure in any local hidden-variable theory —a common-sense framework where information is conveyed locally and deterministically. This classical upper bound is 2. However, quantum mechanics predicts that certain entangled systems can achieve correlation values as high as $2{\sqrt {2}}$. The experimental verification of these violations of the Bell–CHSH inequality has been a profound triumph for quantum mechanics , effectively ruling out the possibility of local hidden variables and confirming that the universe is, indeed, stranger than classical physics could ever have imagined. It means that the “spooky action at a distance” Einstein famously derided is not merely a theoretical quirk, but a fundamental aspect of reality.

Bell basis

The four specific two-qubit states that consistently achieve this maximal correlation value of $2{\sqrt {2}}$ are formally designated as the “Bell states.” These are not merely examples of entanglement ; they are the maximally entangled two-qubit Bell states, forming a complete and orthonormal basis—the Bell basis—for the four-dimensional Hilbert space that describes two qubits . Think of a Hilbert space as the mathematical arena where quantum states reside; for two qubits , it’s a four-dimensional complex vector space.

These four states are:

1. The first Bell state, often called “Phi-plus”: $|\Phi ^{+}\rangle ={\frac {1}{\sqrt {2}}}{\big (}|0\rangle _{A}\otimes |0\rangle _{B}+|1\rangle _{A}\otimes |1\rangle _{B}{\big )}\qquad (1)$ This state represents a superposition where both qubits are either simultaneously in the $|0\rangle$ state or simultaneously in the $|1\rangle$ state, with equal probability. They are perfectly correlated in the computational basis. The tensor product symbol ($\otimes$) explicitly denotes that these are two separate qubits forming a joint state.

2. The second Bell state, “Phi-minus”: $|\Phi ^{-}\rangle ={\frac {1}{\sqrt {2}}}{\big (}|0\rangle _{A}\otimes |0\rangle _{B}-|1\rangle _{A}\otimes |1\rangle _{B}{\big )}\qquad (2)$ Similar to $|\Phi^+\rangle$, but with a relative phase difference between the $|00\rangle$ and $|11\rangle$ components. This seemingly minor change has significant implications for how they correlate in different measurement bases.

3. The third Bell state, “Psi-plus”: $|\Psi ^{+}\rangle ={\frac {1}{\sqrt {2}}}{\big (}|0\rangle _{A}\otimes |1\rangle _{B}+|1\rangle _{A}\otimes |0\rangle _{B}{\big )}\qquad (3)$ Here, the qubits are always in opposite states. If Alice’s qubit is $|0\rangle$, Bob’s is $|1\rangle$, and vice versa. They are perfectly anti-correlated in the computational basis.

4. The fourth Bell state, “Psi-minus”: $|\Psi ^{-}\rangle ={\frac {1}{\sqrt {2}}}{\big (}|0\rangle _{A}\otimes |1\rangle _{B}-|1\rangle _{A}\otimes |0\rangle _{B}{\big )}\qquad (4)$ Again, anti-correlated like $|\Psi^+\rangle$, but with a relative phase difference. This state is particularly notable as it is antisymmetric, a property that often arises in multi-particle quantum systems and has implications for particle statistics.

These four states collectively form a complete set, meaning any arbitrary two-qubit quantum state can be expressed as a linear combination of these Bell states. They are the fundamental building blocks for understanding entanglement in two-qubit systems.

Creating Bell states via quantum circuits

While the theoretical description of Bell states is elegant, their practical realization requires specific manipulations of qubits using quantum circuits . There are, of course, a multitude of intricate ways to construct these entangled states, but the most straightforward, and thus the most commonly illustrated, method involves a simple two-gate sequence. This minimal circuit takes a computational basis state as its initial input and employs a Hadamard gate followed by a Controlled-NOT (CNOT) gate .

Let’s consider the example depicted in typical diagrams: the quantum circuit begins with two qubits in the initial computational state $|00\rangle$. The objective is to transform this into the first Bell state, $|\Phi ^{+}\rangle$.

Here’s the step-by-step transformation:

1. Initial State: The system starts in $|00\rangle$, which represents $|0\rangle_A \otimes |0\rangle_B$.
2. Hadamard Gate Application: The first gate encountered is the Hadamard gate , applied to the first qubit (qubit A). The Hadamard gate is a unitary operation that transforms a definite state into a superposition . Specifically, it transforms $|0\rangle$ into $\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$ and $|1\rangle$ into $\frac{1}{\sqrt{2}}(|0\rangle - |1\rangle)$. Applying it to $|0\rangle_A$ yields $\frac{1}{\sqrt{2}}(|0\rangle_A + |1\rangle_A)$. The second qubit ($|0\rangle_B$) remains untouched. Thus, the state of the system becomes $\left(\frac{|0\rangle_A + |1\rangle_A}{\sqrt{2}}\right) \otimes |0\rangle_B = \frac{1}{\sqrt{2}}(|0\rangle_A \otimes |0\rangle_B + |1\rangle_A \otimes |0\rangle_B)$, which can be more compactly written as $\frac{1}{\sqrt{2}}(|00\rangle + |10\rangle)$. This is now a superposition but not yet entangled.
3. CNOT Gate Application: Next, a Controlled-NOT (CNOT) gate is applied. In this circuit, the first qubit (A) acts as the control, and the second qubit (B) acts as the target. The CNOT gate inverts the state of the target qubit only if the control qubit is in the $|1\rangle$ state. If the control is $|0\rangle$, the target remains unchanged. Applying the CNOT gate to the state $\frac{1}{\sqrt{2}}(|00\rangle + |10\rangle)$:
   • For the $|00\rangle$ component: The control qubit is $|0\rangle$, so the target qubit ($|0\rangle$) remains unchanged. This component stays $|00\rangle$.
   • For the $|10\rangle$ component: The control qubit is $|1\rangle$, so the target qubit ($|0\rangle$) is flipped to $|1\rangle$. This component becomes $|11\rangle$. Combining these, the final state is $\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$. This, as you might have noticed, is precisely the definition of the first Bell state, $|\Phi ^{+}\rangle$.

This sequence elegantly demonstrates how superposition and controlled operations combine to create entanglement . By systematically varying the initial two-qubit inputs—$|00\rangle$, $|01\rangle$, $|10\rangle$, and $|11\rangle$—this exact circuit configuration will produce all four Bell states.

More generally, this circuit transforms an input state $|xy\rangle$ (where $x$ and $y$ are the classical bit values of the input qubits) into a Bell state according to the equation: $|\beta (x,y)\rangle =\left({\frac {|0,y\rangle +(-1)^{x}|1,{\bar {y}}\rangle }{\sqrt {2}}}\right),$ where $\bar{y}$ represents the negation of $y$ (i.e., if $y=0$, $\bar{y}=1$, and vice versa). This compact formula neatly encapsulates how the initial classical bits determine which of the four entangled Bell states is produced. For example, if $x=0, y=0$, then $|\beta(0,0)\rangle = \frac{1}{\sqrt{2}}(|0,0\rangle + (-1)^0|1,1\rangle) = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle) = |\Phi^+\rangle$. A rather efficient way to generate something so profoundly non-classical, wouldn’t you say?

Properties of Bell states

The characteristics of Bell states are what make them so central to quantum information science , simultaneously useful and mind-bending. A key property is that the outcome of a measurement on a single qubit from a Bell pair is fundamentally indeterminate; it’s random, with a 50/50 chance of being 0 or 1 if measured in the standard basis. However, the true quantum magic reveals itself when you consider the second qubit . Upon measuring the first qubit (say, in the z-basis), the result of measuring the second qubit is guaranteed to yield either the same value (for the $|\Phi\rangle$ Bell states) or the opposite value (for the $|\Psi\rangle$ Bell states). This isn’t just a correlation; it’s a perfect, instantaneous link that defies classical explanation.

As John Bell meticulously proved, these measurement correlations within Bell states are unequivocally stronger than any correlations that could ever exist between classical systems. This isn’t a minor distinction; it’s a profound declaration that quantum mechanics enables forms of information processing that are utterly beyond the capabilities of classical physics. This is the bedrock upon which the entire field of quantum computing and quantum communication is built.

Furthermore, the Bell states form an orthonormal basis . This means they are mutually orthogonal (distinct) and individually normalized (total probability of 1). Consequently, they can be precisely identified and distinguished through an appropriate quantum measurement .

Another crucial aspect is the information paradox they present. A Bell state, as a description of the entire two-qubit system, is a pure state —meaning all information about the system is known. Yet, if you consider only one of the qubits in isolation, its individual state is described by a mixed state (specifically, a maximally mixed state). This “mixedness” implies that all information about that individual qubit is not known; it’s as if the information is distributed across the entanglement, rather than localized in either particle. This is a direct consequence of entanglement : you know everything about the pair, but nothing definitive about its individual components until a measurement forces a collapse.

Finally, Bell states exhibit specific symmetries: they are either symmetric or antisymmetric with respect to the exchange of the two subsystems. For instance, $|\Phi^+\rangle$ and $|\Psi^+\rangle$ are symmetric, while $|\Phi^-\rangle$ and $|\Psi^-\rangle$ are antisymmetric (or vice versa, depending on convention and specific definition of symmetry). This property is particularly relevant in the study of identical particles in quantum mechanics , where particle statistics (like fermions and bosons ) are tied to the symmetry of their wave functions . The concept of Bell states can be generalized to multi-qubit systems, leading to the notion of absolutely maximally entangled (AME) states , which represent the ultimate limit of entanglement for a given number of particles, with their reduced density operators being maximally mixed for any single or subset of particles.

Bell state measurement

The Bell measurement is not merely a curious observation; it is a critical operational concept within quantum information science . It constitutes a joint quantum-mechanical measurement performed on two qubits with the explicit goal of determining precisely which of the four Bell states the pair currently occupies. This isn’t trivial, as the states are entangled and cannot be distinguished by measuring the qubits individually.

The circuit that performs a Bell decoding, or Bell state measurement, is, rather conveniently, the adjoint (or inverse) of the circuit used to encode or create the Bell states. This means if you understand how to entangle, you effectively understand how to disentangle and measure. The process typically involves applying a Controlled-NOT (CNOT) gate to the two qubits (say, A and B), followed by a Hadamard gate on the first qubit (A). After these operations, the originally entangled state is transformed into one of the four computational basis states ($|00\rangle, |01\rangle, |10\rangle, |11\rangle$), which can then be measured using standard classical detection techniques. The CNOT gate essentially “un-entangles” the two previously entangled qubits , allowing their quantum information to be projected onto classical, measurable outcomes. This conversion from a purely quantum state to a classical measurement is what allows for the extraction of information.

Understanding quantum measurement in this context benefits from two fundamental principles:

1. Principle of Deferred Measurement : This states that any quantum measurement operation within a circuit can conceptually be postponed and moved to the very end of the circuit without altering the final probabilities of outcomes. This simplifies circuit design and analysis.
2. Principle of Implicit Measurement: At the termination of any quantum circuit , it can be assumed that a measurement is performed on any qubit lines that are not explicitly connected to further operations. This is a convention that streamlines the representation of quantum algorithms.

These principles streamline the theoretical design and analysis of quantum circuits , even if practical implementations often require measurements at specific intermediate points.

Applications of Bell state measurements

Bell state measurements are not just theoretical constructs; they are the lynchpin of several critical quantum information protocols:

• Crucial Step in Quantum Teleportation : This is perhaps the most famous application. The result of a Bell state measurement, performed by Alice on an unknown qubit and her half of an EPR pair , is precisely the classical information she sends to Bob. Bob then uses this classical information to perform a specific unitary operation on his half of the EPR pair , thereby reconstructing the original, unknown quantum state . Without a reliable Bell state measurement, quantum teleportation simply fails.

• Limitations of “Linear Evolution, Local Measurement” (LELM) Techniques: In experimental physics, especially with photons, many detection setups rely on “linear evolution, local measurement” techniques. This implies that the detection apparatus interacts with each particle independently, without its operation being influenced by the state or evolution of other particles. “Local measurement” means each particle triggers a distinct detector, registering a “click.” Devices constructed from passive optical elements like mirrors, beam splitters, and wave plates fall into this category. While experimentally attractive due to their ease of use and high measurement cross-section , such LELM techniques face inherent limitations. For entanglement in a single qubit variable (e.g., polarization ), only three out of the four Bell states can be unambiguously distinguished. This means two of the Bell states are indistinguishable from each other using these methods. If the teleported quantum state happens to correspond to one of these ambiguous Bell states, the teleportation event cannot be completed successfully, leading to a loss of efficiency in quantum communication protocols. This limitation arises from fundamental conservation laws and the non-deterministic nature of linear optical gates.

• Advantages of Hyper-entanglement : To overcome the limitations of LELM, researchers have explored hyper-entanglement , where particles are entangled in multiple degrees of freedom (or “qubit variables”) simultaneously. For photonic systems, this could mean entanglement in both polarization and orbital angular momentum (OAM) states. By tracing over one variable (effectively ignoring its state), experimenters can achieve a complete Bell state measurement in the other variable. For instance, if two photons are hyper-entangled in polarization and OAM, one could perform a measurement that distinguishes all four Bell states in the polarization degree of freedom, even with LELM, by utilizing the OAM entanglement as an auxiliary resource. This significantly enhances the success rate of quantum teleportation . Beyond teleportation, hyper-entanglement also boosts the channel capacity in protocols like superdense coding , allowing more information to be encoded and transmitted per particle.

• Generalization for Multiple Variables: More generally, for a system exhibiting hyper-entanglement across $n$ distinct quantum variables, linear optical techniques are capable of distinguishing at most $2^{n+1}-1$ classes of states out of a total of $4^n$ possible Bell states. This formula underscores the increasing complexity and the persistent, albeit reduced, limitations of linear optics even with multiple degrees of freedom. It means that while hyper-entanglement improves distinguishability, a truly universal and deterministic Bell state measurement for all $4^n$ states remains a formidable experimental challenge.

Bell state correlations

The defining characteristic of Bell states lies in their correlations. Independent measurements performed on two qubits that are entangled in a Bell state exhibit perfect correlations, provided each qubit is measured in the “relevant” basis. What constitutes “relevant” depends on the specific Bell state.

Let’s break this down:

• For the $|\Phi ^{+}\rangle$ state, which is defined as $\frac{1}{\sqrt{2}}(|00\rangle + |11\rangle)$, if both Alice and Bob measure their respective qubits in the same basis (e.g., the standard computational basis ${|0\rangle, |1\rangle}$ or the Hadamard basis ${|+\rangle, |-\rangle}$), their results will be perfectly positively correlated. If Alice measures 0, Bob measures 0. If Alice measures 1, Bob measures 1.

• For the $|\Phi ^{-}\rangle$ state, defined as $\frac{1}{\sqrt{2}}(|00\rangle - |11\rangle)$:

  • If Alice and Bob measure in the standard computational basis ${|0\rangle, |1\rangle}$, their outcomes will still be perfectly positively correlated (0 with 0, 1 with 1).
  • However, if they measure in the Hadamard basis ${|+\rangle, |-\rangle}$, their outcomes will be perfectly anti-correlated. If Alice measures $|+\rangle$, Bob measures $|-\rangle$, and vice versa. This can be mathematically demonstrated, as shown in the note: $|\Phi ^{-}\rangle ={\frac {1}{\sqrt {2}}}(|00\rangle -|11\rangle)$ We know that $|0\rangle = \frac{1}{\sqrt{2}}(|+\rangle + |-\rangle)$ and $|1\rangle = \frac{1}{\sqrt{2}}(|+\rangle - |-\rangle)$. Substituting these into the expression for $|\Phi^-\rangle$: $={\frac {1}{2{\sqrt {2}}}}((|+\rangle _{A}+|-\rangle _{A})(|+\rangle _{B}+|-\rangle _{B})-(|+\rangle _{A}-|-\rangle _{A})(|+\rangle _{B}-|-\rangle _{B}))$ Expanding the terms: $={\frac {1}{2{\sqrt {2}}}}(|++\rangle +|+-\rangle +|-+\rangle +|–\rangle -|++\rangle +|+-\rangle +|-+\rangle -|–\rangle )$ Notice that some terms cancel out (e.g., $|++\rangle$ and $-|++\rangle$), while others reinforce (e.g., $|+-\rangle$ and $|+-\rangle$). $={\frac {1}{\sqrt {2}}}(|+-\rangle +|-+\rangle )$ This final form clearly shows that in the Hadamard basis, the qubits are always found in opposite states ($|+\rangle$ and $|-\rangle$), hence the perfect anti-correlation. In other measurement bases, the correlations would be partial or probabilistic, a testament to the basis-dependent nature of these quantum relationships.

• For the $|\Psi ^{+}\rangle$ state, defined as $\frac{1}{\sqrt{2}}(|01\rangle + |10\rangle)$, the correlations are perfectly anti-correlated if both qubits are measured in the same basis (e.g., computational or Hadamard). If Alice measures 0, Bob measures 1; if Alice measures 1, Bob measures 0. More generally, if Alice measures her qubit in an arbitrary basis $b_1$, and Bob measures his qubit in a basis $b_2 = X.b_1$ (where $X$ represents the Pauli-X gate , which effectively flips the basis vectors), then they would observe perfectly positively correlated results.

The table below summarizes the relationship between the correlated bases for the two qubits within each Bell state, assuming Alice measures her qubit in an arbitrary basis $b_1$:

Here, $Z.b_1$ implies that Bob’s measurement basis is related to Alice’s by a Pauli-Z operation (a phase flip), and $X.b_1$ implies a Pauli-X operation (a bit flip). The combined operation $X.Z.b_1$ indicates a combination of both. This table neatly encapsulates how the specific Bell state dictates the necessary transformation of one measurement basis relative to the other to observe perfect positive correlations.

Applications

The fundamental properties and specific correlations of Bell states are not abstract curiosities but form the backbone of several transformative applications in the nascent field of quantum information .

Superdense coding

Superdense coding is a particularly elegant application that demonstrates the power of entanglement . It enables two distant parties to communicate two bits of classical information by sending only a single qubit . This is an impressive feat, considering a single classical bit can only convey one bit of information. The entire phenomenon hinges on the pre-shared entangled states , specifically one of the Bell states, between the two communicators.

Imagine Alice and Bob, separated by a vast distance, but who have previously shared an EPR pair , with Alice holding one qubit and Bob the other, both initially in the state: $|\psi \rangle ={\frac {|00\rangle +|11\rangle }{\sqrt {2}}}$. This is, in fact, the $|\Phi^+\rangle$ Bell state.

Alice’s objective is to transmit one of four possible two-bit classical messages: ‘00’, ‘01’, ‘10’, or ‘11’. She achieves this by performing a local quantum gate transformation on her single qubit . The specific gate she applies depends on the message she wishes to send:

• To send ‘00’: Alice does nothing to her qubit . This is equivalent to applying the identity gate ($I$). The initial entangled state remains $|\psi \rangle ={\frac {|00\rangle +|11\rangle }{\sqrt {2}}}\equiv |{\Phi ^{+}}\rangle$. $00:I={\begin{bmatrix}1&0\0&1\end{bmatrix}}\longrightarrow |\psi \rangle ={\frac {|00\rangle +|11\rangle }{\sqrt {2}}}\equiv |{\Phi ^{+}}\rangle$

• To send ‘01’: Alice applies the Pauli-X gate (bit flip) to her qubit . This transforms the initial Bell state into a different one. $01:X={\begin{bmatrix}0&1\1&0\end{bmatrix}}\longrightarrow |\psi \rangle ={\frac {|01\rangle +|10\rangle }{\sqrt {2}}}\equiv |{\Psi ^{+}}\rangle$

• To send ‘10’: Alice applies the Pauli-Z gate (phase flip) to her qubit . $10:Z={\begin{bmatrix}1&0\0&-1\end{bmatrix}}\longrightarrow |\psi \rangle ={\frac {|00\rangle -|11\rangle }{\sqrt {2}}}\equiv |{\Phi ^{-}}\rangle$

• To send ‘11’: Alice applies the Pauli-Y gate (a combination of bit and phase flip, or $iY = ZX$) to her qubit . $11:iY=ZX={\begin{bmatrix}0&1\-1&0\end{bmatrix}}\longrightarrow |\psi \rangle ={\frac {|01\rangle -|10\rangle }{\sqrt {2}}}\equiv |{\Psi ^{-}}\rangle$

After Alice performs the chosen transformation, her single qubit (now part of one of the four Bell states) is sent to Bob. Upon receiving Alice’s qubit , Bob now possesses both qubits of the entangled pair. To decipher Alice’s message, Bob performs a Bell measurement on the combined two-qubit system. This measurement effectively projects the entangled state onto one of the four two-qubit computational basis vectors, which directly corresponds to one of the four Bell states Alice prepared. By identifying which Bell state he has, Bob can unambiguously determine the original two-bit classical message Alice intended to send. This protocol neatly circumvents the no-communication theorem because only one qubit is transmitted, and classical information (the choice of operation) is encoded into the quantum state, not transmitted instantaneously.

Quantum teleportation

Quantum teleportation , despite its name, does not involve moving matter from one location to another. Instead, it is the remarkable process of transferring an unknown quantum state from a sender (Alice) to a receiver (Bob) over an arbitrary distance, without physically moving the particle itself. This process, which relies heavily on pre-shared entanglement (an EPR pair ), has become a cornerstone of research in quantum communication and quantum computing , with recent advancements exploring its feasibility for information transfer through practical mediums like optical fibers .

The process unfolds in a series of precise steps:

1. Shared EPR pair : Alice and Bob begin by sharing an EPR pair (one of the Bell states), each taking one qubit before they are physically separated. This shared entangled link acts as the “quantum channel.”
2. Alice’s Unknown Qubit: Alice possesses a third qubit whose quantum state ($|\psi\rangle$) she wishes to transmit to Bob, but crucially, she does not know what this state is. The fundamental principle of quantum mechanics prevents her from simply measuring it to learn its state without disturbing it.
3. Alice’s Joint Operations:
   • Alice first sends her unknown qubit and her half of the EPR pair through a Controlled-NOT (CNOT) gate . This operation entangles her unknown qubit with her part of the EPR pair .
   • Subsequently, Alice sends her first qubit (the original unknown one) through a Hadamard gate .
4. Alice’s Bell Measurement : After these operations, Alice performs a Bell measurement on her two qubits . This joint measurement yields one of four possible classical outcomes, corresponding to which of the four Bell states her two qubits collapsed into.
5. Classical Communication: Alice then sends this classical information (two bits, indicating which of the four outcomes she observed) to Bob, typically via a conventional communication channel (e.g., radio waves, email). This is the only information that travels between them.
6. Bob’s Reconstruction: Based on the two classical bits he receives from Alice, Bob performs one of four specific unitary operations (identity, Pauli-X , Pauli-Z , or Pauli-Y ) on his half of the original EPR pair . This precise operation reconstructs the original, unknown quantum state $|\psi\rangle$ on his qubit .

The following quantum circuit diagram visually describes this process:

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  Unknown Qubit (|psi>) ---H---CNOT----(Measurement)---(Classical Channel)---(Control for Bob's Gate)
                             |     |
                             |     |
  Alice's EPR Qubit (0) -----H-----
                                   |
  Bob's EPR Qubit (0) ------------- (Quantum Channel) --- (Bob's Gate) --- (Reconstructed |psi>)

It’s crucial to appreciate that no actual matter or quantum information (in the sense of the unknown state itself) travels from Alice to Bob faster than light. Only classical information is exchanged, which then allows Bob to “prepare” the quantum state locally using his entangled resource. The original unknown qubit at Alice’s location is destroyed in her Bell measurement , ensuring that the no-cloning theorem is not violated.

Quantum cryptography

Quantum cryptography leverages the fundamental and often counterintuitive principles of quantum mechanics to establish secure communication channels. The core theory underpinning this field is the inescapable fact that it is physically impossible to measure an unknown quantum state without inherently disturbing or altering that state. This quantum fragility serves as a built-in eavesdropping detection mechanism within a system. Any attempt by an unauthorized party to intercept and read the quantum-encoded information will inevitably leave a detectable trace, alerting the legitimate communicators to a breach in security.

The most widely researched and developed form of quantum cryptography is quantum key distribution (QKD). QKD protocols enable two parties, typically referred to as Alice and Bob, to generate and share a genuinely random, secret cryptographic key. This key can then be used with classical encryption methods (like the one-time pad ) to encrypt and decrypt messages with provable security. The private key is established between Alice and Bob over a public channel, but the quantum properties ensure that any eavesdropping attempt is immediately obvious. Bell states play a role in certain QKD protocols (like the E91 protocol ), where the shared entanglement is used to establish correlations, and deviations from expected correlations signal an eavesdropper.

Beyond simple two-state qubits , quantum cryptography can also involve a state of entanglement between multi-dimensional systems, often referred to as two-qudit (quantum digit) entanglement . While a qubit operates with two basis states (0 and 1), a qudit operates with $d$ basis states (0, 1, …, $d-1$). Utilizing higher-dimensional qudits in entanglement offers potential advantages for quantum cryptography , such as increased information capacity per particle and enhanced robustness against certain types of noise and eavesdropping attacks. This makes the exploration of generalized entangled states for qudits a promising avenue for future secure communication technologies.

See also

• Bell test experiments
• Bell’s theorem
• EPR paradox
• GHZ state
• Dicke state
• Superdense coding
• Quantum teleportation
• Quantum cryptography
• Quantum circuits
• Bell diagonal state

Notes

a. $|\Phi ^{-}\rangle ={\frac {1}{\sqrt {2}}}(|00\rangle -|11\rangle )$ $={\frac {1}{2{\sqrt {2}}}}((|+\rangle _{A}+|-\rangle _{A})(|+\rangle _{B}+|-\rangle _{B})-(|+\rangle _{A}-|-\rangle _{A})(|+\rangle _{B}-|-\rangle _{B}))$ $={\frac {1}{2{\sqrt {2}}}}(|++\rangle +|+-\rangle +|-+\rangle +|–\rangle -|++\rangle +|+-\rangle +|-+\rangle -|–\rangle )$ $={\frac {1}{\sqrt {2}}}(|+-\rangle +|-+\rangle )$